Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates. Part II: L 2 (Ω)-estimates

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J. Inv. Ill-Posed Problems, Vol. 11, No. 3, pp. 1–96 (2003)c© VSP 2003

Global Uniqueness, Observability and Stabilization of Nonconserva-

tive Schrodinger Equations Via Pointwise Carleman Estimates

I. LASIECKA,∗ R. TRIGGIANI,∗ and X. ZHANG†

Received November 29, 2002

Abstract — We consider a general non-conservative Schrodinger equation defined on an open bounded domain Ω in Rn,with C2-boundary Γ = ∂Ω = Γ0 ∪ Γ1, Γ0 ∩ Γ1 = ∅, subject to (Dirichlet and, as a main focus, to) Neumann boundaryconditions on the entire boundary Γ. Here, Γ0 and Γ1 are the unobserved (or uncontrolled) and observed (or controlled)parts of the boundary, respectively, both being relatively open in Γ. The Schrodinger equation includes energy-level(H1(Ω)-level) terms, which accordingly may be viewed as unbounded perturbations. The first goal of the paper is toprovide Carleman-type inequalities at the H1-level, which do not contain lower-order terms; this is a distinguishingfeature over most of the literature. This goal is accomplished in a few steps: the paper obtains first pointwise Carlemanestimates for C2-solutions; and next, it turns these pointwise estimates into integral-type Carleman estimates with no

lower-order terms, originally for H2-solutions, and ultimately for H1-solutions. The passage from H2- to H1-solutionsis readily accomplished in the case of Dirichlet B.C., but it requires a delicate regularization argument in the case ofNeumann B.C. This is so since finite energy solutions are known to have L2-normal traces in the case of DirichletB.C., but by contrast do not produce H1-traces in the case of Neumann B.C. From Carleman-type inequalities withno lower-order terms, one then obtains the sought-after benefits. These consist of deducing, in one shot, as a part ofthe same flow of arguments, two important implications: (i) global uniqueness results for H1-solutions satisfying over-determined boundary conditions, and—above all—(ii) continuous observability (or stabilization) inequalities with anexplicit constant. The more demanding purely Neumann boundary conditions requires the same geometrical conditionson the triple Ω, Γ0, Γ1 that arise in the corresponding problems for second-order hyperbolic equations. The mostgeneral result, with weakest geometrical conditions, is, in fact, deferred to Section 9. Sections 1 through 8 provide themain body of our treatment with one vector field under a preliminary working geometrical condition, which is thenremoved in Section 9, by use of two suitable vector fields.

The second and final goal of this paper is to shift the Carleman estimates (Hence, the continuous observabil-ity/stabilization inequalities) by one unit downward to the lower L2(Ω)-level. This is accomplished in Section 10 bymeans of pseudo-differential analysis, and accordingly, it contains lower-order terms. Applications of these L2(Ω)-Carleman estimate includes a new uniform stabilization of the conservative Schrodinger equation in the state spaceL2(Ω), by an attractive boundary feedback.

∗University of Virginia, Department of Mathematics, P. O. Box 400137, Charlottesville, VA 22904-4137. Emails: I.L.,[email protected]; R.T., [email protected].

Research partially supported by the National Science Foundation under Grant DMS-0104305 and by the Army Research Office,under Grant DAAD19-02-1-0179.

†School of Mathematics, Sichuan University, Chengdu 610064, China; and Departamento de Mathematicas, Facultad de Cien-cias, Universidad Autonoma de Madrid, 28049, Madrid, Spain.

Research partially supported by a Foundation for the Author of National Excellent Doctoral Dissertation of P. R. China (ProjectNo.:200119), Grant BFM2002-03345 of the Spanish MCYT, National Natural Science Foundation of China under Grant 10371084,and The Project—sponsored by SRF and ROCS, SEM.

2 I. Lasiecka, R.Triggiani, X. Zhang

1. INTRODUCTION. PROBLEM STATEMENT

1.1 Problem Statement. AssumptionsLet Ω be an open bounded domain in Rn with boundary ∂Ω = Γ of class C2, consisting of the closure of

two disjoint parts: Γ0 (uncontrolled or unobserved part) and Γ1 (controlled or observed part), both relativelyopen in Γ : ∂Ω = Γ ≡ Γ0 ∪ Γ1, Γ0 ∩ Γ1 = ∅. In this paper, we consider the following Schrodinger equation inthe (complex-valued) unknown w(t, x) defined on Ω:

Pw ≡ iwt + ∆w = F (w) + f in Q ≡ (0, T ]× Ω. (1.1.1)

In (1.1.1), we have setF (w) ≡ q1(t, x) · ∇w(t, x) + q0(t, x)w(t, x), (1.1.2)

subject to the following preliminary standing assumption on the coefficients:We let |q1|, q0 ∈ L∞(Q), so that the following pointwise estimate holds true:

|F (w)|2 ≤ CT [|∇w|2 + |w|2], ∀ (t, x) ∈ Q. (1.1.3)

Moreover, we assume throughout that the non-homogeneous term f satisfies

f ∈ L2(Q). (1.1.4)

Remark 1.1.1. The above regularity assumption suffice for the first main result: Theorem 2.1.1. In effect,for this we could relax the standing assumption of the lower-order coefficient q0 and just require q0 ∈ Lp(Q), p =n+ 1 [20, Remark 1.1.1]. Beyond that, in particular to obtain Theorem 2.2.1, stronger regularity assumptionsare needed. 2

Main assumptions. The main focus of the present paper refers to the case where Eqn. (1.1.1) is supple-mented by purely Neumann B.C.: ∂w

∂ν |Σ ≡ 0, while observation (or control) takes place only on a subportion Γ1

of the boundary Γ. Here, ν is the outward unit normal at Γ, so that ∂∂ν is the normal derivative on Γ. This

problem is of interest on both physical grounds (it arises in the structural acoustic problem) and on math-ematical grounds (the Lopatinski’s condition is not satisfied). For this key case, we shall need the followingassumptions just as in [20], in the corresponding wave equation case.

In addition to the standing assumptions (1.1.3) on F (w) and (1.1.4) on f , the following assumptions arepostulated throughout Section 8 of this paper.

(A.1) Given the triple Ω,Γ0,Γ1, ∂Ω = Γ0 ∪ Γ1, there exists a strictly convex (real-valued) non-negativefunction d : Ω → R+, of class C3(Ω), such that, if we introduce the (conservative) vector field h(x) =[h1(x), . . . , hn(x)] ≡ ∇d(x), x ∈ Ω, then the following two properties hold true:

(i)∂d

∂ν

∣

∣

∣

∣

Γ0

= ∇d · ν = h · ν = 0 on Γ0; h ≡ ∇d. (1.1.5N)

(ii) the (symmetric) Hessian matrix Hd of d(x) [i.e., the Jacobian matrix Jh of h(x)] is strictly positivedefinite on Ω: there exists a constant ρ > 0 such that for all x ∈ Ω:

Hd(x) = Jh(x) =

dx1x1 , . . . , dx1xn

......

dxnx1 , . . . , dxnxn

=

∂h1

∂x1, . . . ,

∂h1

∂xn...

...

∂hn∂x1

, . . . ,∂hn∂xn

≥ ρI. (1.1.6)

(A.2) A working assumption throughout Section 8, to be later relaxed (eliminated) in our final results ofSection 9, is that d(x) has no critical point on Ω:

infx∈Ω

|h(x)| = infx∈Ω

|∇d(x)| = p > 0. (1.1.7)

Global Uniqueness, Observability and Stabilization of Nonconservative Schrodinger Equations 3

Remark 1.1.2. Assumption (A.2) will be dispensed with in Section 9, which will provide the most generalresults of the present paper: see Theorem 9.1.1. This can be achieved, as in the case of the wave equation [20,Section 10] by splitting Ω as Ω = Ω1 ∪ Ω2, for two suitably overlapping sets Ω1 and Ω2, and working with twostrictly convex functions d1 and d2, where di satisfies assumption (1.1.7) but only on Ωi, i = 1, 2. Since the fullstatement of results without (A.2) requires a rather lengthy preparatory background, we have opted for thestrategy of retaining assumption (A.2) throughout Section 8, and then introduce the case without assumption(A.2) in Section 9. This will follow the original treatment for the wave equation case in [20, Section 10] in theEuclidean case, later generalized in [46, Section 10] in the Riemannian case ∆g . 2

Remark 1.1.3. (Neumann) Assumption (1.1.5N) is due to the purely Neumann B.C. ∂w∂ν

∣

∣

Σ≡ 0, as in the

case of the wave equation [20, Remark 1.1.2]. It was introduced in [43, Section 5]. Reference [20, AppendicesA-C] provides, by different mathematical techniques, several classes of triples Ω,Γ0,Γ1 in Rn, n ≥ 2, whereassumptions (A.1) and (A.2) are satisfied. (In light of Remark 1.1.2, only assumption (A.1) is the critical one.)For instance, just to quote one general result: one can construct explicitly [20, Theorem A.4.1, p. 301] therequired strictly convex function d(x) satisfying (A.1) if the (Euclidean) bounded domain Ω ∈ Rn is (i) convex(respectively, concave) on the side of the portion Γ0 of its boundary, and (ii) there exists a radial vector field(x − x0) for some x0 ∈ Rn which is entering (respectively, exiting) Ω through Γ0. Moreover, [20, CorollaryA.4.2, p. 306] the previously constructed strictly convex function d(x) has the following additional property:its gradient ∇d|Γ0 , once restricted on the portion Γ0 of the boundary, vanishes at the unique point x ∈ Γ0, ifsuch exists on Γ0, where the vector field x− x0 is orthogonal to Γ0.

The above condition on the existence of such d(x) is only sufficient. It has been recently extended alsoto the case where Ω is a bounded set of a finite-dimensional Riemannian manifold [46, Appendix B]. Otherclasses are given in [20, Appendices A-C] satisfying assumption (A.1): for instance, where the portion Γ0 ofthe boundary is logarithmic convex [20, Lemma A.2.2, p. 294].

(Dirichlet) Even though the purely Neumann B.C. case: ∂w∂ν |Σ ≡ 0 will be the central focus of this paper,

our treatment will allow us to also include the purely Dirichlet B.C. case: w|Σ ≡ 0. Here, however, (1.1.5N)can be dispensed with. It will be replaced by the much weaker condition

h · ν ≤ 0 on Γ0. (1.1.5D)

The canonical example is d(x) = 12‖x−x0‖2, with x0 outside Ω, where then h(x) = ∇d(x) = (x−x0) is radial.

[A combination of Dirichlet/Neumann B.C. is also included in our treatment.] 2

Remark 1.1.4. In effect, assumption (A.2) = (1.1.7) is needed to hold true only for x ∈ Γ0 (uncontrolledor unobserved part of the boundary Γ): inf |∇d(x)| = p > 0, where the inf is taken over Γ0. Indeed, a criticalpoint of d(x) at a point (necessarily interior) of Ω, or at a point x ∈ Γ1 (controlled or observed part of Γ) canalways be eliminated, by smoothly redefining d(x), first locally around the critical point, and then away fromΓ0, as to make the new critical point fall off of Ω, while preserving the positivity condition (1.1.6). 2

At any rate, throughout Section 6, we shall merely deal with smooth solutions of Eqn. (1.1.1) with noB.C. imposed, subject only to hypotheses (1.1.6) and (1.1.7). Then, assumption (1.1.5N):

h · ν ≡ 0 on Γ0 [resp. (1.1.5D) : h · ν ≤ 0 on Γ0]

will be introduced only when analyzing purely Neumann B.C. [resp. purely Dirichlet B.C.].

Pseudo-convex function ϕ(x, t). Having chosen, on the strength of assumption (A.1), a strictly convexpotential function d(x) ≥ 0, we next introduce the pseudo-convex function ϕ : Ω×R → R of class C3 by setting

ϕ(x, t) = d(x) − c

(

t− T

2

)2

; 0 ≤ t ≤ T, x ∈ Ω, (1.1.8a)

where T > 0 is arbitrary, and where then c = cT is chosen large enough as to have

cT 2 > 4 maxx∈Ω

d(x), so that cT 2 > 4 maxx∈Ω

d(x) + 4δ (1.1.8b)


for a suitably small δ > 0, henceforth kept fixed. Unless otherwise explicitly noted, ϕ(x, t) is selected asdescribed above and kept fixed henceforth. Such function ϕ(x, t) has the following properties:

(a) for the constant δ > 0, fixed in (1.1.8b), we have

ϕ(x, 0) ≡ ϕ(x, T ) = d(x) − cT 2

4≤ −δ, uniformly in x ∈ Ω; (1.1.9)

(b) there are t0 and t1, with 0 < t0 <T2 < t1 < T , such that

minx∈Ω,t∈[t0,t1]

ϕ(x, t) ≥ − δ

2, (1.1.10)

since ϕ(

x, T2)

= d(x) ≥ 0 for all x ∈ Ω (in fact, only the weaker property: minϕ(x, t) ≥ σ > −δ isactually needed).

Throughout this paper, we set

E(t) ≡∫

Ω

|∇w(t)|2dΩ; E(t) =

∫

Ω

[|∇w(t)|2 + |w(t)|2]dΩ = ‖w(t)‖2H1(Ω). (1.1.11)

Goals. Literature. As already mentioned, we consider, at first, sufficiently smooth solutions w(t, x), sayin H2,2(Q) ≡ L2(0, T ;H2(Ω)) ∩H2(0, T ;L2(Ω)), of the Schrodinger Eqn. (1.1.1) with no B.C. imposed.

First goal. Then, our first goal is to establish Carleman-type inequalities at the H1(Ω)-level for w—the basic energy level—for these solutions without lower-order terms. Carleman inequalities, however, with(interior) lower-order terms were obtained in [44, Theorems 2.1.1 and 2.1.2, pp. 464–466] for the SchrodingerEqn. (1.1.1), and in [45, Theorems 3.3 and 3.4, pp. 640–641] for Eqn. (1.1.1) with the (Euclidean) Laplacian ∆replaced by a variable coefficient (in space) uniformly elliptic operator; or, with essentially the same effort, forthe Schrodinger Eqn. (1.1.1) with the Euclidean Laplacian ∆ replaced by the Laplace-Beltrami operator ∆g ,defined on a bounded set Ω ⊂ M with boundary, of a Riemannian manifold M, g. In the present work, aswell as in the prior references [44] and [45], the boundary terms (traces of w) of the solutions w of Eqn. (1.1.1)which appear in the Carleman estimates are given explicitly. For our final observability results in Theorem2.3.1 (Dirichlet case) and Theorem 2.4.1 (Neumann case), the explicit constant in estimates (2.3.2) and (2.4.2)

is Ce−C`2

, where ` is the norm of the coefficients q0 and q1.

Second goal. As a consequence of Carleman estimates without lower-order term and explicit boundaryterms (our first goal), we then achieve our second goal: that is, we obtain global uniqueness results as wellas continuous observability/uniform stabilization inequalities in the basic energy level H1(Ω), in one shot, aspart of the same flow of arguments. All this is achieved first for smoother solutions in L2(0, T ;H2(Ω)) ∩H2(0, T ;L2(Ω)), and ultimately for finite energy solutions in L2(0, T ;H1(Ω)) ∩H 1

2 (0, T ;L2(Ω)). This passageis definitely non-trivial in the case of the purely Neumann problem, unlike the case of the Dirichlet problemwhere optimal regularity results [15] are available. The above achievement is in contrast with prior Carlemanestimates, hence continuous observability/uniform stabilization inequalities polluted by lower-order terms asin [44], [45] and, for F (w) ≡ 0, in [15]. In this latter case, a first disadvantage is the necessity to require anindependent global uniqueness result in order to absorb, and hence eliminate, the lower-order term from thesought-after control theoretic estimates. Moreover, a second disadvantage is the lack of control on the constantsarising in the final uniform stabilization/continuous observability estimates, as the aforementioned absorptionprocess proceeds by contradiction. An explicit constant in the uniform stabilization inequality provides anestimate of the resulting exponential decay of the feedback semigroup, via semigroup theory [35, p. 116], [1,p. 178]. Moreover, an explicit constant in the continuous observability inequality is useful in at least twocontexts: (i) in the corresponding semilinear problems [16], [50], and (ii) in yielding the minimal norm among

the steering controls [43, Appendix], [13, Appendix]. In our case, such explicit constant is CeC`2

, the reciprocalof the constant in estimates (2.3.2) and (2.4.2), where ` is the norm of the coefficients q0, q1.


In short, a key aim of the present paper is to eliminate lower-order terms from the sought-after continuousobservability/uniform stabilization estimates at the H1(Ω)-level, therefore avoiding the two disadvantages citedabove.

Third goal. Our third goal in this paper is to obtain also lower-level energy estimate (more precisely,w in L2(Ω) rather than in H1(Ω)), see Theorem 2.6.1, such as they are needed in the problem of uniformstabilization with L2(0,∞;L2(Γ))-Dirichlet feedback control which is solved in Section 11. The shift from theH1(Ω)-energy level estimates down to the L2(Ω)-energy level estimates is accomplished through a pseudo-differential approach in Section 10. This then produces lower-order terms for the L2(Ω)-level, which Remark2.6.1 in dicates how to eliminate. As a consequence of the lower-level, L2(Ω)-estimates of Section 10, we obtainin Section 11 a new result on the uniform stabilization of the conservative Schrodinger equation in the statespace L2(Ω), by means of a pointwise boundary feedback dissipative term involving only w (not its derivative).

Literature. Pure Schrodinger equations. Earlier contributions on the pure Schrodinger equation ona bounded domain of Rn (Eqn. (1.1.1) with F ≡ 0) include: [15] for optimal regularity, exact controllabilityand uniform stabilization in the optimal space H−1(Ω), under L2-Dirichlet control (the regularity result holdstrue also for the general case of variable coefficients in the principal part and energy level terms); [32] for exactcontrollability/uniform stabilization under Neumann L2-control on a portion of the boundary and homogeneousDirichlet B.C. on the complementary part of the boundary, in the spaceH1(Ω) (which is not the space of optimalregularity in this case); [26] for exact controllability under a sharp geometric optic condition; and [47].

More general Schrodinger equations. Subsequent generalizations valid for more general models aregiven in [37], [38], [39] as part of a general theory on a-priori control-theoretic inequalities for evolution equa-tions, requiring the assumption on the existence of a pseudo-convex function, and based on Carleman estimatescontaining boundary traces—unlike prior literature [4] which dealt with solutions compactly supported; [44],[45], for Eqn. (1.1.1) and its generalization on a Riemannian manifold M, g, under the existence of a g-strictlyconvex function. Moreover, papers [6,7]—which deal with the variable coefficient principal part and no energylevel terms in the equation—give an exact controllability result directly, while reading off the boundary controlas a trace of the required solution, without using the dual continuous observability inequality, as in all of theaforementioned literature, but rather the approach of [31].

Estimates with lower-order terms. The works cited above on Schrodinger equations under variousB.C. and degree of generality obtain the continuous observability/uniform stabilization inequalities polluted bylower-order terms. In fact, this is the case for almost all papers on exact controllability/uniform stabilization ofthe literature, regardless of the specific evolution equation. Moreover, no one of the above references considersthe most challenging case of the present paper: Neumann control on a portion of the boundary and homogeneousNeumann B.C. on the complementary portion of the boundary.

Estimates without lower-order terms. To our knowledge, the only exceptions where lower-order termsdo not appear in the observability/stabilization estimates are [11], [20], [10], [48], [49] for second-order hyperbolicequations with ∆ as principal part; with the last two references (which include the Dirichlet case but are) mostlyfocused on the more challenging purely Neumann B.C. case; [46] for ∆g as principal part; also focused on theNeumann B.C. case as in [20]; and [41] for general evolution equations (local theory). In the case of Schrodingerequations, the local theory of [41] excludes the case of Neumann B.C. (since this does not satisfy the strongLopatinski condition) and conjectures that the same unique continuation result continues to hold true also inthis case [41, Remark 5.7, p. 406]. The present paper—and its successor with ∆ replaced by ∆g—establishesthis by providing global results in the purely Neuman B.C. case of Eqn. (1.1.1), as it was the case for [20]for wave-like equations. Moreover, the approach pursued here, as well as in [11], [20], [10], [46], [48], [49], forsecond-order hyperbolic equations, is inspired by [25] and accordingly is different from the pseudo-differentialanalysis for general evolution equations in [41].

It is expected that the present paper will admit a generalization of (1.1.1) where the Laplacian in Rn willbe replaced by the Laplace-Beltrami operator ∆g in a Riemannian manifold M, g (in particular, a variablecoefficient principal part in Rn) the same way the wave equation treatment of [20] was extended in [45]. Seealso [22, 23, 24].


2. MAIN RESULTS UNDER ASSUMPTIONS (A.1) AND (A.2)

2.1 Carleman Estimates without Lower-Order Terms for H2,2(Q)-Solutions of Eqn. (1.1.1)and No B.C. under Assumptions (1.1.6), (1.1.7)

Theorem 2.1.1. (First version) Let T > 0 be arbitrary and let c = cT be defined by (1.1.8b). Letd(x) ∈ C3(Ω) be the non-negative, real, strictly convex function satisfying assumptions (A.1(ii)) = (1.1.6) and(A.2) = (1.1.7). Define accordingly ϕ(x, t) by (1.1.8). Let w be a solution of Eqn. (1.1.1) [with no boundaryconditions imposed] in the following class:

w ∈ H2,2(Q) ≡ L2(0, T ;H2(Ω)) ∩H2(0, T ;L2(Ω)), (2.1.1a)

so that∂w

∂ν

∣

∣

∣

∣

Γ

∈ L2(0, T ;H12 (Γ)); wt ∈ L2(0, T ;H1(Ω)); wt|Γ ∈ L2(0, T ;H

12 (Γ)), (2.1.1b)

where (1.1.1) is subject to the standing assumption (1.1.3) for F (w) and (1.1.4) for f .Then, for all τ sufficiently large, the following one-parameter family of estimates holds true:

BΣ(w) + 4

∫ T

0

∫

Ω

e2τϕ|f |2dΩ dt

≥ mρ,p,τ,CT

∫ T

0

∫

Ω

e2τϕ[|∇w|2 + |w|2]dΩ dt

− Cd,T τ e−2τδ[E(T ) + E(0)] (2.1.2)

≥ mρ,p,τ,CTe−δτ

∫ t1

t0

E(t)dt− Cd,T τ e−2τδ[E(T ) + E(0)]; (2.1.3)

mρ,p,τ,CT≡ min

[

2τρ− 1

2− 4CT

]

, [4τ3ρp2 + O(τ2) − 4CT ]

∞ as τ → ∞, (2.1.4)

where ρ > 0, p > 0, δ > 0 are defined by (1.1.6), (1.1.7), (1.1.8b), and Cd,T is a positive constant dependingon d and T. Moreover, E(t) are defined by (1.1.11), while t0, t1 are as in (1.1.10). Finally, setting h ≡ ∇d as in(1.1.5), then the boundary terms BΣ(w) are given explicitly as follows, where ξ = Re w, η = Im w:

BΣ(w) = 2τ

∫ T

0

∫

Γ

e2τϕ [2τ2|h|2 + Φ]|w|2h · ν dΓ dt− 4cτ

∫ T

0

∫

Γ

e2τϕ(

t− T

2

)[

η∂ξ

∂ν− ξ

∂η

∂ν

]

dΓ dt

− 2τ

∫ T

0

∫

Γ

e2τϕ[ξtη − ξηt]h · ν dΓ dt+∫ T

0

∫

Γ

e2τϕ [2τ2|h|2 − τ∆d]

[

w∂w

∂ν+ w

∂w

∂ν

]

dΓ dt

+ 2τ

∫ T

0

∫

Γ

e2τϕh ·[

∇w ∂w

∂ν+ ∇w ∂w

∂ν

]

dΓ dt− 2τ

∫ T

0

∫

Γ

e2τϕ|∇w|2h · ν dΓ dt. (2.1.5)

Here, c is the constant in (1.1.8b), while the function Φ occurring in (2.1.5) may be taken to satisfy: eitherΦ ≡ 0, or else Φ = τ∆d(x), see (4.3b). Thus, from (2.1.5), the following estimate holds true:

BΣ(w) ≤ Cφ,τ

∫ T

0

e2τϕ

[

‖w(t)‖2H1(Γ) +

∥

∥

∥

∥

∂w

∂ν

∥

∥

∥

∥

2

L2(Γ)

+ ‖wt‖2H−1(Γ)

]

dt. 2 (2.1.6)

Theorem 2.1.1 is proved in Sections 3 through 5 (for H2,2(Q)-solutions). In particular, inequalities (2.1.2),(2.1.3) are established in inequalities (5.1), (5.2) below.

Remark 2.1.1. The Carleman estimate (2.1.2) of Theorem 2.1.1 is essentially the one in [44, Theorem 2.1.1,p. 464] later generalized in [45, Theorem 3.3, p. 640] to the case where the Euclidean Laplacian ∆ is replaced


by the Laplace-Beltrami operator ∆g on a Riemannian manifold M, g, except for the critical improvementthat our present version (2.1.2) does not include an interior lower-order term, unlike the estimates of theaforementioned references [44], [45]. 2

Corollary 2.1.2 (Global uniqueness). Assume the setting of Theorem 2.1.1. In particular, let w be asolution of Eqn. (1.1.1) in the class H2,2(Q) defined in (2.1.1).

(i) Neumann case: Assume further that such w satisfies, in addition, the following B.C.:

∂w

∂ν

∣

∣

∣

∣

Σ

≡ 0 and w|Σ1 ≡ 0, where h · ν = 0 on Γ0, (2.1.7)

with Σ = (0, T ]× Γ, Σ1 = (0, T ]× Γ1. Then, in fact, such solution must vanish: w ≡ 0 in (0, T )× Ω.(ii) Dirichlet case: Assume further that such solution w satisfies, in addition, the following B.C.:

w|Σ ≡ 0 and∂w

∂ν

∣

∣

∣

∣

Σ1

≡ 0, where h · ν ≤ 0 on Γ0. (2.1.8)

Then, in fact, such solution must vanish: w ≡ 0 in (0, T )× Ω. 2

Corollary 2.1.2 is established at the end of Section 5, after completion of the proof of Theorem 2.1.1.As in [44], [45], in the case dim Ω ≥ 2, in order to refine Theorem 2.1.1, we need to specialize the first-order

differential operator F (w) by imposing a structural property, as stated by the following assumption, whereby,eventually, without loss of generality, we can assume that q1 be purely imaginary (magnetic potential [36,Theorem X.22, p. 173]):

(A.3) We assume, at first, that q1(t, x) = −ir1(t, x) for a real-valued function r1(t, x); that is, that q1 ispurely imaginary (as in the case of magnetic potential). In this case, we make the following additional regularityassumptions (over |r1|, q0 ∈ L∞(Q) in (1.1.3)) on the coefficients r1 and q0:

r1 ∈ L∞((0, T ;W 1,∞((Ω)n)), so that∂r1i∂xj

∈ L∞(Q), i, j = 1, . . . , n; and (2.1.9a)

either q0 ∈

L1(0, T ;W 1,2(Ω))

L1(0, T ;W 1,2+ε(Ω))

L1(0, T ;W 1,n(Ω))

, or else (q0)t ∈

L1(0, T ;L1(Ω)), n = 1;

L1(0, T ;L1+ε(Ω)), n = 2;

L1(0, T ;Ln2(Ω)), n ≥ 3

. (2.1.9b)

Sufficient conditions are |∇q0|, (q0)t ∈ L∞(Q), respectively. The proof of Lemma 6.1(v) will use q0 on the LHSof (2.1.9b), while Remark 6.1 will use (q0)t on the RHS of (2.1.9b).

Next, regarding the more general form q1(t, x) = ∇π − ir1(t, x) for a scalar real-valued function π and areal-valued vector function r1(t, x), Appendix A shows what follows:

(i) Case dim Ω = 1. In this case, Theorem A.1 of Appendix A shows that: after a change of variablebased on q1, we can always assume without loss of generality that q1 ≡ 0, and so F (w) = q0(t, x)w, so that Fhas no first-order term in this case. Moreover, q0 ∈ L∞(Q) provided that q0, q1, q1t, q1x ∈ L∞(Q). Additionalregularity of q0 can be traced in the explicit change of variable of Theorem A.1 of Appendix A.

(ii) Case dim Ω ≥ 2. In this case, Theorem A.2 of Appendix A shows that: after appropriate changes ofvariables as well as time-space rescaling, we can always assume without loss of generality that the followingproperties for q1 be achieved:

∇π ≡ 0 in Q;

and either |r1 · µ|2 ≤ r1 · ν < 1 on Γ,

or else r1 · ν ≡ 0 on Γ0,(2.1.9c)

where µ is any unit tangent vector ( ∂∂µ = tangential gradient) and ν is the unit outward normal at Γ. Moreover,

the new (transformed) q0 ∈ L∞(Q), provided that πt, |∇π|, ∆π, |r1|, (old) q0 ∈ L∞(Q), with additional


regularity of q0 to be traced in the explicit change of variable of Theorem A.2 of Appendix A. The first“w.l.o.g.” assumption will be used in the case of Neumann B.C.; the second “w.l.o.g.” assumption will be usedin the case of Dirichlet B.C. In conclusion, under (A.3), we can always assume that (1.1.1) be of the form(magnetic potential [36, Theorem X.22, p. 173]),

iwt + ∆w = −ir1(t, x) · ∇w + q0(t, x)w, (2.1.9d)

i.e., with q1 = −ir1, where r1 is real-valued and with q0 satisfies (2.1.9a)–(2.1.9c). 2

The structural property iq1 = r1 (real) is critical.The use of assumption (A.3) = (2.1.9) will be seen in Lemma 6.1(ii),(iv), Eqns. (6.7), (6.9), (6.11), estimate

(6.12). Such assumption will permit us to obtain a second version, more refined, of the Carleman estimate(2.1.2) of Theorem 2.1.1.

Theorem 2.1.3. (Second version) Assume the setting (in particular, (1.1.6), (1.1.7)), and the notation ofTheorem 2.1.1. In addition, we assume hypothesis (A.3) = (2.1.9) and that f ∈ L2(0, T ;H1(Ω)). Let w bea solution of Eqn. (1.1.1) in the class H2,2(Q) in (2.1.1). Then, for all τ > 0 sufficiently large, the followingone-parameter family of estimates holds true, where kϕ,τ > 0,

BΣ(w) + 4

∫ T

0

∫

Ω

e2τϕ|f |2dΩ dt+ Cp,q,ρ,τ‖f‖2L2(0,T ;H1(Ω))

≥

mρ,p,τ,cTe−δτ

(t1 − t0)

2e−cTT − Cd,T τe−2τδ

[E(T ) + E(0)] (2.1.10)

≥ kϕ,τ [E(T ) + E(0)], (2.1.11)

with explicit mρ,p,τ,cTwhich is noted in Eqn. (2.1.4) or Eqn. (5.3) below, E(t) is defined in (1.1.11). Moreover,

the boundary terms BΣ(w) in (2.1.10) are given by

BΣ(w) = BΣ(w) + Cϕ,p,q

∫ T

0

∫

Γ

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

[

|wt| +∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

|r1 · ν + Oc| + |q0w| + |w| + |f |]

dΓ dt

, (2.1.12)

with constant Cϕ,p,q which may be given explicitly via the LHS of (2.1.10) and G(T ) in (6.13). Moreover,Oc = 0 in the case of Dirichlet B.C. w|Σ = 0, and Oc = 1 otherwise. 2

Theorem 2.1.3 is proved in Section 6 (for H2,2(Q)-solutions).

2.2 Extension of Carleman estimates to finite energy solutions

Theorem 2.2.1. Assume the hypotheses of Theorem 2.1.3: (1.1.3), (A.3) = (2.1.9) for F (w), f ∈L2(0, T ;H1(Ω)), in addition to (A.1(ii)) = (1.1.6) and (A.2) = (1.1.7). Then, estimate (2.1.2) of Theorem2.1.1 and estimate (2.1.11) of Theorem 2.1.3 can be extended to finite energy solutions in the class

w ∈ C([0, T ];H1(Ω));∂w

∂ν, wt ∈ L2(0, T ;L2(Γ)), (2.2.1)

provided that the (unbounded) term F (w) = ∇π− ir1 ·∇w+ q0w satisfies the following additional hypotheses:(A.4) (this is only for convenience)

the coefficients r1 and q0 are time-independent. (2.2.2)

[In the time-dependent case, our analysis could still be carried out with additional terms to be analyzed,likely none of them critical.] The proof of Theorem 2.2.1 is given in Section 7.3, Theorem 7.3.1.


2.3 Continuous Observability. Global Uniqueness. Dirichlet B.C.We first list our main results in our treatment of Sections 1–8.

Purely Dirichlet problem. Here we consider the following problem:

iwt + ∆w = F (w) + f in (0, T ]× Ω ≡ Q;

w(0, · ) = w0 in Ω;

w|Σ = 0 in (0, T ]× Γ = Σ,

(2.3.1a)

(2.3.1b)

(2.3.1c)

In the case of Dirichlet B.C. the extension of Theorem 2.1.3 as given by Theorem 2.2.1 is not needed.Indeed the extension of the final sought-after continuous observability inequality (2.3.2) below can be readilyaccomplished from H2,2(Q)-solutions to H1,1(Q)-solutions just by virtue of the regularity theorem (‘reverseinequality’) obtained in [15] and reported also in [18, Chapter 10, Section 9]: the Neumann trace ∂w

∂ν isdominated by the H1(Ω)-norm of the I.C. w0. Thus the additional assumption (2.2.2) of Theorem 2.2.1 maybe dispensed with. We obtain

Theorem 2.3.1. Let w be a solution of problem (2.3.1) with I.C. w0 ∈ H10 (Ω), and with f ∈

L2(0, T ;H1(Ω)), under the standing assumption (1.1.3) on F (w), as well as (A.3) = (2.1.9). Assume, fur-ther, hypotheses (A.1)(ii) = (1.1.6), (A.2) = (1.1.7), for d(x), see Remark 1.1.2. Define now Γ0 as in (1.1.5D),i.e., by: h · ν ≤ 0, h = ∇d, on Γ0. Let Γ1 = Γ \ Γ0. Let T > 0 be arbitrary. Then:

(a) there exists a constant CT > 0, such that the following continuous observability inequality holds true:

CTE(0) ≤∫ T

0

∫

Γ1

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

dΓ1dt+ ‖f‖2L2(0,T ;H1(Ω)). (2.3.2a)

The constant CT in (2.3.2) is explicit: is of the order Ce−C`2

, where ` is the appropriate norm of thecoefficients q1 and q0. For instance, under (1.1.3) and (2.1.9a–b) for n ≥ 3, we have

` = |q0|L∞(Q) + |q0|L1(0,T ;W 1,n(Ω)) + |q1|L∞(0,T ;W 1,∞((Ω)n), (2.3.2b)

and analogously for n = 1, 2.

(b) Let now f = 0 in (2.3.1). Then, the following global uniqueness result holds true: with T > 0, let w bea H1,1(Q)-solution of Eqn. (2.3.1a) and over-determined B.C.’s:

w|Σ ≡ 0 and∂w

∂ν

∣

∣

∣

∣

Σ1

= 0, where h · ν ≤ 0 on Γ0 = Γ \ Γ1, Σ1 = (0, T ]× Γ1. (2.3.3)

Then, in fact, w ≡ 0 in Q, indeed in R+t × Ω. 2

The above global uniqueness result appears to be new over (mostly local) results of the literature [5], [8],[9], [40], also in view of the passage from local to global results given in [30].

Theorem 2.3.1(a) is established as Theorem 8.2(c) in Section 8.2 below.

2.4 Continuous Observability. Global Uniqueness. Neumann B.C.We first list our main continuous observability inequalities, and related global uniqueness results in our

treatment of Sections 1 through 8.

Purely Neumann problem. Here we consider the following problem:

iwt + ∆w = F (w) + f in (0, T ]× Ω ≡ Q;

w(0, · ) = w0 in Ω;

∂w

∂ν

∣

∣

∣

∣

Σ

= 0 in (0, T ]× Γ ≡ Σ,

(2.4.1a)

(2.4.1b)

(2.4.1c)


As explained in more detail in Section 7, it is for the Neumann B.C. case that the extension Theorem 2.2.1is called for. Accordingly, we inherit its additional (non-critical) assumption (2.2.2).

Theorem 2.4.1. Let w be the solution of problem (2.4.1) with I.C. w0 ∈ H1(Ω), and with f ∈L2(0, T ;H1(Ω)), under the standing assumption (1.1.3) on F (w). Assume, further, hypotheses (A.1), (A.2),(A.3) [that is, (1.1.5N), (1.1.6), (1.1.7), (2.1.9)], as well as the additional hypothesis (A.4) = (2.2.2). LetΓ1 = Γ \ Γ0, where h · ν = 0 on Γ0 as defined by (1.1.5N), and let T > 0 be arbitrary. Then:

(a) there exists a constant CT > 0, such that the following continuous observability inequality holds true:

CTE(0) ≤∫ T

0

∫

Γ1

[|w|2 + |wt|2]dΓ1 dt+ ‖f‖2L2(0,T ;H1(Ω)). (2.4.2)

The constant CT in (2.4.2) is explicit: it is given by the expression in (2.3.2b).

(b) Let now f = 0 in (2.4.1). Then, the following global uniqueness result holds true: with T > 0, let w bea H1,1(Q)-solution of Eqn. (2.4.1a) and over-determined B.C.’s:

∂w

∂ν

∣

∣

∣

∣

Σ

≡ 0 and w|Σ1 ≡ 0, where h · ν = 0 on Γ0 = Γ \ Γ1, Σ1 = (0, T ]× Γ1. (2.4.3)

Then, in fact, w ≡ 0 in Q, indeed in R+t × Ω. 2

Indeed, we shall first prove the global uniqueness statement of part (b) of Theorem 2.4.1 in Section 8, asa direct consequence of the Carleman estimate without lower-order terms of Theorem 2.1.3, first for smoothsolutions and next extended to H1,1(Q)-solutions in Section 7. Next, part (b) will be used to establish part(a), in Theorem 8.4 of Section 8 by virtue also of the trace result of Theorem 8.3.

The above global uniqueness result appears to be new over (mostly local) results of the literature [5], [8],[9], [40], also in view of the passage from local to global results given in [30].

It is well known that there exists a duality between continuous observability and exact controllability [13],[15]. We thus omit, for lack of space, to the corresponding exact controllability/uniform stabilization results.

2.5 The Euler-Bernoulli EquationIn this section, we transfer the estimates of the Schrodinger problem, given in the previous sections, into

estimates for the Euler-Bernoulli (E-B) equation with ‘hinged’ B.C. Thus we consider the following E-B plateproblem:

wtt + ∆2w = f in (0, T ]× Ω ≡ Q;

w(0, · ) = w0, wt(0, · ) = w1 in Ω;

w|Σ = 0, ∆w|Σ = 0 in (0, T ]× Γ = Σ,

(2.5.1a)

(2.5.1b)

(2.5.1c)

where now w(t, x) is real-valued. Writing the E-B equation as an iteration of two Schrodinger equations, inthe usual way [15, Section 5]

wtt + ∆2w = (∆ + i∂t)(∆ − i∂t)w = f, (2.5.2)

and setting

v = iwt − ∆w;

|v|2 = w2t + |∆w|2; |∇v|2 = |∇wt|2 + |∇∆w|2,

(2.5.3a)

(2.5.3b)

we rewrite problem (2.5.1) via (2.5.3) as

ivt + ∆v = f in Q;

v(0) = iw1 − ∆w0 in Ω;

v|Σ = 0 in Σ,

(2.5.4a)

(2.5.4b)

(2.5.4c)


Setting in this section

Ew(t) ≡∫

Ω

[|∇∆w(t)|2 + |∇wt(t)|2]dΩ

= Ev(t) =

∫

Ω

|∇v(t)|2dΩ,

(2.5.5a)

(2.5.5b)

via (2.5.3b), we can apply Theorem 2.3.1 (Dirichlet case) to the v-problem (2.5.4) and obtain

Theorem 2.5.1. Let w be the solution of problem (2.5.1) with w0, w1 ∈ H3(Ω) × H1(Ω). Let f ∈L2(0, T ;H1(Ω)). Let d(x) be the function satisfying properties (1.1.6) and (1.1.7) [e.g., the one in Remark1.1.2]. Let, as in Theorem 2.3.1, Γ0 be defined by (1.1.5D), i.e., h · ν ≤ 0 on Γ0, h = ∇d. Let Γ1 = Γ \ Γ0 andlet T > 0 arbitrary. Then, there exists a constant CT > 0, such that the following continuous observabilityinequality holds true:

CTEw(0) ≤∫ T

0

∫

Γ1

[

(

∂∆w

∂ν

)2

+

(

∂wt∂ν

)2]

dΓ1dt+ ‖f‖2L2(0,T ;H1(Ω)). (2.5.6)

The constant CT in (2.5.6) is explicit and is given by the expression in (2.3.2b). 2

Proof. Apply Theorem 2.3.1 with v(0) ∈ H1(Ω), |∇v(0)|2 = |∇w1|2 + |∇∆w0|2 . 2

Remark 2.5.1. Theorem 2.5.1 reproduces a classical result [14], [27], except for the improvement thatthe constant CT in estimate (2.5.6) is explicit in the present approach. This feature justifies the insertion ofthis result here. Classical results all produce estimate (2.5.6) with a non-explicit constant CT , resulting from acompactness/uniqueness argument by contradiction which loses control of the constant. See also [50]. 2

2.6 Lower-level (L2(Ω))-energy estimatesThe final goal of this paper is to provide an energy level estimate at the L2(Ω) level, down one unit from

the results of previous subsections. To this end, we introduce anisotropic Sobolev spaces:

Hsa(Σ) ≡ Hs,s/2(Σ) ≡ L2(0, T ;Hs(Γ)) ∩Hs/2(0, T ;L2(Γ)),

the latter Sobolev spaces being classical time-space spaces defined in [28, vol. II]. Then, H−1a (Σ) is the dual

space to H1a(Σ), with respect to L2(Σ) as a pivot/space. We then have

Theorem 2.6.1. Assume (1.1.3) and (2.1.9a) on F . Let w be a sufficiently smooth solution of theSchrodinger equation (1.1.1) with f ∈ L2(Q). Let w|Σ0 ≡ 0 where h · ν ≤ 0 on Γ0, as in (1.1.5D). Then, forany T > 0, the following inequality holds true: there exists a constant CT such that

∫ T

0

[

‖w‖2L2(Ω) + ‖wt‖2

H−2(Ω)

]

dt+ ‖w(0)‖2L2(Ω) + ‖wt(0)‖2

H−2(Ω)

≤ CT

‖w‖2L2(Σ1)

+

∥

∥

∥

∥

∂w

∂ν

∥

∥

∥

∥

2

H−1a (Σ1)

+

∫ T

0

∫

Γ1

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

|w|dΓ1dt+ ‖w‖2H−1(Q) + ‖f‖2

L2(Q)

. (2.6.1)

The proof of Theorem 2.6.1 will be given in Section 10.

Remark 2.6.1. It is expected that the use of an integral transform to lift the problem in time would allowone to obtain a general global unique continuation result also at the L2(Ω)-level, the same as it is done in thispaper at the H1(Ω)-level. This global uniqueness theorem would then permit one to eliminate the l.o.t. inestimate (2.6.1) by a compactness/uniqueness argument. 2

Remark 2.6.2. Theorem 2.6.1 holds true also with ∆ replaced by ∆g , the Riemannian Laplacian on aRiemannian manifold M, g. To this end, one simply invokes [45] rather than [44].

3. A FUNDAMENTAL LEMMA

As in the case of second-order hyperbolic equations—of which the paper is the counterpart as relevant to the


Schrodinger equation—the starting point of our proofs is the following pointwise estimate. Though it sharesthe same philosophy as in [25], [20], it requires new technicalities to account for the different dynamics.

With t ∈ Rt and x = [x1, . . . , xn] ∈ Rnx , we shall indicate the partial derivatives as follows: ∂∂xj

∂∂tf = ftxj

,etc.

Lemma 3.1. Letw(t, x) ∈ C2(Rt × Rnx ; C); `(t, x) ∈ C3(Rt × Rnx : R);

Ψ(t, x), Φ(t, x) ∈ C1(Rt × Rnx : R),(3.1)

be given functions, all real-valued, except for w(t, x) which is, instead, complex-valued. Let

`txj≡ 0, j = 1, . . . , n; θ(t, x) = e`(t,x); v(t, x) = θ(t, x)w(t, x), (3.2)

where the condition on ` is an assumption, while the other two relations in (3.2) are definitions. Let ε > 0 bearbitrary. Then, the following pointwise inequality holds true:

(

1 +1

ε

)

θ2|iwt + ∆w|2 − ∂M

∂t+ div V

≥

2(Ψ + ∆`)|∇`|2 − 2|Ψ + ∆`| |∇`|2 + 4∑

j,k

`xkxj`xj`xk

− 2(Ψ + ∆`)∆`− 2∇` · ∇(Ψ + ∆`)

− ε(Ψ + ∆`)2 − n

ε

∑

j,k

`2xjxjxk+ 2∇` · ∇(Φ − ∆`) − (Ψ2 + Φ2) + 2Φ∆`+ `tt

|v|2

+ 2

∑

j,k

[`xjxk(vxj

vxk+ vxk

vxj)] − (Ψ + ∆`)|∇v|2

− ε|∇v|2 + 2θ2[(Ψ + ∆`) − |Ψ + ∆`|]|∇w|2. (3.3)

The left side of (3.3) is defined as follows. Call ξ ≡ Re w and η ≡ Im w. Then, we set:

M = M(w) ≡ θ2[2(∇` · ∇ξ)η − ξ∇` · ∇η − `t|w|2]; (3.4)

V ≡ [V1, . . . , Vj , . . . , Vn]; (3.5a)

Vj = Vj(w) ≡ 2θ2

(2|∇`|2 − ∆`− Ψ + Φ)`xj|w|2 + `t(ξxj

η − ξηxj) − `xj

(ξtη − ξηt)

+1

2(2|∇`|2 − ∆`)(wxj

w + wxjw) +

∑

k

`xk(wxj

wxk+ wxj

wxk) − `xj

|∇w|2

. (3.5b)

Proof. Step 1. From v = θw, we obtain via (3.1) [20],

θt = θ`t; vt = θtw + θwt = `tv + θwt; or θwt = (vt − `tv)

θwxjxj= vxjxj

− 2`xjvxj

+ (`2xj− `xjxj

)v, j = 1, . . . , n;

θ∆w = ∆v − 2∇` · ∇v + (|∇`|2 − ∆`)v.

(3.6a)

(3.6b)

(3.6c)

Thus, by (3.6a), (3.6c), we compute

|θ(iwt + ∆w)|2 = |i(vt − `tv) + ∆v − 2∇` · ∇v + (|∇`|2 − ∆`)v|2 (3.7)

= |I1 − I2 + I3|2 (3.8)

= |I1|2 + |I2|2 + |I3|2 − (I1I2 + I2I1) − (I2I3 + I3I2) + (I1I3 + I3I1), (3.9)


where, after adding and subtracting Φv and Ψv in (3.7), we have set

I1 ≡ ivt + ∆v + (|∇`|2 − ∆`)v − Ψv,

I2 = 2∇` · ∇v + Φv + i`tv; I3 ≡ (Φ + Ψ)v.

(3.10a)

(3.10b)

Thus, dropping |I1|2 + |I2|2 in (3.9), we obtain

|θ(iwt + ∆w)|2 ≥ |I3|2 − (I1I2 + I2I1) − (I2I3 + I3I2) + (I1I3 + I3I1). (3.11)

Step 2. With reference to (3.10b), in this Step 2 we shall establish that

I2I3 + I3I2 = 2∑

j

[(Φ + Ψ)`xj|v|2]xj

+ 2(Φ + Ψ)(Φ − ∆`)|v|2 − 2[∇(Φ + Ψ) · ∇`]|v|2. (3.12)

Proof of (3.12). By invoking the definitions in (3.10b), we compute

I2I3 + I3I2 = [2∇` · ∇v + Φv + i`tv](Φ + Ψ)v + (Φ + Ψ)v[2∇` · ∇v + Φv − i`tv]

= 2(Φ + Ψ)[∇` · ∇vv + ∇` · ∇vv] + 2Φ(Φ + Ψ)|v|2 (3.13)

= 2(Φ + Ψ)∇` · ∇(|v|2) + 2Φ(Φ + Ψ)|v|2, (3.14)

after a cancellation of [i`t(Φ + Ψ)|v|2] to arrive at (3.13). We next observe that

2∑

j

[(Φ + Ψ)`xj|v|2]xj

= 2∇(Φ + Ψ) · ∇`|v|2 + 2(Φ + Ψ)∆`|v|2 + 2(Φ + Ψ)∇` · ∇(|v|2). (3.15)

Extracting 2(Φ + Ψ)∇` · ∇(|v|2) from (3.15) and substituting into the right side of (3.14) yields (3.12), asdesired.

Step 3. With reference to (3.10), in this Step 3 we shall establish that

I1I3 + I3I1 = Φ(I1v + I1v) + iΨ(vtv − vtv) +∑

j

[Ψ(vxjv + vxj

v)]xj

− ∇Ψ · (∇vv + ∇vv) − 2Ψ|∇v|2 + 2Ψ(|∇`|2 − ∆`− Ψ]|v|2. (3.16)

Proof of (3.16). By invoking the definitions in (3.10), we compute

I1I3 + I3I1 = I1(Φ + Ψ)v + (Φ + Ψ)vI1 = Φ(I1v + I1v) + I1Ψv + ΨvI1

= Φ(I1v + I1v) + [ivt + ∆v + (|∇`|2 − ∆`)v − Ψv]Ψv

+ Ψv[−ivt + ∆v + (|∇`|2 − ∆`)v − Ψv]

= Φ(I1v + I1v) + iΨ(vtv − vtv) + Ψ[∆vv + ∆vv] + 2Ψ[|∇`|2 − ∆`− Ψ]|v|2. (3.17)

On the other hand,∑

j

[Ψ(vxjv + vxj

v)]xj=∑

j

Ψ(vxjxjv + vxj

vxj+ vxjxj

v + vxjvxj

) +∑

j

Ψxj(vxj

v + vxjv)

= Ψ[∆vv + ∆vv] + 2Ψ|∇v|2 + ∇Ψ · [∇vv + ∇vv]. (3.18)


Extracting Ψ[∆vv+∆vv] from (3.18) and substituting into the right side of (3.17) yields (3.16), as desired.

Step 4. With reference to (3.10), in this Step 4 we shall establish that

I1I2 + I2I1 = Φ(I1v + I1v) + i

2(∇` · ∇vv)t − 2∑

j

(`xjvtv)xj

+ 2∆`(vvt) +∑

j

[`t(vvxj− vvxj

)]xj

+ (`t|v|2)t − `tt|v|2 + (C);(3.19a)

(C) ≡ (B) − 2∑

j,k

[`xkxj(vxj

vxk+ vxj

vxk)] + 2∆`|∇v|2 − 2[|∇`|2 − ∆`− Ψ]∆`|v|2

− 2

∑

j

[

∑

k

(2`xk`xjxk

− `xjxkxk)

]

`xj−∇Ψ · ∇`

|v|2; (3.19b)

(B) ≡ 2∑

j

∇` · [∇vvxj+ ∇vvxj

] − `xj|∇v|2 + [|∇`|2 − ∆`− Ψ]`xj

|v|2

xj

. (3.19c)

Proof of (3.19). We shall accomplish this in a few steps. (i) First, we show that

I1I2 + I2I1 = Φ(I1v + I1v) + 2i∇` · [∇vvt −∇vvt] + `t(|v|2)t + 2∇` · [∇v∆v + ∇v∆v]

+ 2∇` · ∇(|v|2)[|∇`|2 − ∆`− Ψ] + i`t[v∆v − v∆v]. (3.20)

In fact, to prove (3.20), we recall (3.10) and write

I1I2 + I2I1 = I1[2∇` · ∇v + Φv − i`tv] + [2∇` · ∇v + Φv + i`tv]I1

= Φ[I1v + I1v] + [ivt + ∆v + (|∇`|2 − ∆`)v − Ψv][2∇` · ∇v − i`tv]

+ [−ivt + ∆v + (|∇`|2 − ∆`)v − Ψv][2∇` · ∇v + i`tv]

= Φ[I1v + I1v] + ivt2∇` · ∇v + ivt(−i`tv) − ivt2∇` · ∇v − ivt(i`tv)

+ [∆v + (|∇`|2 − ∆`)v − Ψv]2∇` · ∇v + [∆v + (|∇`|2 − ∆`)v − Ψv]2∇` · ∇v

− i`tv[∆v + (|∇`|2 − ∆`)v − Ψv] + i`tv[∆v + (|∇`|2 − ∆`)v − Ψv] (3.21)

= Φ[I1v + I1v] + 2i[∇` · ∇vvt −∇` · ∇vvt] + `t[vtv + vtv]

+ 2∇` · [∇v∆v + ∇v∆v] + 2(|∇`|2 − ∆`− Ψ)∇` · (∇vv + ∇vv) + i`t[∆vv − ∆vv], (3.22)

after a cancellation of terms, i`t(|∇`|2 − ∆`)|v|2 and i`tΨ|v|2. Then, (3.22) readily yields (3.20).(ii) Next, we shall show that (3.20) can be rewritten as

I1I2 + I2I1 = Φ[I1v + I1v] + 2i∑

j

`xj[(vvxj

)t − (vtv)xj]

+ `t(|v|2)t

+ 2∑

j,k

`xj(vxk

vxj+ vxk

vxj)xk

− 2∇` · ∇(|∇v|2)

+ 2∇` · ∇(|v|2)[|∇`|2 − ∆`− Ψ] + i`t∑

j

(vvxj− vvxj

)xj. (3.23)


Indeed, adding and subtracting, we obtain

vxjvt − vxj

vt = (vvxj)t − (vtv)xj

;

∇` · [∇vvt −∇vvt] =∑

j

`xj[(vvxj

)t − (vtv)xj];

(3.24a)

(3.24b)

vvxjxj− vvxjxj

= (vvxj− vvxj

)xj;

v∆v − v∆v =∑

j

(vvxj− vvxj

)xj;

(3.25a)

(3.25b)

vxkxkvxj

+ vxkxkvxj

= (vxkvxj

+ vxkvxj

)xk− (|vxk

|2)xj;

∇` · [∇v∆v + ∇v∆v] =∑

j,k

`xj[(vxk

vxj+ vxk

vxj)xk

− (|vk |2)xj].

(3.26a)

(3.26b)

We now substitute (3.24b), (3.25b), (3.26b) into the 2nd, 6th, and 4th terms on the right side of (3.20), andwe readily obtain (3.23).

(iii) Finally, starting from (3.23) we shall establish the final sought-after identity (3.19), as desired. Thisis the step where we shall use the assumption `txj

≡ 0, j = 1, . . . , n, on `. To this end, we use the followingidentities, valid since `txj

≡ 0:

(`xjvvxj

)t − (`xjvtv)xj

+ `xjxjvvt = `xj

[(vvxj)t − (vtv)xj

];

∑

j

`xjvvxj

)t − (`xjvtv)xj

+ ∆` vvt =∑

j

`xj[(vvxj

)t − (vtv)xj]

;

(3.27a)

(3.27b)

`t(vvxj− vvxj

)xj= [`t(vvxj

− vvxj)]xj

;

∑

j

`t(vvxj− vvxj

)xj=∑

j

[`t(vvxj− vvxj

)]xj.

(3.28a)

(3.28b)

We now substitute (3.27b), (3.28b) into the 2nd and 6th on the right side of (3.23), and we readily obtain

I1I2 + I2I1 = Φ[I1v + I1v] + 2i∑

j

(`xjvvxj

)t − (`xjvtv)xj

+ 2i∆` vvt + `t(|v|2)t + i∑

j

[`t(vvxj− vvxj

)]xj+ (A); (3.29)

(A) ≡ 2∑

j,k

`xk(vxj

vxk+ vxj

vxk)xj

− 2∇` · ∇(|∇v|2) + 2∇` · ∇(|v|2)[|∇`|2 − ∆`− Ψ]. (3.30)

We now recall the term (B) from (3.19c) and relate it to (A). Set and compute

(B) ≡ 2∑

j

∑

k

[`xk(vxj

vxk+ vxj

vxk) − `xj

|vxk|2] + [|∇`|2 − ∆`− Ψ]`xj

|v|2

xj

(3.31)

= 2∑

j

∑

k

[`xkxj(vxj

vxk+ vxj

vxk) + `xk

(vxjvxk

+ vxjvxk

)xj− `xjxj

|vxk|2 − `xj

(|vxk|2)xj

] + [|∇`|2

− ∆`− Ψ]xj`xj

|v|2 + [|∇`|2 − ∆`− Ψ]`xjxj|v|2 + [|∇`|2 − ∆`− Ψ]`xj

(|v|2)xj

; (3.32)


(B) = (A) + 2∑

j,k

[`xkxj(vxj

vxk+ vxj

vxk)] − 2∆`|∇v|2

+ 2∑

j

[|∇`|2 − ∆`− Ψ]xj`xj

|v|2 + 2[|∇`|2 − ∆`− Ψ]∆`|v|2, (3.33)

since the 2nd, 4th, and 7th term in (3.32) comprise (A). Finally, we extract (A) from (3.33) and substitute itinto the right side of (3.29) and obtain

I1I2 + I2I1 = Φ(I1v + I1v) + 2i∑

j

(`xjvvxj

)t − (`xjvtv)xj

+ 2i∆`vvt + `t(|v|2)t

+ i∑

j

[`t(vvxj− vvxj

)]xj+ (B) − 2

∑

j,k

[`xkxj(vxj

vxk+ vxj

vxk)] − 2∆`|∇v|2

− 2∑

j

[|∇`|2 − ∆`− Ψ]xj`xj

|v|2 − 2[|∇`|2 − ∆`− Ψ]∆`|v|2. (3.34)

Finally, to obtain (3.19) from (3.34), we substitute the definition (3.31) for (B) in the right side of (3.34),and we use here the identities

`t(|v|2)t = (`t|v|2)t − `tt|v|2; [|∇`|2 − ∆`− Ψ]xj=∑

k

[2`xk`xjxk

− `xkxkxj] − Ψxj

. (3.35)

Step 5. In this step we shall establish that

|θ(iwt + ∆w)|2 ≥ |I3|2 − (I1I2 + I2I1) − (I2I3 + I3I2) + (I1I3 + I3I1) (3.36)

= X1 +X2 +X3 +X4; (3.37)

X1 ≡ [2(Ψ + ∆`)(|∇`|2 − ∆`) + 4∑

j,k

`xk`xj`xjxk

+ 2∇Φ · ∇`

− 2∑

j,k

(`xj`xjxkxk

) − Ψ2 − Φ2 + 2Φ∆`+ `tt]|v|2

+ 2

∑

j,k

[`xjxk(vxj

vxk+ vxj

vxk)] − (Ψ + ∆`)|∇v|2

= real-valued; (3.38a)

X2 ≡ − ∂

∂t(`t|v|2) − 2

∑

j

∂

∂xj

[|∇`|2 − ∆`+ Φ]`xj|v|2 − Ψ

2(vxj

v + vxjv)

+ ∇` · [∇vvj + ∇vvj ] − `j |∇v|2

= real-valued; (3.38b)

X3 = −∇Ψ · [∇vv + ∇vv] + i(Ψ + ∆`)(vtv − vtv) = real-valued; (3.39a)

X4 ≡ −i∑

j

∂

∂t[2`xj

vvxj+ `xjxj

|v|2] +∂

∂xj[`t(vvxj

− vvxj) − 2(`xj

vtv)]

= real-valued. (3.39b)

In fact, to obtain (3.37), we need to invoke (3.10b) (for I3), (3.19), (3.12), (3.16) in the right side of (3.36)


(which is the same as (3.11)). We obtain

|I3|2 − (I1I2 + I2I1) − (I2I3 + I3I2) + (I1I3 + I3I1)

= |(Φ + Ψ)v|2 −

Φ(I1

v + I1v) + i

(2∇` · ∇vv)t − 2∑

j

(`xjvtv)xj

+ 2∆`(vvt) +∑

j

[`t(vvxj− vvxj

)]xj

+ 2∑

j

∇` · [∇vvxj+ ∇vvxj

] − `xj|∇v|2 + (|∇`|2 − ∆`)`xj

|v|2

xj

− 2∑

j

Ψ`xj@@@I |v|2xj

− 2∑

j,k

`xkxj(vxj

vxk+ vxj

vxk) + 2∆`|∇v|2 − 2(|∇`|2 − ∆`)∆`|v|2 + 2

Ψ∆`|v|2

− 2∑

j

[

∑

k

2`k`xjxk

]

`xj|v|2 + 2

∑

j

[

∑

k

`xjxkxk

]

`xj|v|2 + 2∇Ψ

· ∇`|v|2 + (`t|v|2)t − `tt|v|2

−[[

2∑

j

[Φ`xj|v|2]xj

+ 2∑

j

[Ψ`xj

@@

@I|v|2]xj

+ 2Φ2 + ΨΦ − Φ∆`− Ψ∆`−∇Φ · ∇`−∇Ψ

· ∇`|v|2]]

+

((

Φ(I1

v + I1v) + iΨ(vtv − vtv) +∑

j

[Ψ(vxjv + vxj

v)]xj−∇Ψ · (∇vv + ∇vv)

− 2Ψ|∇v|2 + 2Ψ(|∇`|2 − Ψ)|v|2 − 2Ψ

∆`|v|2

))

. (3.40)

In (3.40), is (3.19), [[ ]] is (3.12) and (( )) is (3.16). The resulting cancellations are noted in apairwise fashion (4 pairs cancel out). If we now, in (3.40), collect the two terms pre-multiplied by “i”, and usethe identities

(vvt) = (|v|2)t − (vtv − vtv), and `xjxj(|v|2)t = (`xjxj

|v|2)t, (3.41)

by invoking `txj≡ 0 from (3.1), we then obtain the term X4 in (3.39b) for identity (3.37). Likewise, if we

collect the remaining terms in (3.40), which are not pre-multiplied by “i”, we obtain the terms

X1 +X2 +X3

in (3.38a-b), (3.39a) for identity (3.37). Thus, (3.37) is established.

Step 6. Orientation. So far all terms in estimate (3.36)–(3.39) are expressed in terms of the variablev = θw, rather than the original variable w. Beginning with this step, we return from the variable v to thevariable w in terms under the time-derivative ∂

∂t and the space-derivative ∂∂xj

(space gradient). [This is the

same strategy that was used in [20] for hyperbolic equations.] This means the terms comprising X2 and X4 in(3.38b) and (3.39b). More precisely, in our present Schrodinger case, we shall proceed as follows: (i) we shallleave X1 unchanged until the very end of the proof; (ii) we shall pass from the variable v to the variable win the terms X2 and X4: these will still yield—though now in the variable w—time derivative terms ∂

∂t and

space derivative terms ∂∂xj

. This is accomplished in (3.56) for X2 and in (3.46) for X4, below. The ∂∂t -terms

will define ∂∂tM , see (3.4); while the ∂

∂xj-terms will contribute to the definition of div V , see (3.5b), (iii) from

X3—or better from the lower bound on X3—we shall extract one more term in ∂∂xj

to complete the definition

of div V in (3.5). This is accomplished in (3.54). In conclusion: the full terms X2 and X4 as well as onecomponent term of X3 are those that contribute to

[

−∂M∂t + div V

]

on the left side of (3.3). 2


To return from v to w in selected terms, as identified in the above Orientation, we shall use the followingidentities:

v = θw, θ = e`, hence θt = θ`t and vt = θ[wt + `tw]; θxj= θ`xj

;

vxj= θ[wxj

+ `xjw]; |∇v|2 = θ2

∑

j

[wxj+ `xj

w]2.

(3.42a)

(3.42b)

Step 7. In this step we shall show that X4 as defined by (3.39b) in terms of v can be rewritten in termsof w = ξ + iη, ξ = Re w, η = Im w as follows:

X4 =∂

∂t

∑

j

2[θ2`xj(ξxj

η − ξηxj)]

+∑

j

∂

∂xj

2θ2[`t(ξxjη − ξηxj

) − `xj(ξtη − ξηt)]

. (3.43)

Proof of (3.43). We return to X4 given by (3.39b) and rewrite it as follows:

X4 ≡ −i∑

j

µj = −i∑

j

2[(`xjvvxj

)t − (`xjvtv)xj

] + (`xjxj|v|2)t + [`t(vvxj

− vvxj)]xj

, (3.44)

where the following identity may be verified by direct use of (3.42),

µj ≡ 2[(`xjvvxj

)t − (`xjvtv)xj

] + (`xjxj|v|2)t + [`t(vvxj

− vvxj)]xj

= 2[(θ`2xj|w|2 + θ2`xj

wwxj)t − (θ2`xj

`t|w|2 + θ2`xjwwt)xj

] + (θ2`xjxj|w|2)t + [θ2`t(wwxj

− wwxj)]xj

;

(3.45)

X4 =∂

∂t

2∑

j

[θ2`xj(ξxj

η − ξηxj)]

+∑

j

∂

∂xj



, (3.46)

and identity (3.43) is established, as desired.

Step 8. We next consider the last term in the definition of X3 and (3.39a) and for it prove the followingidentity which returns from v to w:

i(Ψ + ∆`)(vtv − vtv) = θ2(Ψ + ∆`)[(Pw)w + w(Pw)] + 2θ2(Ψ + ∆`)|∇w|2

−θ2(Ψ + ∆`)∑

k

(wxkw + wxk

w)xk, (3.47)

recalling Pw = iw + ∆w, see (1.1.1). Indeed, to prove (3.47), we perform a direct computation using (3.42a);in particular, we use, in succession the following three identities, the first of which follows from (3.42) after acancellation of θ2`t|w|2:

i(Ψ + ∆`)(vtv − vtv) = θ2(Ψ + ∆`)(iwtw + (iwt)w); (3.48)

iwtw + (iwt)w = (Pw)w + w(Pw) − (∆ww + ∆ww); (3.49)

∑

k

(wxkw + wxk

w)xk= ∆ww + ∆ww + 2|∇w|2. (3.50)

Identity (3.49) is obtained by using Pw = iwt + ∆w from (1.1.1). Then, using (3.50) in (3.49) and the latterin (3.48) yields (3.47).


Step 9. Regarding the last term on the right side of (3.47), in this step we observe that

θ2(Ψ + ∆`)∑

k

(wxkw + wxk

w)xk=∑

k

[θ2(Ψ + ∆`)(wxkw + wxk

w)]xk

− 2θ2(Ψ + ∆`)∑

k

[`xk(wxk

w + wxkw)] −

∑

k

[θ2(Ψxk+ ∆`xk

)(wxkw + wxk

w)], (3.51)

where, with regard to the 2nd term on the right of (3.51), we have

∑

k

[wxk(`xk

w) + wxk(`xk

w)] =∑

k

2 Re(wxk(`xk

w)) ≥ −∑

k

(|wxk|2 + `2xk

|w|2)

= −[|∇w|2 + |∇`|2|w|2]. (3.52)

The validity of (3.51) is an immediate verification.

Step 10. In this step, we return to (3.47) and establish the following estimate:

i(Ψ + ∆`)(vtv − vtv) ≥ θ2(Ψ + ∆`)[(Pw)w + w(Pw)]

+ 2θ2[(Ψ + ∆`)|∇w|2 − |Ψ + ∆`| |∇w|2] −∑

k

[θ2(Ψ + ∆`)(wxkw + wxk

w)]xk

− 2θ2|Ψ + ∆`| |∇`|2|w|2 + θ2∑

k

[(Ψxk+ ∆`xk

)(wxkw + wxk

w)]. (3.53)

In fact: Using inequality (3.52) for the second term on the right of (3.51) produces an estimate for (3.51),which once inserted for the last term on the right of (3.47), finally yields (3.53).

Step 11. In this step we establish the following estimates for the term X3 defined in (3.39a):

X3 ≡ i(Ψ + ∆`)(vtv − vvt) −∑

k

[Ψxk(vxk

v + vxkv)]

≥ θ2(Ψ + ∆`)[(Pw)w + w(Pw)] + 2θ2[(Ψ + ∆`)|∇w|2 − |Ψ + ∆`| |∇w|2]

−∑

k

∂

∂xk[θ2Ψ(wxk

w + wxkw)] −

∑

k

∂

∂xk[θ2∆`(wxk

w + wxkw)]

+∑

k

[∆`xk(vxk

v + vxkv)] − 2

∑

k

[(Ψxk+ ∆`xk

)`xk]|v|2 − 2|Ψ + ∆`| |∇`|2|v|2. (3.54)

To show (3.54), we return to (3.53): here we leave everything unchanged, except that, for its last term, we usethe identity

∑

k

([Ψk + ∆`xk)(vxk

v + vxkv − 2`xk

|v|2)] =∑

k

θ2[(Ψxk+ ∆`xk

)(wxkw + wxk

w)]. (3.55)

Identity (3.55) follows directly, by (3.42b), after a cancellation of 2`xkθ2|w|2. Thus, substituting the right side

of (3.55) into the last term of (3.53) yields (3.54).

Step 12. In this step, with reference to X2 given by (3.38b), we establish the following identity for theterm X2 in (3.3) which returns from v to w:


X2 ≡ −(`t|v|2)t − 2∑

j

(|∇`|2 − ∆`+ Φ)`xj|v|2 − Ψ

2(vxj

v + vxjv)

+∑

k

[`xk(vxj

vxk+ vxj

vxk) − `xj

|vxk|2]

xj

= − ∂

∂t(`tθ

2|w|2) − 2∑

j

∂

∂xj

θ2[2|∇`|2 − ∆`+ Φ − Ψ]`xj|w|2

+ θ2(

|∇`|2 − Ψ

2

)

(wxjw + wxj

w) + θ2[

∑

k

`xk(wxj

wxk+ wxj

wxk) − `xj

|∇w|2]

. (3.56)

Proof of (3.56). Using (3.42b), we first obtain the following identities by direct computations

vxjv + vxj

v = θ2[2`xj|w|2 + wxj

w + wxjw]; (3.57)

vxjvxk

+ vxjvxk

= θ2[

2`xj`xk

|w|2 + `xj(wwxk

+ wwxk) + `k(wxj

w + wxjw) + (wxj

wxk+ wxj

wxk)

]

; (3.58)

|vxk|2 = vxk

vxk= θ2[`2xk

|w|2 + `xk(wwxk

+ wwxk) + |wxk

|2]. (3.59)

Finally, we substitute (3.57), (3.58), (3.59) on the left side of (3.56) and readily obtain the right side of(3.56) after a cancellation of `xj

`xk(wwxk

+ wwxk).

Step 13. We return to inequality (3.37), rewritten here

|θ(iwt + ∆w)|2 ≥ X1 +X2 +X3 +X4. (3.60)

Now, we leave X1 untouched; we invoke identity (3.56) for X2; we invoke inequality (3.54) for X3; and, finally,we invoke identity (3.43) for X4. In doing so, the term

∑

j

∂

∂xjθ2[Ψ(wxj

w + wxjw)]

in X2 cancels out with the term of opposite sign in X3. We obtain from (3.60),

θ2|iwt + ∆w|2 ≥ X1 +

[[

∂

∂t[−`tθ2|w|2] − 2

∑

j

∂

∂xj

θ2[2|∇`|2 − ∆`+ Φ − Ψ]`xj|w|2

− 2∑

j

∂

∂xj

θ2|∇`|2(wxjw + wxj

w)

− 2∑

j

∂

∂xj

θ2[

∑

k

`xk(wxj

wxk+ wxj

wxk) − `xj

|∇w|2]]]

+

θ2(Ψ + ∆`)[(Pw)w + w(Pw)] + 2θ2[(Ψ + ∆`)|∇w|2 − |Ψ + ∆`| |∇w|2]

−∑

j

∂

∂xj[θ2∆`(wxj

w + wxjw)] +

∑

j

[∆`xj(vxj

v + vxjv)]

− 2∑

j

[(Ψxj+ ∆`xj

)`xj]|v|2 − 2|Ψ + ∆`| |∇`|2|v|2

+

((

∂

∂t

∑

j

2[θ2`xj(ξxj

η − ξηxj)]

+∑

j

∂

∂xj



))

. (3.61)


Step 14. Finally, for any ε > 0, we note the following two inequalities:

(Ψ + ∆`)[(Pw)w + (Pw)w] ≥ −ε|Ψ + ∆`|2|w|2 − 1

ε|Pw|2; (3.62)

∑

j

∆`xj(vxj

v + vxjv) ≥ −ε|∇v|2 − 1

ε

∑

j

|∆`xj|2|v|2 (3.63)

(both being specializations, as (3.52) before, of the inequality [ab+ ab] = 2 Re(ab) ≥ −ε|a|2 − 1ε |b|2). Finally,

using these two inequalities for the first and fourth terms of the group , we readily obtain, afterrecalling the definitions of X1 in (3.38a):

θ2(

1 +1

ε

)

|iwt + ∆w|2

≥[

2(Ψ + ∆`)(|∇`|2 − ∆`) + 4∑

j,k

`xk`xj`xjxk

+ 2∇φ · ∇`− 2∇` · ∇∆`− (Ψ2 + Φ2) + 2Φ∆`+ `tt

]

|v|2

+ 2

∑

j

∇`j · (∇vvxj+ ∇vvxj

) − (Ψ + ∆`)|∇v|2

+

[[

− ε|Ψ + ∆`|2θ2|w|2 + 2θ2[(Ψ + ∆`)|∇w|2 − |Ψ + ∆`| |∇w|2] − ε|∇v|2 − n

ε

∑

j,k

`2xjxjxk|v|2

− 2[∇Ψ · ∇`+ ∇∆` · ∇`]|v|2 − 2|Ψ + ∆`| |∇`|2|v|2]]

+∂

∂t[−`tθ2|w|2] +

∂

∂t

2θ2[∇` · ∇ξη −∇ηξ]

− 2∑

j

∂

∂xj

`xj[2|∇`|2 − ∆`+ Φ − Ψ]θ2|w|2

− 2∑

j

∂

∂xj

θ2|∇`|2(wxjw + wxj

w)

− 2∑

j

∂

∂xj

θ2[

∑

k

`xk(wxj

wxk+ wxj

wxk) − `xj

|∇w|2]

− 2∑

j

∂

∂xj[θ2∆`(wxj

w + wxjw)] +

∑

j

∂

∂xj



. (3.64)

Finally, recalling the definition of M in (3.4), and of V in (3.5), we recognize that inequality (3.64) can berewritten more concisely as in (3.3). The proof of Lemma 3.1 is complete. 2

Remark 3.1. Inspection of the proof reveals that we may relax by 1 unit the regularity in time of thequantities in (3.1), and thus require w to be C1 in time and C2 in space; ` to be C2 in time and C3 in space;Ψ and Φ to be C0 in time and C1 in space. The above proof never uses wtt, `ttt, Ψt, Φt. 2

4. A BASIC POINTWISE INEQUALITY

We now make suitable choices of the functions `(t, x), Ψ(t, x) and Φ(t, x) occurring in Lemma 3.1.

Theorem 4.1. Letw(t, x) ∈ C2(Rt × Rnx ; C); d(x) ∈ C3(Rnx ; R) (4.1)

be two given functions, w being complex-valued, while d being real-valued. [At this stage, w and d need not bethe solution of Eqn. (1.1.1) and the function provided by assumptions (A.1) and (A.2), respectively.] If τ > 0


is a parameter, we select the functions `(t, x), Ψ(t, x) and Φ(t, x) in Lemma 3.1 as follows:

`(t, x) ≡ τ

[

d(x) − c

(

t− T

2

)2]

≡ τϕ(t, x) ∈ C3(Rt × Rnx ; R); (4.2)

Ψ(t, x) ≡ −∆`(t, x) ∈ C1(Rt × Rnx ; R); (4.3a)

either Φ(t, x) = ∆`(t, x); or else Φ(t, x) ≡ 0, (4.3b)

so that the required assumption `txj≡ 0, j = 1, . . . , n in (3.2) is satisfied. The function ϕ(t, x) in (4.2) is

defined consistently with (1.1.8a). In particular, T > 0 is arbitrary, while the constant c appearing in (4.2) isselected in (1.1.8b).

(a) Then, with the above choices, Lemma 3.1 specializes as follows. Set h ≡ ∇d, then:

`xj≡ τdxj

; |∇`|2 = τ2|∇d|2; `xkxj≡ τdxkxj

; `xjxjxk≡ τdxjxjxk

;∑

j,k

`2xjxjxk= τ2

∑

j,k

d2xjxjxk

; (4.4)

4H`∇` · ∇` = 4∑

j,k

`xkxj`xj

`xk≡ 4τ3

∑

j,k

dxkxjdxj

dxk≡ 4τ3Hd∇d · ∇d; (4.5)

2H`[∇v · ∇v + ∇v · ∇v] = 2∑

j,k

`xjxk(vxj

vxk+ vxk

vxj) ≡ 2τ [Hd∇v · ∇v + Hd∇v · ∇v]; (4.6)

∇` = τ∇d; ∆` ≡ τ∆d; `t = −2cτ

(

t− T

2

)

; `tt ≡ −2cτ ; `txj≡ 0; (4.7)

−(Ψ2 + Φ2) + 2Φ∆` = −(Φ − ∆`)2, for Ψ = −∆`, (4.8)

where Hd is the n × n Hessian matrix of d(x) defined by (1.1.6) and H` is the n × n Hessian matrix of `.Moreover, in (4.8), we have only used Ψ = −∆` from (4.3a). Moreover, still with Ψ = −∆`, we obtain

2∇[Φ − ∆`] · ∇`− (Ψ2 + Φ2) + 2Φ∆` = 2∇[Φ − ∆`] · ∇`− (Φ − ∆`)2

=

0, if Φ = ∆`,

−τ2[2∇∆d · ∇d+ (∆d)2], if Φ ≡ 0,

(4.9a)

(4.9b)

according to the two choices in (4.3b).

(b) As a consequence of (4.2)–(4.9), the pointwise estimate (3.3) of Lemma 3.1 becomes as follows:

(b1) Under the choice Ψ = −∆`, we obtain from (3.3)

(

1 +1

ε

)

θ2|iwt + ∆w|2 − ∂M

∂t+ div V

≥ 2H`∇v · ∇v + H`∇v · ∇v − ε|∇v|2 +

4H`∇` · ∇`−n

ε

∑

j,k

`2xjxjxk− (Ψ2 + Φ2) + 2Φ∆`+ `tt

|v|2

(4.10)

= 2τHd∇v · ∇v + Hd∇v · ∇v − ε|∇v|2 +

4τ3Hd∇d · ∇d−n

ετ2∑

j,k

d2xjxjxk

− (Φ − ∆`)2 − 2cτ

|v|2.

(4.11)


(b2) Either with the further choice Φ = 0 in (4.11) (whereby (Φ−∆`)2 = τ2(∆d)2) or else with the furtherchoice Φ = ∆` in (4.11), we likewise obtain in both cases from (4.11) the final resulting pointwise estimate

(

1 +1

ε

)

θ2|iwt + ∆w|2 − ∂M

∂t+ div V ≥ 2τ [Hd∇v · ∇v + Hd∇v · ∇v] − ε|∇v|2

+[4τ3Hd∇d · ∇d+ O(τ2)]θ2|w|2, ∀ x, t ∈ Q, (4.12)

for Ψ = −∆`, and either Φ = 0 or else Φ = ∆`, where the constant in O depends on d, n, c, ε.(c) Moreover, only under the choice (4.2) for ` and (4.3a) for Ψ = −∆`, and leaving Φ uncommitted, the

component Vj in (3.5b) of the vector V in (3.5a) specializes to the following expression:

Vj = Vj(w) = 2θ2

(2τ2|∇d|2 + Φ)τdxj|w|2 + (`tη)ξxj

− (`tξ)ηxj− (ξtη − ξηt)τdxj

+

[

τ2|∇d|2 − 1

2τ∆d

]

[wwxj+ wwxj

] + τ∇d · [∇wwxj+ ∇wwxj

] − τdxj|∇w|2

. (4.13)

Proof. The proof is a direct verification. The choice Ψ + ∆` ≡ 0 causes the vanishing of five terms forthe coefficient of |v|2; the vanishing of one term for the coefficient of |∇v|2; and the vanishing of the coefficientof |∇w|2. The remaining terms then yield (4.12) by virtue of the obvious identities (4.4)–(4.9). Notice thatO(τ2) comes from n

ε

∑

j,k `2xjxjxk

regardless of the choice of Φ. The verification of (4.13) is immediate from(3.5b). 2

The pointwise estimate of interest in Corollary 4.2 below is then obtained for the choice of the functiond(x) ∈ C3(Rnx) coming from assumptions (A.1)(ii) = (1.1.6) and (A.2) = (1.1.7).

Corollary 4.2. Let d(x) ∈ C3(Rnx) satisfy assumptions (A.1)(ii) = (1.1.6) and (A.2) = (1.1.7). Definethen `, Ψ, and Φ as in (4.2), (4.3), with constant c > 0 in (4.2) selected as in (1.1.8b) for T > 0 arbitrary. Letw ∈ C2(Rt × Rnx ; C) as in (4.1).

(i) Then, with these choices, inequality (4.12) specializes as follows: for any ε > 0 small, we have forsufficiently large τ :

(

1 +1

ε

)

θ2|iwt + ∆w|2 − ∂M

∂t+ div V ≥ [4τρ− ε]|∇v|2 + [4τ3ρp2 + O(τ2)]θ2|w|2 (4.14)

≥ δ0

[

2τρ− ε

2

]

θ2|∇w|2 + [4τ3ρp2(1 − δ0) + O(τ2)]θ2|w|2, (4.15)

for some 1 > δ0 > 0, where the constant in O depends on n, ε, d, c.(ii) Moreover, for future use below, we note that on the boundary Γ = ∂Ω with outward unit normal

ν = [ν1, . . . , νn], (4.13) yields the following identity, where ∇` = τ∇d = τh, |h| = |∇d|, and where Φ is leftuncommitted:

∫

Ω

div V dΩ =

∫

Γ

V · ν dΓ =

∫

Γ

∑

j

VjνjdΓ

= 2

∫

Γ

θ2 [2τ2|h|2 + Φ] τ |w|2h · ν dΓ + 2

∫

Γ

θ2`t

[

η∂ξ

∂ν− ξ

∂η

∂ν

]

dΓ

− 2

∫

Γ

θ2[ξtη − ξηt]τh · ν dΓ +

∫

Γ

θ2 [2τ2|h|2 − τ∆d]

[

w∂w

∂ν+ w

∂w

∂ν

]

dΓ

+ 2

∫

Γ

θ2τh ·[

∇w∂w∂ν

+ ∇w∂w∂ν

]

dΓ − 2

∫

Γ

θ2|∇w|2τh · ν dΓ. (4.16)


(iii) Finally, as for the M -term in (3.4), we likewise note for future reference that

∫

Ω

∫ T

0

∂M

∂tdt dΩ =

[∫

Ω

M dΩ

]T

0

≤ τCd,T

[∫

Ω

e2τϕ[|∇w|2 + |w|2]dΩ]T

0

(4.17)

≤ Cd,T τ e−2τδ[E(T ) + E(0)], (4.18)

where E(t) = ‖w(t)‖2H1(Ω), as defined in (1.1.11).

Proof. (i) Inequality (4.14) follows at once from estimate (4.12) by direct use of assumptions (A.1)(ii) =(1.1.6) and (A.2) = (1.1.7). Next, (4.15) is then readily obtained from (4.14), by use of the inequality [20],

2|∇v|2 ≥ θ2|∇w|2 − 2τ2|∇d|2|v|2 = θ2|∇w|2 − 2τ2|∇d|2θ2|w|2,

which follows from θwxj= vxj

−θτ dxjw, see (3.2). (ii) Identity (4.16) follows at once from (4.13) with h = ∇d.

(iii) Recalling M in (3.4), we obtain

∣

∣

∣

∣

∣

∫

Ω

∫ T

0

∂M

∂tdt dΩ

∣

∣

∣

∣

∣

=

∣

∣

∣

∣

∣

[∫

Ω

M dΩ

]T

0

∣

∣

∣

∣

∣

(by (3.4)) =

∣

∣

∣

∣

∣

[∫

Ω

θ2[2τ(∇d · ∇ξ)η − τξ∇d · ∇η − `t|w|2]dΩ]T

0

∣

∣

∣

∣

∣

(4.19)

≤ τcd,T

∣

∣

∣

∣

∣

[∫

Ω

θ2 [|∇w| |w| + |w|2]dΩ]T

0

∣

∣

∣

∣

∣

(4.20)

≤ τcd,T

∣

∣

∣

∣

∣

[∫

Ω

θ2[|∇w|2 + |w|2]dΩ]T

0

∣

∣

∣

∣

∣

, (4.21)

where in going from (4.19) to (4.20) we have recalled ξ = Re w, η = Im w, and `t from (4.7). Inequality(4.21) establishes (4.17) via (3.2) on θ, from which (4.18) follows by recalling property (1.1.9) for ϕ and(1.1.11). 2

5. PROOF OF THEOREM 2.1.1 AND COROLLARY 2.1.2. CARLEMAN ESTIMATESFOR H2,2(Q)-SOLUTIONS OF EQN. (1.1.1). FIRST VERSION

The next key result yields a Carleman-type estimate. This is Theorem 2.1.1, rewritten here for convenience,however, at this stage, obtained only for H2,2(Q)-solutions of Eqn. (1.1.1). Achievement of its extensionTheorem 2.2.1 in its full strength, i.e., as extended to solutions of Eqn. (1.1.1) in the class (2.2.1) will takeplace in Section 7.

Theorem 5.1. Let T > 0, and let c be defined accordingly by (1.1.8b). Let d(x) ∈ C3(Rnx) be a functionsatisfying (A.1)(ii) = (1.1.6) and (A.2) = (1.1.7). Define ϕ(t, x) as in (1.1.8a) = (4.2). Let w ∈ C2(Rt×Rnx ; C)be a solution of Eqn. (1.1.1) [and no B.C.] under the standing assumption (1.1.3) for F (w) with constant cTand (1.1.4) for f . Then: the following one-parameter family of estimates hold true for all τ > 0 sufficiently


large and all ε > 0:

BΣ(w) + 2

(

1 +1

ε

)∫ T

0

∫

Ω

e2τϕ|f |2dΩ dt

≥ mρ,p,τ,cT ,ε

∫ T

0

∫

Ω

e2τϕ[|∇w|2 + |w|2]dΩ dt− cd,T τ e−2τδ[E(T ) + E(0)] (5.1)

≥ mρ,p,τ,cT ,ε e−δτ

∫ t1

t0

E(t)dt− Cd,T τe−2τδ[E(T ) + E(0)] (5.2)

mρ,p,τ,cT ,ε ≡ min

[

δ0

(

2τp− ε

2

)

− 2CT

(

1 +1

ε

)]

,

[

4τ3ρp2(1 − δ0) + O(τ2)

−2CT

(

1 +1

ε

)]

, ∞ as τ ∞. (5.3)

On the LHS of (5.1), the boundary terms BΣ(w) are defined by (recall h ≡ ∇d)

BΣ(w) ≡∫ T

0

∫

Ω

div V dΩ dt =

∫ T

0

∫

Γ

V · ν dΓ dt

= 2

∫ T

0

∫

Γ

e2τϕ [2τ2|h|2 + Φ] τ |w|2h · ν dΓ dt− 2

∫ T

0

∫

Γ

e2τϕ2cτ

(

t− T

2

)[

η∂ξ

∂ν− ξ

∂η

∂ν

]

dΓ dt

− 2

∫ T

0

∫

Γ

e2τϕ[ξtη − ξηt]τh · ν dΓ dt+∫ T

0

∫

Γ

e2τϕ [2τ2|h|2 − τ∆d]

[

w∂w

∂ν+ w

∂w

∂ν

]

dΓ dt

+ 2

∫ T

0

∫

Γ

e2τϕτh ·[

∇w ∂w

∂ν+ ∇w ∂w

∂ν

]

dΓ dt− 2

∫ T

0

∫

Γ

e2τϕ|∇w|2τh · ν dΓ dt.

(5.4a)

(5.4b)

(ii) The above inequality (5.3) may then be extended by density to all w ∈ H2,2(Q) = L2(0, T ;H2(Ω)) ∩H2(0, T ;L2(Ω)).

Proof. We return to estimate (4.15) and integrate it overQ = (0, T ]×Ω. On the LHS we invoke Eqn. (1.1.1),as well as identity (4.16) for div V and inequality (4.18) for ∂M

∂t . We thus obtain recalling θ and `t from (3.2),(4.2), (4.7), and BΣ(w) from (5.4a):

(

1 +1

ε

)∫ T

0

∫

Ω

θ2|F (w) + f |2dΩ dt+ τCd,T e−2τδ[E(T ) + E(0)] +BΣ(w)

≥(

1 +1

ε

)∫ T

0

∫

Ω

θ2|iwt + ∆w|2dΩ dt−[∫

Ω

M dΩ

]T

0

+

∫ T

0

∫

Ω

div V dΩ dt

≥ δ0

[

2τρ− ε

2

]

∫ T

0

∫

Ω

θ2|∇w|2dΩ dt+ [4ρ(1− δ0)τ3p2 + O(τ2)]

∫ T

0

∫

Ω

θ2|w|2dΩ dt. (5.5)

Next, invoking estimate (1.1.3) for F (w) on the LHS of (5.5), we finally obtain

2

(

1 +1

ε

)∫ T

0

∫

Ω

e2τϕ|f |2dΩ dt+BΣ(w) ≥[

δ0

(

2τρ− ε

2

)

− 2CT

(

1 +1

ε

)]∫ T

0

∫

Ω

e2τϕ|∇w|2dΩ dt

+

[

4τ3ρp2(1 − δ0) + O(τ2) − 2CT

(

1 +1

ε

)]∫ T

0

∫

Ω

e2τϕ|w|2dΩ dt− τCd,T e−2τδ[E(T ) + E(0)], (5.6)


with CT the constant in (1.1.3) and (5.6) establishes (5.1) by using (5.3). Finally, using property (1.1.10) onϕ and E(t) in (1.1.11) in the first integral term in the RHS of (5.1), we obtain estimate (5.2), by taking τsufficiently large. 2

Theorem 5.1 coincides with Theorem 2.1.1 by taking ε = 1.

Proof of Corollary 2.1.2. Step 1. We return to the definition (2.1.5) (or (5.4b)) of BΣ(w) and verifydirectly that

conditions (2.1.7) ⇒ BΣ(w) ≡ 0

conditions (2.1.8) ⇒ BΣ(w) = 2τ

∫ T

0

∫

Γ0

e2τϕ|∂w∂ν

|2h · ν ≤ 0.(5.7)

Indeed, in the case, say, of (2.1.7), the individual integral terms in (2.1.5) (or (5.4b)) vanish either because∂w∂ν ≡ 0 on all of Γ; or else because of the combination: h · ν = 0 on Γ0 and w ≡ 0 on Γ1, whereby then

|∇w| = ∂w∂ν = 0 on Γ1, where Γ = Γ0 ∪ Γ1. Similarly, for (2.1.8).

Step 2. With BΣ(w) ≤ 0, Σ = (0, T ) × Γ1, in either case then (2.1.3) (or (5.2)) yields with f ≡ 0,

0 ≥ mρ,p,τ,CTe−δτ

∫ t1

t0

E(t)dt− Cd,T τ e−2τδ[E(T ) + E(0)]; (5.8)

Cd,T τe−τδ[E(T ) + E(0)]

mρ,p,τ,CT

≥∫ t1

t0

E(t)dt. (5.9)

Letting τ ∞, whereby mρ,p,τ,CT ∞ at the rate of τ , see (5.3), we conclude that

0 =

∫ t1

t0

E(t)dt, hence w ≡ 0 on (t0, t1) × Ω, (5.10)

recalling E(t) in (1.1.11), where we recall from (1.1.10) that 0 < t0 <T2 < t1 < T . With T > 0 given and fixed,

for which (2.1.7) or (2.1.8) hold true, we may repeat the above argument for all intervals smaller than T andget accordingly, at least, w ≡ 0 in (0, T2 ) × Ω. Finally, we recall from the line below (1.1.10) that actually wemay take as interval (t0, t1) any interval where ϕ(t, x) ≥ σ > −δ uniformly in Ω, with σ any number arbitrarilyclose to −δ. We then conclude that the preceding argument actually yields w ≡ 0 on (0, T ) × Ω, hence w ≡ 0on [0, T ]× Ω, since w ∈ C([0.T ];L2(Ω)), a-fortiori from w ∈ H2,2(Q). 2

6. PROOF OF THEOREM 2.1.3 CARLEMAN ESTIMATE FOR H2,2(Q)-SOLUTIONS OFEQN. (1.1.1). SECOND VERSION

Our starting point is Theorem 5.1, Eqn. (5.2), which coincides with Theorem 2.1.1, Eqn. (2.1.3), forH2,2(Q)-solutions. This is a key contribution of the present paper, which forms the basis upon which the sought-after results are based. The subsequent development is now more in line with [44], [46]. Thus, from now on inthis section, our present proof parallels closely the treatment of [44, Section 2.3] for the Euclidean Schrodingerequation (1.1.1), a subsequent (slightly different) counterpart of which, as generalized to the correspondingRiemann Schrodinger equation on a Riemann manifold M, g, is given in [46, Section 5]. Indeed, our presenttreatment relaxes the assumptions on the coefficient q1 of F (w) of these references, by virtue of the (standard)preliminary changes of variables or time/space rescaling, described below in this section, and documented inAppendix A.

Further assumptions (A.3) = (2.1.9) imposed on the coefficients q0, q1 of F over hypothesis(1.1.3). Resulting model without loss of generality. To obtain Theorem 2.1.3 from Theroem 2.1.1, weneed to impose additional assumptions over the preliminary hypothesis that |q1(t, x)| ∈ L∞(Q) and q0(t, x) ∈L∞(Q), see (1.1.3) and Remark 1.1.1. These additional requirements are combined and listed as assumption(A.3) = (2.1.9). They include: (i) a main structural assumption on the coefficient q1 (when dim Ω ≥ 2), as well


as (ii) further smoothness assumptions on both q0 and q1. Our assertions are documented in Appendix A, towhich we refer for a complete analysis. We distinguish two cases.

Case: dim Ω = 1. Here, with reference to

iwt + wxx = F (w) = q1(t, x)wx + q0(t, x)w, (6.1a)

we further assume over (1.1.3) that

q0, q1, q1t, q1x ∈ L∞(Q). (6.1b)

In this case, Theorem A.1 in Appendix A shows that we can always assume without loss of generality, moduloa change of variable, that q1 = 0 and so F (w) = q0w, so that F has no first-order term.

Case: dim Ω ≥ 2. Here, we specialize the coefficient q1 of F (w) = q1(t, x) · ∇w + q0(t, x)w by requiringthat

q1(t, x) = ∇π(t, x) − ir1(t, x), (6.2a)

with π and r1 being real-valued vector functions, subject to the assumptions

πt, |∇π|,∆π, |r1| ∈ L∞(Q), in addition to q0 ∈ L∞(Q). (6.2b)

Moreover, for dim Ω ≥ 2, under these assumptions (6.2), it is documented in Theorem A.2 of Appendix Athat we can always assume without loss of generality, modulo change of variables and time-space rescaling that

∇π ≡ 0 in Q; |r1 · µ|2 ≤ r1 · ν < 1 on Γ, (6.3)

where µ is a unit tangent vector ( ∂∂µ = tangential gradient) and ν is the unit outward normal at Γ.

Accordingly, under assumption (6.2), we can always assume without loss of generality that the originalSchrodinger model (1.1.1) be specialized to

iwt + ∆w = −ir1(t, x) · ∇w + q0(t, x)w + f. (6.4)

On the new real-valued vector r1(t, x) and new function q0 (after the change of variable), we impose the specificregularity hypotheses detailed in (2.1.9a), (2.1.9b).

Step 1. Lemma 6.1. (i) Let w be a H2,2(Q)-solution of the Schrodinger equation (1.1.1), with f ∈ L2(Q)and F (w) = q1 · ∇w + q0w satisfying (1.1.3). Then:

(i1) with reference to E(t) in (1.1.11), we have for all t, s in [0, T ]:

E(t) − E(s) = 2 Re

∫ t

s

∫

Γ

∂w

∂νwt dΓ dσ

+ 2 Re

i

∫ t

s

∫

Γ

∂w

∂νw dΓ dσ

+ 2 Re

∫ t

s

∫

Ω

[F (w) + f ][i∆w − iw]dΩ dσ

; (6.5)

(i2) Let q0 be as in (2.1.9b) (left), then:

Re

i

∫

Ω

∆wq0w dΩ

= Re

i

∫

Γ

∂w

∂νq0w dΓ

− Re

i

∫

Ω

q0|∇w|2dΩ

− Re

i

∫

Ω

w∇w · ∇q0dΩ

. (6.6)

(ii) Assume further (A.3) = (2.1.9), so that, in particular, we can assume that q1 = −ir1, r1 real-valued, forF (w) = q1 ·∇w+ q0w, for dim Ω ≥ 2 with r1, q0 satisfying (2.1.9a) and (2.1.9b). [See also paragraph preceding(6.4).] Then, for a unit tangent vector µ [see (2.1.19) and also paragraph leading to (6.4)]:


(ii1)Re

i

∫

Ω

∆w(q1 · ∇w)dΩ

= Re

∫

Ω

∆w(r1 · ∇w)dΩ

=1

2

∫

Γ

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

r1 · ν dΓ

+ Re

∫

Γ

∂w

∂ν

∂w

∂µr1 · µ dΓ

− 1

2

∫

Γ

∣

∣

∣

∣

∂w

∂µ

∣

∣

∣

∣

2

r1 · ν dΓ

−∫

Ω

Re

J trr1∇w · ∇w

dΩ − 1

2

∫

Ω

|∇w|2div(r1)dΩ, (6.7)

Re

i

∫

Ω

∆w(q1 · ∇w)dΩ

= Re

∫

Ω

∆w(r1 · ∇w)dΩ

≤∫

Γ

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

dΓ

− 1

2

∫

Γ

∣

∣

∣

∣

∂w

∂µ

∣

∣

∣

∣

2

[r1 · ν − |r1 · µ|2]dΓ −∫

Ω

Re

J trr1∇w · ∇w

dΩ − 1

2

∫

Ω

|∇w|2div(r1)dΩ, (6.8a)

where J trr1 (x) is the transpose of the Jacobian matrix of the real-valued vector r1 = [r11, r12, . . . , r1n] = iq1:

J trr1 (x) =

∂r11∂x1

, · · · , ∂r1n∂x1

......

∂r11∂xn

∂r1n∂xn

, (6.8b)

consistently with the definition in (1.1.6).(ii2)

Re

i

∫

Ω

∆wF (w)dΩ

= Re

i

∫

Ω

∆w [q1 · ∇w + q0w]dΩ

=1

2

∫

Γ

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

r1 · ν dΓ

+ Re

∫

Γ

∂w

∂ν

∂w

∂µr1 · µ dΓ

− 1

2

∫

Γ

∣

∣

∣

∣

∂w

∂µ

∣

∣

∣

∣

2

r1 · ν dΓ + Re

i

∫

Γ

∂w

∂νq0w dΓ

− Re

∫

Ω

J trr1∇w · ∇wdΩ

− 1

2

∫

Ω

|∇w|2div(r1)dΩ

− Re

i

∫

Ω

q0|∇w|2dΩ

− Re

i

∫

Ω

w∇w · ∇q0 dΩ

. (6.9)

(iii) Assume further that f ∈ L2(0, T ;H1(Ω)). Then:(iii1)

Re

i

∫

Ω

∆wf dΩ

= −Re

i

∫

Γ

∂w

∂νf dΓ

+ Re

i

∫

Ω

∇w · ∇f dΩ

. (6.10)

(iv) Under the assumptions of parts (i) through (iii), we then have:

E(t) − E(s) = 2 Re

∫ t

s

∫

Γ

∂w

∂ν[wt + iq0w − if ] dΓ dσ

+

∫ t

s

∫

Γ

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

r1 · ν dΓ dσ

+ 2 Re

i

∫ t

s

∫

Γ

∂w

∂νw dΓ dσ

+ 2 Re

∫ t

s

∫

Γ

∂w

∂ν

∂w

∂µr1 · µ dΓ dσ

−∫ t

s

∫

Γ

∣

∣

∣

∣

∂w

∂µ

∣

∣

∣

∣

2

r1 · ν dΓ dσ − 2 Re

∫ t

s

∫

Ω

[

J trr1∇w · ∇w + iq0|∇w|2 + iw∇w · ∇q0

− i∇w · ∇f + i(F (w) + f)w

]

dΩ dσ

−∫ t

s

∫

Ω

|∇w|2div(r1)dΩ dσ. (6.11a)


where, for purposes other than the Dirichlet B.C., we estimate

2 Re

∫ t

s

∫

Γ

∂w

∂ν

∂w

∂µr1 · µ dΓ dσ −

∫ t

s

∫

Γ

∣

∣

∣

∣

∂w

∂µ

∣

∣

∣

∣

2

r1 · ν dΓ dσ

≤∫ t

s

∫

Γ

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

dΓ dσ

+

∫ t

s

∫

Γ

∣

∣

∣

∣

∂w

∂µ

∣

∣

∣

∣

2

[|r1 · µ|2 − r1 · ν]dΓ dσ ≤∫ t

s

∫

Γ

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

dΓ dσ, (6.11b)

while in the Dirichlet case w|Σ ≡ 0, the LHS of (6.11b) vanishes.(v) Finally, under the assumptions of parts (i) through (iii), we then have the following estimate for all

0 ≤ s ≤ t ≤ T :

E(t) − E(s) = 2 Re

∫ t

s

∫

Γ

∂w

∂ν[wt + iq0w − if ]dΓ dσ

+

∫ t

s

∫

Γ

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

r1 · ν dΓ dσ

+ 2 Re

i

∫ t

s

∫

Γ

∂w

∂νw dΓ dσ

+ Oc

(∫ t

s

∫

Γ

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

dΓ dσ

)

+ O(∫ t

s

E(σ)dσ + ‖f‖2L2(s,t;H1(Ω))

)

; (6.12a)

Oc = 0 in the Dirichlet case w|Σ ≡ 0, and Oc = 1 otherwise.

|E(t) − E(s)| ≤ G(T ) + cT

∫ t

s

E(σ)dσ; (6.12b)

G(T ) = 2

∫ T

0

∫

Γ

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

[

|wt| +1

2

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

|r1 · ν + Oc| + |q0| |w| + |w| + |f |]

dΓ dt+

∫ T

0

∫

Ω

[|∇f |2+|f |2]dΩ dt (6.13)

cT = cq0,q1 = constant depending on: constant CT in (1.1.3); norms |div r1|L∞(0,T ), |J trr1 |L∞(0,T ). (6.14)

Proof. (i) We multiply Eqn. (1.1.1) by [i∆w − iw], take real parts—whereby 2 Rewwt = ∂∂t |w|2 while

the term i|∆w|2 drops off—and integrate by parts, using similarly 2 Re∇w · ∇wt = ∂∂t |∇w|2. We readily

obtain (6.5). Identity (6.6) is obtained by Green’s first theorem.(ii) Identity (6.7) is where the key structural hypothesis (A.3) = (2.1.9) is used, thus we provide details

following [44]. We shall use the identity (Appendix B)

Rei∇w · ∇(q1 · ∇w) = ReJ trr1∇w · ∇w +r12

· ∇(|∇w|2), (6.15)

which holds true precisely for q1 = −ir1, r1 real-vector, as assumed in (A.3) = (2.1.9), with the matrix J trr1defined by (6.8b). Thus we compute the critical term in (6.7), initially by Green’s first theorem:

Re

i

∫

Ω

∆w(q1 · ∇w)dΩ

= Re

i

∫

Γ

∂w

∂ν(q1 · ∇w)dΓ

−∫

Ω

Re i∇w · ∇(q1 · ∇w) dΩ (6.16)

(by (6.15)) = Re

i

∫

Γ

∂w

∂ν(q1 · ∇w)dΓ

− Re

∫

Ω

J trr1∇w · ∇w dΩ

− 1

2

∫

Ω

r1 · ∇(|∇w|2)dΩ, (6.17)

after using identity (6.15). We next evaluate the last term on the RHS of (6.17) by the divergence theorem, toobtain

∫

Ω

r1 · ∇(|∇w|2)dΩ =

∫

Γ

|∇w|2r1 · ν dΓ −∫

Ω

|∇w|2div r1 dΩ. (6.18)


Finally, regarding the boundary term on the RHS of (6.17), we recall that iq1 = r1 (real) by (A.3) = (2.1.9)and compute with µ a unit tangent vector

on Γ : i∂w

∂ν(q1 · ∇w) =

∂w

∂ν

[

r1 · ν∂w

∂ν+ r1 · µ

∂w

∂µ

]

= r1 · ν∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

+ r1 · µ∂w

∂ν

∂w

∂µ. (6.19)

Finally, substituting (6.18) and (6.19) into (6.17) yields identity (6.7) using also that |∇w|2 =∣

∣

∂w∂ν

∣

∣

2+∣

∣

∣

∂w∂µ

∣

∣

∣

2

.

Then, inequality (6.8a) follows readily from (6.7): in (6.8a), we have evidenced the term |r1 ·µ|2−r1 ·ν ≤ 0, via(6.3), or (2.1.9c) without loss of generality, for future use in Appendix C, Eqn. (C.7). Identities (6.9), (6.10)follow readily: the first from (6.6) and (6.7); the second by Green’s first theorem.

(iv) In obtaining identity (6.11a), we use (6.5), (6.9), (6.10). Details are given in Appendix B. The termon the LHS of (6.11b) plainly vanishes in the case of Dirichlet B.C. w|Σ ≡ 0, and otherwise gives the estimatenoted there by using that |r1 · µ|2 − r1 · ν ≤ 0 without loss of generality by virtue of property (6.3), or (2.1.9c).

[This has the implication of removing the term∣

∣

∣

∂2w∂µ2

∣

∣

∣|r1 · ν| from the definition of G(T ) in (6.13).]

Part (v) is a consequence of (iv), as follows: The critical term (in (6.11a) to analyze is

2

∣

∣

∣

∣

Re

i

∫ t

s

∫

Ω

w∇w · ∇q0dΩ dσ∣

∣

∣

∣

≤∫ t

s

∫

Ω

|∇w|2dΩ dσ +

∫ t

s

∫

Ω

|w|2|∇q0|2dΩ dσ. (6.20)

We shall show that

RHS of (6.20) ≤ C

∫ t

s

∫

Ω

|∇w|2dΩ dσ, (6.21)

as desired, provided that

q0 ∈ L1(0, T ;W 1,2(Ω)) for n = 1; q0 ∈ L1(0, T ;W 1,2+ε(Ω)) for n = 2, ε > 0;

q0 ∈ L1(0, T ;W 1,n(Ω)) for n ≥ 3, (6.22)

as assumed in (2.1.9b) (left). Indeed, focusing first on the space variable, by embedding H1(Ω) ⊂ Lρ(Ω) weobtain:

w ∈ H1(Ω) ⇒ w ∈ Lρ(Ω) for ρ ≤ ∞, n < 2; ρ <∞, n = 2; ρ ≤ 2n

n− 2, n > 2. (6.23)

Thus,|w|2 ∈ L∞(Ω), n < 2; |w|2 ∈ Lρ(Ω), n = 2, ρ <∞; |w|2 ∈ L n

n−2(Ω), n > 2. (6.24)

According to (6.24), the second integral term in (6.21) yields (6.22) provided that

|∇q0|2 ∈ L1(Ω), m < 2; |∇q0|2 ∈ L1+ε(Ω), n = 2; |∇q0| ∈ Ln2(Ω), n = 3, (6.25)

since ρ′ = n2 is the conjugate index of ρ = n

n−2 , ρ′ = ρρ−1 . Then, (6.25) yields

q0 ∈ W 1,2(Ω), n < 2; q0 ∈W 1,2+ε(Ω), n = 2; q0 ∈W 1,n(Ω), n > 2. (6.26)

Finally, starting from (6.26) and incorporating in the analysis also the time-derivative, we see that (6.20) yields(6.21), provided that (6.22) holds true.

Then (6.21), used in (6.11a), yields O(∫ t

s E(σ)dσ), as stated in (6.12a), provided that, in addition, q0 ∈L∞(Q) and |∇r1| ∈ L∞(Q), ∇r1 = [∇r11, . . . ,∇r1n], i.e., r1i ∈ L∞(0, T ;W 1,∞(Ω)), as assumed in (2.1.9a).

Remark 6.1. In the proof of Lemma 6.1, instead of multiplying Eqn. (1.1.1) by [i∆w − iw], one couldalternatively multiply Eqn. (1.1.1) by wt (= −i∆w + iF (w) + if) as done in [44, p. 481]. In this case, onewould integrate in time rather in space. This way, one would have to handle a term like

∫ t

s

∫

Ω

q0wwtdΩ dσ, hence

∫ t

s

∫

Ω

(q0)t|w|2dΩ dσ (6.27)


[see [44, Eqn. (2.3.8), p. 482]], rather than the term∫ t

s

∫

Ω|∇q0|2|w|2dΩ dσ as in (6.20). Arguing as above, we

see that this method would likewise yield

∫ t

s

∫

Ω

(q0)t|w|2dΩ dσ ≤ C

∫ t

s

∫

Ω

|∇w|2dΩ dσ, (6.28)

as desired, provided that

(q0)t ∈ L1(0, T ;L1(Ω)), n < 2; (q0)t ∈ L1(0, T ;L1+ε(Ω)), n = 2, ε > 0;

(q0)t ∈ L1(L1(0, T ;Ln2(Ω)), (6.29)

the counterpart (in space) of (6.25). 2

Step 2. Lemma 6.2. Let w be a solution of Eqn. (1.1.1) in the class (2.1.1). Let f ∈ L2(0, T ;H1(Ω)).Assume further hypothesis (A.3) = (2.1.9). Then,

(ii) for any 0 ≤ s ≤ t ≤ T :

E(t) ≤ [E(s) +G(T )]ecT (t−s); E(s) ≤ [E(t) +G(T )]ecT (t−s); (6.30)

(iii)

E(t) ≥ E(0) + E(T )

2e−cTT −G(T ); 0 ≤ t ≤ T. (6.31)

Proof. As usual we apply Gronwall inequality to the two inequalities that result from (6.12) and obtain(6.30), from which (6.31) follows [44]. 2

Remark 6.1. The above Lemmas 6.1 and 6.2 are the counterpart for the H1(Ω)—energy E(t) of [44,Lemma 2.3.1, p. 480], [46, Lemma 5.1 and Proposition 5.2, p. 650], which were instead stated for the ‘gradient-energy’ E(t), see (1.1.11). In the present paper, where the effort is aimed at obtaining the Carleman estimateof Theorem 2.1.3 without lower-order term, it is necessary to work with the energy E(t) in (1.1.11), ratherthan the energy E(t) in (1.1.11). This will be clear in Step 3 below. In [44], [46], where lower-order terms inthe Carleman estimate (hence in observability/stabilization inequalities) were tolerated, it was more focusedto work with E(t). 2

Step 3. We invoke inequality (6.31) for E(t) in Eqn. (2.1.3) = (5.2) and obtain

BΣ(w) + 4

∫ T

0

∫

Ω

e2τϕ|f |2dΩ dt+mρ,p,τ,CTe−δτG(T )(t1 − t0)

≥

mρ,p,τ,CTe−δτ

e−cTT

2(t1 − t0) − Cd,T τ e

−2τδ

]

[E(T ) + E(0)] (6.32)

= kφ,τ,CT[E(T ) + E(0)], (6.33)

for all τ sufficiently large, so that kφ,τ,CT> 0 [recall (5.3)]. Recalling, on the LHS of (6.22), the definition

(6.13) for G(T ), we then see that (6.33) yields (2.1.5) with BΣ(w) defined by (2.1.7). The proof of Theorem2.1.3 is complete. 2

7. EXTENSION OF ESTIMATES TO FINITE ENERGY SOLUTIONS

So far, our estimates have been stated and proved only for C2(Rt×Rnx ; C)-solutions, henceH2,2(Q)-solutionsto Eqn. (1.1.1) with f ∈ L2(Q) (Theorem 5.1(ii)) or f ∈ L2(0, T ;H1(Ω)) (Section 6). In the present section, we


extend all our previous estimates from H2,2(Q)-solutions to finite energy solutions in the class (2.2.1), rewrittenhere

w ∈ C([0, T ];H1(Ω)); wt,∂w

∂ν∈ L2(0, T ;L2(Γ)). (7.0)

In order to achieve this goal, it suffices to extend the validity of estimate (5.2) of Theorem 5.1 from H 2,2(Q)-solutions to finite energy solutions defined by the above class (7.0) = (2.2.1). Here, the main difficulty is thefact that finite energy solutions subject to Neumann B.C. do not produce H1-traces on the boundary. [Bycontrast in the case of Dirichlet B.C. the regularity result of [15], [18, Chapter 10, Section 9] allows one to obtain∂w∂ν ∈ L2(Σ) for w0 ∈ H1

0 (Ω). To overcome this difficulty, we shall invoke a regularization procedure, which wasalready employed in [20, Section 8], with inspiration coming from [12]. Here we shall follow the same schemeof steps as in the case of second-order hyperbolic equations, [20, Section 8], except that the present situation ismore challenging. This is so since the term F (w) in the Schrodinger Eqn. (1.1.1) is an unbounded perturbationof the basic generator on the state space H1(Ω) (see Appendix C), while in the case of second-order hyperbolicequations in [20, Section 8] the corresponding term F (w) is a bounded perturbation of the basic generator onH1(Ω)×L2(Ω). All this requires a more attentive analysis, given in Appendix C and the present Section 7. Tobegin with, the starting point, is the following regularity result—in the interior and on the boundary traces—ofthe non-homogeneous, boundary dissipative problem of the next subsection.

7.1 Regularity Estimates of the Non-homogeneous Boundary Dissipative ProblemIn this section, we consider the well-posedness issue of the following non-homogeneous problem

iwt + ∆w = F (w) + f in (0, T ]× Ω ≡ Q;

w(0, · ) = w0 in Ω;

∂w

∂ν+ wt = g in (0, T ]× Γ ≡ Σ,

(7.1.1a)

(7.1.1b)

(7.1.1c)

with dissipative B.C. The homogeneous case g = 0 is dealt with in Appendix C (Theorem C.2) under the as-sumption (A.4) = (2.2.2) [called(C.0) in Appendix C] that the coefficients q1 = ir1 and q0 are time-independent.

Theorem 7.1.1. Consider problem (7.1.1), under the standing assumptions (1.1.3) and (A.3) = (2.1.9) onF , and, moreover, the additional hypothesis (A.4) = (2.2.2).

(a) Assume, at first, that|r1 · µ|2 ≤ r1 · ν ≤ const < 1. (7.1.2)

Assume the following hypotheses on the data:

w0 ∈ H1(Ω); f ∈ L2(0, T ;H1(Ω)); g ∈ L2(0, T ;L2(Γ)). (7.1.3)

Then: (i) problem (7.1.1) has a unique solution, which, moreover, satisfies the following estimates:(i1) for any ε > 0, there is a positive constant Cε such that

E(t) + ε

∫ t

0

∫

Γ

|wt|2dΓ dσ +

∫ t

0

∫

Γ

∣

∣

∣

∣

∂w

∂µ

∣

∣

∣

∣

2

[r1 · ν − |r1 · µ|2]dΓ dσ

≤ E(0) + Cε

‖g‖2L2(0,t;L2(Γ)) + ‖f‖2

L2(0,t;H1(Ω)) +

∫ t

0

E(σ)dσ

, (7.1.4)

where E(t) is defined by (1.1.11), as the square of the H1(Ω)-norm of the solution w(t).(i2) Consequently, there is a constant CT > 0 such that

E(t) ≤ cT

E(0) + ‖g‖2L2(0,T ;L2(Γ)) + ‖f‖2

L2(0,T ;H1(Ω)

, 0 ≤ t ≤ T ; (7.1.5)


∫ t

0

∫

Γ

|wt|2dΓ dσ ≤ cT

E(0) + ‖g‖2L2(0,T ;L2(Γ)) + ‖f‖2

L2(0,T ;H1(Ω)

, 0 ≤ t ≤ T. (7.1.6)

(ii) More specifically, the following regularity properties hold true for problem (7.1.1), under present as-sumptions: the data in (7.1.3) produce a solution of (7.1.1) satisfying

w,∆w ∈ C([0, T ];H1(Ω) ×H−1(Ω)), F (w) ∈ C([0, T ];L2(Ω)), wt ∈ L2(0, T ;H−1(Ω)); (7.1.7)

∂w

∂ν

∣

∣

∣

∣

Σ

, wt

∣

∣

∣

∣

Σ

∈ L2(0, T ;L2(Γ)) (7.1.8)

continuously.

(b) Assumption (7.1.2) is redundant and can be eliminated. Thus, all conclusions (7.1.4)–(7.1.8) hold truefor F satisfying just assumptions (1.1.3), and (A.3) = (2.1.9), with, moreover, r1 and q0 time independent(non-critical assumption (A.4)). 2

Proof. (a) For g ≡ 0, Theorem C.2 in Appendix C shows, under present assumptions, that problem (7.1.1)generates a s.c. semigroup eAt on the space H ≡ H1(Ω)/const., with infinitesimal generator A. Hence, forg 6= 0, problem (7.1.1) admits a unique solution, at least on the space C([0, T ]; [D(A∗)]′), [ ]′ denotingduality w.r.t. H. We now prove the relevant estimates.

(i1) Under assumptions (1.1.3) and (A.3) = (2.1.4), we return to identity (6.11a), more specifically (B.7)–(B.8) of Appendix B, with s = 0 and obtain

E(t) − E(0) = 2 Re

∫ t

0

∫

Γ

∂w

∂ν

[

wt +∂w

∂µr1 · µ+ iq0w − if

]

dΓ dσ

+ 2Re

i

∫ t

0

∫

Γ

∂w

∂νwdΓ dσ

+

∫ t

0

∫

Γ

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

r1 · ν dΓ dσ

−∫ t

0

∫

Γ

∣

∣

∣

∣

∂w

∂µ

∣

∣

∣

∣

2

r1 · ν dΓ dσ + O∫ t

0

E(σ)dσ + ‖f‖2L2(0,t;H1(Ω))

. (7.1.9)

Substituting ∂w∂ν = −wt + g on the right side of (7.1.9) yields

E(t) + 2

∫ t

0

∫

Γ

|wt|2dΓ dσ +

∫ t

0

∫

Γ

∣

∣

∣

∣

∂w

∂µ

∣

∣

∣

∣

2

r1 · ν dΓ dσ = E(0)

+ 2 Re

∫ t

0

∫

Γ

(−wt + g)

(

∂w

∂µr1 · µ

)

dΓ dσ

+ 2 Re

∫ t

0

∫

Γ

[

(−wt + g)(iq0w − if) + g wt

]

dΓ dσ

+ 2 Re

i

∫ t

0

∫

Γ

(−wtw + gw)dΓ dσ

+

∫ t

0

∫

Γ

[|wt|2 + |g|2 − 2 Rewtg]r1 · ν dΓ dσ

+ O∫ t

0

E(σ)dσ + ‖f‖2L2(0,t;H1(Ω))

,

(7.1.10a)where we have deliberately isolated the following term which we now estimate

2 Re

∫ t

0

∫

Γ

(−wt + g)

(

∂w

∂µr1 · µ

)

dΓ dσ

≤∫ t

0

∫

Γ

| − wt + g|2 +

∣

∣

∣

∣

∂w

∂µ

∣

∣

∣

∣

2

|r1 · µ|2dΓ dσ

=

∫ t

0

∫

Γ

[

|wt|2 + |g|2 − 2 Rewtg +

∣

∣

∣

∣

∂w

∂µ

∣

∣

∣

∣

2

|r1 · µ|2]

dΓ dσ. (7.1.10b)


Substituting the above inequality in (7.1.10a) and moving the boundary terms involving |wt|2 to the LHSyields a cancellation and we arrive at

E(t) +

∫ t

0

∫

Γ

|wt|2(1 − r1 · ν)dΓ dσ +

∫ t

0

∫

Γ

∣

∣

∣

∣

∂w

∂µ

∣

∣

∣

∣

2

[r1 · ν − |r1 · µ|2]dΓ dσ ≤ E(0)

+ 2 Re

∫ t

0

∫

Γ

[

(−wt + g)(iq0w − if) + gwt

]

dΓ dσ

+ 2 Re

i

∫ t

0

∫

Γ

(−wtw + gw)dΓ dσ

+

∫ t

0

∫

Γ

[|g|2 − 2 Rewtg](1 + r1 · ν)dΓdσ

+ O∫ t

0

E(σ)dσ + ‖f‖2L2(0,t;H1(Ω))

. (7.1.11a)

At this point, on the LHS of (7.1.11) we use 0 < ε0 ≤ 1− r1 · ν on Γ, by assumption (7.1.2); while on the RHSof (7.1.11) we use 2|a| |b| ≤ ε′|a|2 + 1

ε′ |b|2, with 0 < ε′ << ε0, |a| = |wt|, so that these terms are absorbed by thecorresponding term on the LHS, and b either g|Γ, or f |Γ or else w|Γ, along with trace theory on the latter twoterms to obtain interior terms to be incorporated in O . We thus obtain for any ε > 0 sufficiently small:

E(t) + ε

∫ t

0

∫

Γ

|wt|2dΓ dσ +

∫ t

0

∫

Γ

∣

∣

∣

∣

∂w

∂µ

∣

∣

∣

∣

2

[r1 · ν − |r1 · µ|2]dΓ dσ

≤ E(0) + Oε

∫ t

0

∫

Γ

|g|2dΓ dσ

+ O∫ t

0

E(σ)dσ + ‖f‖2L2(0,t;H1(Ω))

, (7.1.11b)

which is (7.1.4), as desired.

(i2) Setting

α(T ) ≡ Cε‖g‖2L2(0,T ;L2(Γ)) + ‖f‖2

L2(0,T ;H1(Ω)), (7.1.12)

we obtain the inequality

E(t) ≤ [E(0) + α(T )] + Cε

∫ t

0

E(σ)dσ, 0 ≤ t ≤ T, (7.1.13)

from (7.1.4) via assumption (7.1.2). Applying the Gronwall’s inequality to (7.1.13) yields estimate (7.1.5), withcT = eCεT .

(ii) Thus, (7.1.5) implies that w ∈ L∞(0, T ;H1(Ω)) continuously with respect to the data given in (7.1.3).By an approximation/density argument, we then refine this result to obtain w ∈ C([0, T ];H1(Ω)), continuouslywith respect to the data in (7.1.3), as claimed in (7.1.7). From here, returning to Eqn. (7.1.1a), we readily obtainthe rest of (7.1.7). Furthermore, returning now to (7.1.4) via (7.1.2), we obtain wt|Σ ∈ L2(Σ), continuouslywith respect to the data given in (7.1.3), as claimed in (7.1.8). Finally, returning to (7.1.1c), we obtain (7.1.8)for ∂w

∂ν ∈ L2(Σ) as well. Theorem 7.1.1, part (a), is proved.

(b) We now eliminate (7.1.2). Let F satisfy just assumptions (1.3.3), (A.3) = (2.1.9) and (A.4) = (2.2.2).Then, Appendix A shows that we can always manage to obtain q1 = −ir1, with r1 satisfying property (7.1.2).This is achieved by the smooth changes of variables given in (A.11), (A.18), (A.29) of Appendix A, sequentially.The final result is that the transformed problem, say in w, satisfies the same structural equations (7.1.1a–b) as

w, with (7.1.1c) replaced now by ∂w∂ν + wt+h w = gm for a smooth function h and a modified gm = ae−

π2 e−

i2pg.

Moreover, as noted, the new w-problem satisfies (7.1.2). Set gnew = gm−hw. Then Eqn. (7.1.4) holds true forthe new data, critically w in place of w, and gnew in place of g. but then, playing 2|h(w|Γ)| ≤ ε|w|Γ|2 + 1

ε |h|2and invoking trace theory on (w|Γ), we can absorb its contribution in the energy on the LHS of (7.1.4). Inconclusion (7.1.4) holds true for the transformed w-variable, hence for the original w-variable which is relatedto w by smooth functions. Then (7.1.5)–(7.1.8) hold true, in general, for the transformed w, hence for theoriginal w. 2


Theorem 7.1.2. Consider problem (7.1.1) with F (w) subject to assumptions (1.3.3), (A.3) = (2.1.4), aswell as (A.4) = (2.2.2). Assume further the following hypotheses on the data

w0 ∈ D(A); f ∈ H2,2(Q), g ∈ H1,1(Σ), (7.1.14)

where A is the generator of the s.c. semigroup corresponding to the homogeneous problem with f ≡ 0, g ≡ 0as in Theorem C.2 in Appendix C. [In effect, we shall only use the weaker properties

f ∈ C([0, T ];L2(Ω)); ft ∈ L2(0, T ;H1(Ω));

g ∈ C([0, T ];H12 (Γ)); gt ∈ L2(0, T ;L2(Γ)),

which are implied by (7.1.14).] Then, the following interior and boundary regularity properties are true:

w ∈ C([0, T ];H2(Ω)); wt ∈ C([0, T ];H1(Ω)); wtt ∈ L2(0, T ;H−1(Ω)); (7.1.15)

∂wt∂ν

∣

∣

∣

∣

Σ

, wtt

∣

∣

∣

∣

Σ

∈ L2(Σ). (7.1.16)

Proof. Let at first w0 = 0. We differentiate problem (7.1.1) in time (using the convenient assumption(C.0) that r1 and, less critically, q0 are time-independent), and obtain

i(wt)t + ∆(wt) − F (wt) = ft ∈ L2(0, T ;H1(Ω)) in Q;

wt(0, · ) = 0 in Ω;

∂(wt)

∂ν+ (wt)t = gt ∈ L2(Σ) in Σ.

(7.1.17a)

(7.1.17b)

(7.1.17c)

Invoking Theorem 7.1.1 on problem (7.1.17) we obtain all regularity results in (7.1.15), (7.1.16), except for

the first one. To obtain this, we observe that wt|Σ ∈ C([0, T ];H12 (Γ)) by trace theory on the result in (7.1.15)

for wt, and, moreover, g ∈ C([0, T ];H12 (Γ)) as a consequence of assumption (7.1.14) in g. We can then consider

the elliptic problem

∆w − F (w) = −iwt + f ∈ C([0, T ];L2(Ω)) in Ω;

∂w

∂ν= −wt + g ∈ C([0, T ];H

12 (Γ)) in Γ

(7.1.18a)

(7.1.18b)

(where the regularity on the RHS of (7.1.18a) is conservative) and obtain that w ∈C([0, T ];H2(Ω)), as desired. Thus, (7.1.15) is also fully proved.

Let now w0 ∈ D(A) and f ≡ 0, g ≡ 0 in (7.1.1): we then obtain that w,wt ∈ C([0, T ];D(A) ×H1(Ω))by the semigroup generation result of Theorem C.2, Appendix C, and Eqn. (C.20), (C.21). This result thenyields w,wt ∈ C([0, T ];H2(Ω) × H1(Ω)), as claimed in (7.1.15). Next, we return to the correspondingtime-derivative problem, i.e., problem (7.1.17) with f ≡ g ≡ 0 and wt(0) ∈ H1(Ω):

i(wt)t + ∆(wt) − F (wt) ≡ 0 in Q;

wt(0, · ) = Aw0 ∈ H1(Ω) in Ω;

∂(wt)

∂ν+ (wt)t = 0 in Σ,

(7.1.19a)

(7.1.19b)

(7.1.19c)

and readily obtain wtt ∈ C([0, T ];H−1(Ω)). Moreover, we apply Theorem 7.1.1 to problem (7.1.19) andobtain wtt|Σ ∈ L2(Σ), as desired. Finally, w ∈ C([0, T ];D(A)) implies by definition of D(A) in Appendix C,Eqn. (C.20b) that ∂w

∂ν |Γ = −wt, hence ∂wt

∂ν |Γ = −wtt ∈ L2(Σ). Theorem 7.1.2 is proved. 2


7.2 RegularizationThe present section is a counterpart of [20, Section 8] from second-order hyperbolic equations to Schrodinger

equations with the additional difficulty that now F (w) is an unbounded term, unlike the case of [20]. Accordingly,the spirit of the approximation/regularization argument is similar in the two cases, but there arise some relevanttechnical differences. In this section we consider a solution of the Schrodinger Eqn. (1.1.1), however, withf ∈ L2(0, T ;H1(Ω)), in the class (2.2.1), rewritten here

w ∈ C([0, T ];H1(Ω));∂w

∂ν, wt ∈ L2(0, T ;L2(Γ)). (7.2.1)

Assumptions (1.1.3), (A.3) = (2.1.9) as well as (A.4) = (2.2.2) on the unbounded term F (w) are in force,as we shall invoke Theorem 7.1.1. The goal is to show that any such solution w can be approximated bysmooth solutions of corresponding Schrodinger equations, and that these solutions converge faithfully also atthe boundary Σ, which need not be the case for classical approximations via smooth initial data.

Setting, hypotheses. Let w we a solution of the Schrodinger Eqn. (1.1.1) in the class (7.2.1), where weassume that f ∈ L2(0, T ;H1(Ω)) and that F (w) satisfies hypotheses (1.1.3), (A.3) = (2.1.9), as well as (A.4)= (2.2.2). Define

∂w

∂ν+ wt ≡ g ∈ L2(Σ). (7.2.2)

Thus, such w satisfies

iwt + ∆w − F (w) = f ∈ L2(0, T ;H1(Ω)) in Q;

w(0, · ) = w0 ∈ H1(Ω) in Ω;

∂w

∂ν+ wt = g ∈ L2(Σ) in Σ.

(7.2.3a)

(7.2.3b)

(7.2.3c)

Next, consider the following approximating problem, where k denotes a real parameter that tends to infinity;k → ∞:

iwkt + ∆wk = F (wk) + fk in Q;

wk(0, · ) = wk0 in Ω;

∂wk

∂ν+ wkt = gk in Σ

(7.2.4a)

(7.2.4b)

(7.2.4c)

[here we write for convenience wkt to mean (wk)t = the time-derivative of wk ], where the data on problem(7.2.4) are assumed to satisfy the following hypotheses:

(a)fk ∈ H2,2(Q), fk → f in L2(0, T ;H1(Ω)); (7.2.5)

(b)gk ∈ H1,1(Σ), gk → g in L2(Σ); (7.2.6)

(c)wk0 ∈ D(A), wk0 → w(0) in H1(Ω). (7.2.7)

The first part of the next result is essentially Theorem 7.1.2 restated for problem (7.2.4).

Theorem 7.2.1. Assume the above setting and hypotheses. Then, the following properties hold true:(i) for each k, the approximating/regularizing problem (7.2.4) possesses the following interior regularity

properties:(i1)

wk ∈ C([0, T ];H2(Ω)); ∆wk ∈ C([0, T ];L2(Ω)),

F (wk) ∈ C([0, T ];H1(Ω)); wkt ∈ L2(0, T ;L2(Ω)),

(7.2.8)

(7.2.9)


as well as the following boundary regularity properties:(i2)

∂wkt∂ν

, wktt ∈ L2(Σ); (7.2.10)

(i3)

∫ T

0

∫

Γ

|∇tanwk |2dΣ ≤ C

∫ T

0

∫

Γ

[∣

∣

∣

∣

∂wk

∂ν

∣

∣

∣

∣

2

+ |wk|2]

dΣ +

∫ T

0

‖wkt ‖2H−1(Γ)dt

+

∫ T

0

Ek(t)dt + Ek(T ) + Ek(0) + ‖fk‖2L2(Q)

, (7.2.11)

with constant C > 0 independent of k, where Ek(t) is defined by (1.1.11) with wk instead of w, [|∇tanwk| =

|∂wk

∂µ |].Proof. (i1), (i2) The interior regularity properties (7.2.8), (7.2.9), as well as the trace regularity properties

(7.2.10) on the time-derivatives of ∂wk

∂ν and wkt are the results precisely established in Theorem 7.1.2.(i3) Estimate (7.2.11) on the tangential gradient ∇tanw follows now from Proposition D.2, Appendix D.

2

Theorem 7.2.2. Assume the above setting and hypotheses. Then: (i) the following convergence propertiesconcerning problem (7.2.4) hold true: there exists w∗ ∈ C([0, T ];H1(Ω)), such that

wk → w∗ in C([0, T ];H1(Ω)); (7.2.12)

wkt → w∗t in L2(0, T ;H−1(Ω)); (7.2.13)

wkt |Σ → w∗t |Σ in L2(0, T ;L2(Γ)); (7.2.14)

∂wk

∂ν→ ∂w∗

∂νin L2(0, T ;L2(Γ)); (7.2.15)

∂wk

∂µ→ ∂w∗

∂µin L2(0, T ;L2(Γ)). (7.2.16)

(ii) The limit function w∗, identified in part (i), satisfies the problem

iw∗t + ∆w∗ − F (w∗) = f ∈ L2(0, T ;H1(Ω)) in Q;

w∗(0, · ) = w(0) ∈ H1(Ω) in Ω;

∂w∗

∂ν+ w∗

t = g ∈ L2(Σ) in Σ

(7.2.17a)

(7.2.17b)

(7.2.17c)

(iii) Thus, by uniqueness, w∗ = w, with w the original given function in the class (7.2.1), which satisfiesproblem (7.2.3).

Proof. (i) We apply Theorem 7.1.1 to a sequence wk − w` of solutions of (7.2.4) with data fk − f `,wk0 −w`0, gk − g`, which are Cauchy sequences in L2(0, T ;H1(Ω)), H1(Ω), and L2(Σ), respectively. Then,estimate (7.1.5) with such Cauchy data yields that wk −w` is a Cauchy sequence in L∞(0, T ;H1(Ω)), hencein C([0, T ];H1(Ω)), and thus (7.2.12) is established. From (7.2.12), via (7.2.4a) written with such Cauchy datafk− f `, we obtain (7.2.13) as well. Then, estimate (7.1.4) again written for all such Cauchy data f k− f `,wk0 − w`0, gk − g`, yields (7.2.14). From here, via (7.2.3c), we obtain (7.2.15) as well.

Finally, to obtain (7.2.16), we apply estimate (D.5) of Appendix D, again written for Cauchy data f k−f `,wk0 − w`0.


(ii) Standard passage to the limit on system (7.2.4) (in a weak form) yields then that the function w∗,identified in part (i), solves problem (7.2.17) by virtue of the convergence assumptions (7.2.5)–(7.2.7), and theconsequent properties (7.2.12)–(7.2.15).

(iii) Since Eqn. (7.2.17) admits a unique solution w∗, and Eqn. (7.2.17) coincides with Eqn. (7.2.3), weconclude that w∗ = w, as claimed. The proof of Theorem 7.2.2 is complete.

7.3 Theorem 2.2.1: Extension of Estimates (5.2) to Finite Energy SolutionsAs a consequence of Theorem 7.2.2, we obtain

Theorem 7.3.1. Assume hypotheses (1.3.3), (A.3) = (2.1.9), (A.4) = (2.2.2) on F . Let f ∈L2(0, T ;H1(Ω)). Then, estimates (5.2) of Theorem 5.1 hold true also for solutions of Eqn. (1.1.1) in theclass (7.2.1).

Proof. The approximating regularizing solution wk of problem (7.2.4) is in H2,2(Q). Hence, such wk

satisfies estimate (5.2) of Theorem 5.1. Then the convergence properties of Theorem 7.2.2 (Eqns. (7.2.12)–(7.2.16)) guarantee that such estimates hold true also for solutions of Eqn. (1.1.1) in the class (7.2.1). 2

8. GLOBAL UNIQUENESS THEOREMS. CONTINUOUS OBSERVABILITY

We begin with the main case of interest in the present paper.

The case with pure homogeneous Neumann B.C. on Σ. We consider the following over-determinedproblem with Γ0 as yet unspecified and Γ1 = Γ \ Γ0:

iwt + ∆w = F (w) in (0, T ]× Ω ≡ Q;

∂w

∂ν

∣

∣

∣

∣

Σ

= 0 in (0, T ]× Γ ≡ Σ;

w|Σ1 = 0 in (0, T ]× Γ1 ≡ Σ1.

(8.1a)

(8.1b)

(8.1c)

Thus, f = 0 here. As a corollary of Theorem 2.1.3, once extended as in Theorem 7.3.1, that is, of estimate(2.1.11), we obtain the following global uniqueness theorem.

Theorem 8.1. Assume hypotheses (1.1.6) on Hd and (A.2) = (1.1.7), as well as assumption (A.3) = (2.1.9)on F (w).

(a) Let, at first, w be a solution of Eqn. (8.1a) in the class H2,2(Q) in (2.1.1) and of the Neumann B.C.(8.1b) on Σ. Then, with reference to the boundary terms BΣ(w) and BΣ(w) in (2.1.12) and (2.1.5), we have:

BΣ(w) = BΣ(w) = 2 τ

∫ T

0

∫

Γ

e2τϕ[2τ2|h|2 + Φ]|w|2h · ν dΣ

− 2τ

∫ T

0

∫

Γ

e2τϕ[ξtη − ξηt]h · ν dΣ − 2τ

∫ T

0

∫

Γ

e2τϕ∣

∣

∣

∣

∂w

∂µ

∣

∣

∣

∣

2

h · ν dΣ.

(8.2)

(8.3a)

Furthermore, assume now hypothesis (1.1.5N): h · ν = 0 on Γ0, h = ∇d. Then, we obtain

C

∫ T

0

∫

Γ1

[

|w|2 + |wt|2 +

∣

∣

∣

∣

∂w

∂µ

∣

∣

∣

∣

2]

dΣ1 ≥ BΣ(w). (8.3b)

(b) Let now w be a solution of Eqn. (8.1a) in the finite energy class (2.2.1) satisfying the Neumann B.C.(8.1b). Assume the additional hypotheses (A.4) = (2.2.2), so that the extension Theorem 7.3.1 holds true.Hypothesis h · ν = 0 on Γ0 is still in force. Then the following inequality holds true:

C

∫ T

0

∫

Γ1

[

|w|2 + |wt|2 +

∣

∣

∣

∣

∂w

∂µ

∣

∣

∣

∣

2]

dΣ1 ≥ BΣ(w) ≥ kϕ,τ [E(T ) + E(0)]. (8.4)


(c) Let now w be a solution in the finite energy class (2.2.1) of the over-determined problem (8.1a-b-c).Furthermore, assume hypothesis (1.1.5N): h · ν = 0 on Γ0, h = ∇d, in addition to (A.4) = (2.2.2). Then,in fact, w ≡ 0 in Q. [This conclusion improves upon Corollary 2.1.2, by enlarging the lass of solutions, from(2.1.1) to (2.2.1).]

Proof. (a) We return to BΣ(w) given by (2.1.12) and see that (8.2) holds true under hypothesis (8.1b).Moreover, passing to BΣ(w) given by (2.1.5) we see that its 2nd, 4th, 5th integral terms vanish while |∇w|2 =∣

∣

∂w∂ν

∣

∣

2+∣

∣

∣

∂w∂µ

∣

∣

∣

2

. This leads to identity (8.3a).

Under the additional hypothesis h · ν = 0 on Γ0, (8.3a) yields (8.3b) at once.(b) Here, the starting point is that, under present assumptions, the extension Theorem 7.3.1 holds true.

Thus, estimate (2.1.11) of Theorem 2.1.3 holds true also for finite energy solutions in the class (2.2.1).

(c) Now all boundary terms in (8.4) vanish, so that BΣ(w) = 0, since w|Γ1 = 0, hence ∂w∂µ

∣

∣

∣

Γ1

= 0. But then

BΣ(w) = 0 implies E(0) = 0 by (8.4). Then, by definition (1.1.11), E(0) = 0 implies w0 ≡ 0, as desired. Sincethe problem is forward well-posed [see Remark C.1, Appendix C; or else (6.20) with s = 0, s = 0, E(0) = 0,G(T ) = 0], it follows that w ≡ 0 in R+

t × Ω. 2

The case of homogeneous Dirichlet B.C. on Σ. We consider the following overdetermined problemwith Γ0 as yet unspecified and Γ1 = Γ \ Γ0:

iwt + ∆w = F (w) in (0, T ]× Ω ≡ Q;

w|Σ = 0 in (0, T ]× Γ ≡ Σ;

∂w

∂ν

∣

∣

∣

∣

Σ1

= 0 in (0, T ]× Γ1 ≡ Σ1.

(8.5a)

(8.5b)

(8.5c)

Again, f ≡ 0 here. As a corollary of Theorem 2.1.3, that is, of estimate (2.1.11), we obtain the following globaluniqueness/continuous observability theorem.

A big difference with respect to the Neumann B.C. case is that, as detailed in Section 7, there is no need ofthe regularizing treatment of Section 7, in particular of the extension Theorem 7.3.1. Thus, we do not need toinherit the additional assumptions (A.4) = (2.2.2) on which Section 7 rests. Thus, the situation now is morefavorable. We obtain:

Theorem 8.2. Assume hypotheses (1.1.6) on Hd and (A.2) = (1.1.7), as well as assumption (A.3) = (2.1.9)on F (w).

(a) Let w be a solution of Eqn. (8.5a) in the class H2,2(Q) in (2.1.1), and of the Dirichlet B.C. (8.5b) onΣ. Then, with reference to the boundary terms BΣ(w) and BΣ(w) in (2.1.12) and (2.1.5), we have:

BΣ(w) = BΣ(w) + C

∫ T

0

∫

Γ

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

|r1 · ν|dΣ (8.6)

= 2τ

∫ T

0

∫

Γ

e2τϕ∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

h · ν dΣ + C

∫ T

0

∫

Γ

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

|r1 · ν|dΣ. (8.7)

(b) Let now w be the solution of Eqn. (8.5a) in the finite energy class (2.2.1) satisfying the Dirichlet B.C.(8.5b) and w0 ∈ H1(Ω). Furthermore, assume h · ν ≤ 0 on Γ0, h = ∇d. Finally, we recall that without lossof generality we can achieve the additional property that r1 · ν = 0 on Γ0, see (2.1.9c). Then, the followingcontinuous observability inequality holds true:

∫ T

0

∫

Γ1

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

dΣ1 ≥ BΣ(w) ≥ kϕ,τ [E(T ) + E(0)]. (8.8)


(c) Let now w be a solution of the over-determined problem (8.5a-b-c) in the finite energy class (2.2.1).Moreover, let h · ν ≤ 0 and w.l.o.g. r1 · ν = 0 on Γ0, see (2.1.9c), h = ∇d. Then, in fact, w ≡ 0 in Q.

Proof. (a) By (8.5b), hence ∂w∂µ

∣

∣

∣

Σ= 0, we see by (2.1.12) that (8.6) holds true. Moreover, all terms in

BΣ(w) given by (2.1.5) vanish by (8.5b), except the last two terms, where we use, as usual, as a consequenceof (8.5b), that

on Σ : h · ∇w =∂w

∂νh · ν; h · ∇w =

∂w

∂νh · ν; |∇w| =

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

. (8.9)

Then, conclusion (8.7) follows at once from (8.6) and (8.9).(b) Theorem 2.1.1 (or Theorem 5.1) and Theorem 2.1.3 continue to hold true for finite energy solutions

in the class (2.2.1), of these satisfy the Dirichlet B.C. (8.5b). This is due to the (‘reverse’) trace regularityinequality of [15], reported also in [18, Chapter 10, Sect 9]. Moreover, since h · ν ≤ 0 on Σ0 and w.l.o.g. see(2.1.9c), r1 · ν = 0 on Σ0, then (8.7) combined with the basic estimate (2.1.11) of Theorem 2.1.3, yields (8.8).

(c) The additional assumption (8.5c) used in (8.8) implies E(0) = 0, or w0 ≡ 0. Since problem (8.5) isforward well-posed, then it follows that w ≡ 0 in R+

t × Ω. 2

Remark 8.1. Theorem 8.2, parts (b) and (c) hold true with Γ1 ≡ Γ, in which case the assumptions h ·ν ≤ 0on Γ0, and w.l.o.g. see (2.1.9c), r1 · ν = 0 on Γ0 are empty. 2

While inequality (8.8) is the Continuous Observability Inequality (COI) for the Dirichlet problem (8.5), bycontrast inequality (8.4) is not yet the COI of the Neumann problem (8.1) because of the presence of the term∣

∣

∣

∂w∂µ

∣

∣

∣

2

on Γ1.

To control the tangential gradient∣

∣

∣

∂w∂µ

∣

∣

∣

2

= |∇tanw|2, we shall use the following result [44], [17].

Theorem 8.3. Let f ∈ L2(Q) and let w be a solution of Eqn. (1.1.1) in the class (2.1.1).(a) Given a non-empty portion Γ1 of Γ of positive measure, given T > 0 and given T > ε > 0 and ε0 > 0

arbitrarily small, there exists a constant Cε,ε0,T > 0, such that

∫ T−ε

ε

∫

Γ1

∣

∣

∣

∣

∂w

∂µ

∣

∣

∣

∣

2

dΓ1 dt ≤ Cε,ε0,T

∫ T

0

∫

Γ1

|wt|2dΣ1 dt+

∫ T

0

∫

Γ

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

dΓ dt

+‖w‖2

L2(0,T ;H12+ε0 (Ω))

+ ‖f‖2

H−12+ε0 (Q)

. (8.10)

Remark 8.2. Inequality (8.10) will suffice for our present purposes, but, in fact, on the RHS of (8.10), wtcould be penalized in L2(0, T ;H−1(Γ1)) rather than in L2(0, T ;L2(Γ1)). See estimate (B.5), which, however,contains the energy E( · ). By contrast, (8.10) is much sharper as it contains w as a lower-order term, belowenergy level (= H1(Ω)).

As a corollary we then obtain the COI in the Neumann case.

Theorem 8.4. Assume the hypotheses of Theorem 8.1(b): that is hypothesis (1.1.6) in Hd and (A.2) =(1.1.7), as well as assumption (A.3) = (2.1.9), (A.4) = (2.2.2) on F (w), and finally, h · ν = 0 on Γ0.

Let w be the finite energy solution in the class (2.2.1) of Eqn. (8.1a) satisfying the Neumann B.C. (8.1b)and w0 ∈ H1(Ω). Then the following COI holds true: there is a constant CT > 0 such that

∫ T

0

∫

Γ1

[|w|2 + |wt|2]dΣ1 ≥ CT [E(T ) + E(0)]. (8.11)

Proof. Step 1. Under present hypotheses, we first establish the weaker conclusion: there exists CT > 0such that

∫ T

0

∫

Γ1

[|w|2 + |wt|2]dΣ1 + ‖w‖2

L2(0,T ;H12+ε0 (Ω))

≥ CT [E(0) + E(T )]. (8.12)


To this end, we substitute (8.10) with f ≡ 0 in estimate (8.3b), this time rewritten over [ε, T − ε] ratherthan over [0, T ], thereby obtaining by virtue of (8.1b),

C

∫ T−ε

ε

∫

Γ1

[|w|2 + |wt|2]dΓ1dt+ ‖w‖2

L2(0,T ;H12+ε0 (Ω))

≥ kϕ,τE(ε). (8.13)

Next, we recall from (6.21), where G(T ) = 0 by (6.13), that

E(ε) ≥ E(0) + E(T )

2e−cTT . (8.14)

Substituting (8.14) into the RHS of (8.13) yields (8.12), as desired.

Step 2. To eliminate the interior l.o.t. in estimate (8.12), we apply the by now standard compact-ness/uniqueness argument [29], [13], [27]. To this end, we need to invoke the global uniqueness Theorem8.1(c). This way we obtain (8.11). 2

9. REPLACEMENT OF ASSUMPTION (A.2) = (1.1.7) BY VIRTUE OF TWO VECTORFIELDS

Orientation. This section is the counterpart, to Schrodinger equations, of the treatment for second-orderhyperbolic equations in [20] (constant coefficient principal part) and in [46] (on a Riemannian manifold). Thegoal is to dispense with the working assumption (A.2) = (1.1.7). This is done, as in the aforementionedreferences, by writing Ω as the overlapping union of two “suitable” subdomains Ω1 and Ω2, in correspondenceof two functions di, i = 1, 2, each strictly convex in Ω and thus satisfying (1.1.6), where now, however, each dihas no critical point on Ωi, i = 1, 2. This way, two radial vector fields are employed.

9.1 Basic setting using two conservative vector fields as in (1.1.6). Statement of main results

Postulated setting. We divide the original open bounded set Ω into two overlapping subdomains Ω1 andΩ2: Ω = Ω1∪Ω2, Ω1∩Ω2 6= ∅, chosen (in infinitely many ways) to fulfill all the conditions, in particular (a)–(c)below (following the setting of Section 1).

We assume that there exist two functions di: Ω ⇒ R of class C3, i = 1, 2, which are strictly convex, suchthat the following preliminary conditions (a) and (b) are satisfied:

(a)

Hdi(x) = Jhi

(x) =

di;x1x1 , . . . , di;x1xn

...

di;xnx1 , . . . , di;xnxn

=

∂hi;1∂x1

, . . . ,∂hi;1∂xn

∂hi;n∂x1

, . . . ,∂hi;n∂xn

≥ ρI ; (9.1.1)

minΩi

di(x) ≥ m > 0, i = 1, 2; (9.1.2)

(b)

infΩi

|hi(x)| = infΩi

|∇di(x)| ≥ 2p > 0. (9.1.3)

Next, following Section 1, we define the functions for i = 1, 2:

ϕi(x, t) = di(x) − c

(

t− T

2

)2

, x ∈ Ω, 0 ≤ t ≤ T, (9.1.4)


wehre, for T > 0 assigned arbitrary, the constant c is selected as follows:

cT 2 > 4 maxΩ

di(x) + 4δ, (9.1.5)

for some δ > 0 suitably small and kept fixed henceforth. Such functions ϕi thus have the following properties:(i)

ϕi(x, 0) = ϕi(x, T ) = di(x) − cT 2

4≤ −δ, uniformly in x ∈ Ω; (9.1.6)

(ii) there are t0, t1, with 0 < t0 <T2 < t1 < T , such that

minx∈Ω,t∈[t0,t1]

ϕi(x, t) ≥ σ > 0, 0 < σ < m, (9.1.7)

since ϕi(x,T2 ) = di(x) ≥ m > 0 for all x ∈ Ω.

Since T > 0 arbitrarily small but fixed, we see from (9.1.5) that w.l.o.g. we may take c ≥ 1, so that c ≥ √c.

Accordingly, we define the functions

ϕ∗i (x, t) ≡ di(x) −

√c

(

t− T

2

)2

, x ∈ Ω, 0 ≤ t ≤ T, (9.1.8)

so that, by (9.1.4), (9.1.8), we have

ϕ∗i (x, t) ≥ ϕi(x, t), x ∈ Ω, 0 ≤ t ≤ T, i = 1, 2. (9.1.9)

Furthermore, we define the sets (subsets of Ω × [0, T ]):

Qi(σ) = (x, t); x ∈ Ω, 0 ≤ t ≤ T, ϕi(x, t) ≥ σ > 0; (9.1.10)

Q∗i (σ

∗) = (x, t); x ∈ Ω, 0 ≤ t ≤ T, ϕ∗i (x, t) ≥ σ∗ > 0, 0 < σ∗ < σ, (9.1.11)

with constant σ∗ selected as to satisfy 0 < σ∗ < σ < m. Recalling (9.1.7), (9.1.10), and (9.1.11), we obtain

Ωi × [t0, t1] ⊂ Qi(σ) ⊂ Q∗i (σ

∗) ⊂ Ω × [0, T ] ≡ Q;

by (9.1.6), at t = 0 and t = T : no point of Ω belongs to Qi(σ).(9.1.12)

The above setting implies the following critical property:(p) By taking σ∗ sufficiently close to m, we may achieve that

for any (x, t) ∈ Q∗i (σ

∗), we then have|∇di(x)| ≥ p > 0 by (9.1.3).

(9.1.13)

Justification of property (p). It will suffice to take the worst case scenario. This occurs when—in thesetting of the original Section 1—the critical point x0 where ∇d(x0) = 0 (thus violating (1.1.7)) for the originalstrictly convex function d(x) falls on the boundary Γ0 (Remark 1.1.4). Thus, let such x0 ∈ Γ0. We take,for concreteness, the canonical strictly convex function d1(x) = ‖x − x0‖2, though any other strictly convexd1(x) with ∇d1(x0) = 0 will do. We consider the boundary surface of the corresponding set Q∗

1(σ∗) defined by

(9.1.11); that is, the level set

ϕ∗1(x, t) = d1(x) −

√c

(

t− T

2

)2

≡ σ∗ > 0. (9.1.14)

For t = T2 , this level set yields a surface S in Ω of all points x ∈ Ω whose distance ‖x − x0‖ from the

aforementioned critical point x0 ∈ Γ0 is√σ∗:

S ≡ x ∈ Ω :√

d1(x) = ‖x− x0‖ =√σ∗ > 0. (9.1.15)


On the other hand, for t 6= T2 , 0 < t ≤ T , the points x on the boundary surface of the set Q∗

1(σ∗) defined by

(9.1.14) are further away from the point x0 ∈ Γ0, at the larger distance

√

d1(x) = ‖x− x0‖ =

σ∗ +√c

(

t− T

2

)2

12

>√σ∗

from it. This means that the orthogonal projection of the time-space set Q∗1(σ

∗) onto Ω is a subset O1(σ∗) of

Ω,O1(σ

∗) = x ∈ Ω : ‖x− x0‖ ≥√σ∗, (9.1.16)

having (i) the surface S as part of its boundary near x0, and having (ii) all of its points lie on the side of Sfurther away from x0. We take, with σ∗ < m:

Ω1 = x ∈ Ω :√

d1(x) = ‖x− x0‖ ≥ √m ⊂ O1(σ

∗). (9.1.17)

On Ω1: we have |∇d1(x)| = 2‖x−x0‖ ≥ 2p by (9.1.3). Thus, by choosing σ∗ sufficiently close to m, we see via(9.1.16), (9.1.17) that we can have: |∇d1(x)| ≥ p on O1(σ

∗), since ∇d1 is continuous. By definition of O1(σ∗),

this conclusion is precisely (9.1.13). 2

6

T2 Q∗

1(σ∗)

Ω1

O1(σ∗)

@@@I

HHHHHY-√

σ∗

-√m

x0 •

S PPPPPPPq

Fig. 9.1: On Ω1 : |∇d1(x)| ≥ 2p > 0. On O1(σ∗) : |∇d1(x)| ≥ p > 0.

The main result of the present paper (at the H1(Ω)-energy level) is the following Theorem 9.1.1,which extends all previous results.

Theorem 9.1.1. All main results in Section 2 continue to hold true with assumption (A.2) = (1.1.7)removed and replaced by the setting of the present Section 9 based on hypotheses (a) = (9.1.1), (b) = (9.1.3)[which then imply property (p) = (9.1.13)]. 2

9.2 Cut-off functions χi and corresponding subproblems for wi = χiw

Cut-off functions χi. Let χi(t, x) be smooth functions, i = 1, 2. At this stage, it is not important tospecify how χi is constructed. Eventually, in the case of purely Neumann B.C. associated with (1.1.1), χi(t, x)will be the complicated function constructed in [20, Section 10.2], which has the important feature, amongothers, of being only time-dependent (but not space-dependent) on a small interior layer of the boundary Γ.This latter goal is dictated by the Neumann B.C. and would not be necessary when dealing with the DirichletB.C. At any rate, we only assume here at present that such cut-off functions fulfill the requirement

|χi| ≤ const and χi(t, x) ≡ 1 on Qi(σ), (9.2.1)


which is one of the properties satisfied by the cut-off functions in [20, Section 10.2].

Dynamical systems for wi = χiw. Let w ∈ C2(Rt × Ω) be a solution of (1.1.1). We introduce newvariables on [0, T ]× Ω:

wi(t, x) ≡ χi(t, x)w(t, x); fi(t, x) ≡ χi(t, x)f(t, x), i = 1, 2. (9.2.2)

We then see that each term wi satisfies the following problem:

√−1wi,t + ∆wi = F (wi) + fi + Kiw;

wi(0, · ) = wi,0 = χi(0, · )w(0, · );

(9.2.3a)

(9.2.3b)

Ki = [√−1Dt + ∆ − F, χi] = commutator active only on (supp χi). (9.2.3c)

|F (wi) + fi + Kiw|2 ≤ cT [|∇wi|2 + |wi|2 + |fi|2 + (|∇w|2 + |w|2)|suppχi], (t, x) ∈ Q. (9.2.4)

Preliminary estimate: Counterpart of Corollary 4.2, Eqn. (4.15). As constructed above, eachproblem wi in (9.2.3), i = 1, 2, satisfies the counterpart of Theorem 4.1, in particular estimate (4.12). Wetake (4.12) as our present starting point. Our next result is the counterpart of [20, Proposition 10.3.1], or [46,Proposition 10.2.1].

Proposition 9.2.1. Let w ∈ C2(Rt × Ω) be a solution of (1.1.1). Let the setting of Section 9.1 based onassumptions (a) = (9.1.1), (9.1.2), (b) = (9.1.3), and (c) = (9.1.13) be in force. Then each problem (9.2.3),i = 1, 2, satisfies the following pointwise inequality for ε > 0 small:

BΣ(wi) +

(

1 +1

ε

)

C

∫

Q

e2τϕi |fi|2dQ+ Ce2τσ∫ T

0

E(t)dt

≥[

δ0

(

2ρτ − ε

2

)

−(

1 +1

ε

)

CT

]∫ T

0

∫

Ω

e2τϕi [|∇wi|2 + |wi|2]dΩ dt

+ [4ρ(1− δ0)p2τ3 + O(τ2)]

∫

Qi(σ)

e2τϕi |wi|2dx dt− τCdi,T e−2τσ[E(T ) + E(0)]. (9.2.5)

Remark 9.2.1. For τ sufficiently large as to make [4ρ(1− δ0)p2τ3 +O(τ2)], we reach one of our goals and

drop the integral term involving |wi|2 in (9.2.5). 2

Proof. Step 1. Here we shall show that wi satisfies the estimate (counterpart of (4.15):

(

1 +1

ε

)

θ2i

∣

∣

∣

∣

√−1

∂

∂twi + ∆wi

∣

∣

∣

∣

2

− ∂Mi

∂t+ div Vi

≥ δ0

(

2ρτ − ε

2

)

θ2i |∇wi|2 +Bi(τ)θ2i |wi|2, ∀ (t, x) ∈ Q, (9.2.6)

where 0 < δ0 < 1 is a suitable constant, and where the coefficient Bi(τ) in front of the lower-order term isdefined by

Bi(τ) ≡ [4ρτ3(1 − δ0)|∇di|2 + O(τ2), (9.2.7)

and satisfies the estimatesBi(τ) ≥ 4ρτ3(1 − δ0)p

2 + O(τ2) in Q∗i (σ

∗)

Bi(τ) = O(τ3) in [Q∗i (σ

∗)]c,

(9.2.8a)

(9.2.8b)


recalling the set (9.1.11), where [Q∗i (σ

∗)]c is the complement of Q∗i (σ

∗) in [0, T ]× Ω ≡ Q.Indeed, we first write the present version of estimate (4.12) for wi:

(

1 +1

ε

)

θ2i

∣

∣

∣

∣

√−1

∂

∂twi + ∆wi

∣

∣

∣

∣

2

− ∂Mi

∂t+ div Vi

≥ 2τ [Hdi∇vi · ∇vi + Hdi

∇vi · ∇vi] − ε|∇vi|2 + [4τ3Hdi∇di · ∇di + O(τ2)]θ2i |wi|2, ∀ (t, x) ∈ Q. (9.2.9)

Invoking (9.1.1) on Hdi, we obtain for the RHS of (9.2.9)

RHS of (9.2.9) ≥ (4ρτ − ε)|∇vi|2 + [4τ3ρ|∇di|2 + O(τ2)]θ2i |wi|2, (9.2.10)

the counterpart of (4.14). Using the estimate

(4ρτ − ε)|∇vi|2 ≥ δ0

(

2ρτ − ε

2

)

2|∇vi|2

≥ δ0

(

2ρτ − ε

2

)

|[θ2i |∇wi|2 − 2τ2|∇di|2θ2i |wi|2] (9.2.11)

(recall the Eqn. below (4.18)) on the RHS of (9.2.10), where 0 < δ0 < 1, we arrive at the estimate

RHS of (9.2.9) ≥ δ0

(

2ρτ − ε

2

)

θ2i |∇wi|2 +Bi(τ)θ2i |wi|2, (9.2.12)

recalling the definition (9.2.7). Using (9.2.12) on (9.2.9), we obtain (9.2.6), as desired. Then definition (9.2.7)yields (9.2.8a) on Q∗

i (σ∗) by virtue of property (9.1.13), and (9.2.8b) on its complement.

Step 2. Integrating the pointwise estimate (9.2.6) on all of (0, T ] × Ω ≡ Q, we obtain via (9.2.3a):

(

1 +1

ε

)∫ T

0

∫

Ω

e2τϕi |F (wi) + fi + Kiw|2dΩ dt+ τCdiT e−2τδ[Ei(T ) + Ei(0)] +BΣ(wi)

≥ δ0

(

2ρτ − ε

2

)

∫ T

0

∫

Ω

e2τϕi |∇wi|2dQ+

∫

Q

Biθ2i |wi|2dQ (9.2.13)

counterpart of (5.5).

Step 3. Regarding the LHS of (9.2.13), the following estimate holds true:

∫ T

0

∫

Ω

e2τϕi |F (wi) + fi + Kiw|2dQ

≤ C

∫

Q

e2τϕi |fi|2dQ+ cT

∫

Q

e2τϕi [|∇wi|2 + |wi|2]dQ+ ce2τσ∫ T

0

E(t)dt. (9.2.14)

In fact, we recall (9.2.4) and obtain

LHS of (9.2.14) ≤ cT

∫

Q

e2τϕi |fi|2dQ+

∫

Qi(σ)

e2τϕi [|∇wi|2 + |wi|2 + |∇w|2 + |w|2]dx dt

+

∫

[Qi(σ)]ce2τϕi [|∇wi|2 + |wi|2 + |∇w|2 + |w|2]dx dt

, (9.2.15)

since Q = Qi(σ) ∪ [Qi(σ)]c by definition. On Qi(σ): we have χi ≡ 1, by (9.2.1), hence wi ≡ w. On [Qi(σ)]c:we use |∇wi| ≤ c|∇w|, |wi| ≤ c|w| by (9.2.1) and moreover, ϕi ≤ σ by the definition (9.1.10) of Qi(σ), thuse2τϕi ≤ e2τσ. This way (9.2.15) leads to (9.2.14), via (1.1.11).


Step 4. Regarding the last integral term on the RHS of (9.2.13), we have by (9.2.8),∫

Q

Biθ2i |wi|2dQ ≥ [4ρ(1 − δ0)p

2τ3 + O(τ2)]

∫

Q∗

i(σ∗)

e2τϕi |wi|2dx dt− cτ3e2τσ∗

∫

[Q∗

i(σ∗)]c

|wi|2dx dt (9.2.16)

≥ [4ρ(1 − δ0)p2τ3 + O(τ2)]

∫

Qi(σ)

e2τϕi |wi|2dx dt− cτ3e2τσ∗

∫

Q

|w|2dQ, (9.2.17)

recalling (9.1.11). Since Qi(σ) ⊂ Q∗i (σ

∗), [Q∗i (σ

∗)]c ⊂ Q, by (9.1.12) and |wi| ≤ c|w|, then (9.2.16) yields(9.2.17) at once. It therefore remains to show (9.2.16). To this end, we return to (9.2.8a–b) for Bi(τ) andwrite, recalling (9.1.11):

∫

Q

Biθ2i |wi|2dQ =

∫

Q∗

i(σ∗)

Biθ2i |wi|2dx dt+

∫

[Q∗

i(σ∗)]c

Biθ2i |wi|2dx dt

≥ [4ρτ3(1 − δ0)p2 + O(τ2)]

∫

Q∗

i(σ∗)

e2τϕi |wi|2dx dt− cτ3

∫

[Q∗

i(σ∗)]c

e2τϕi |wi|2dx dt. (9.2.18)

But on [Q∗i (σ

∗)]c we have that ϕi ≤ σ∗, (by (9.1.11), hence e2τϕi ≤ e2τσ∗

. Using this in the last integral termof (9.2.18) leads to (9.2.16), as desired.

Step 5. We return to inequality (9.2.13). On its LHS we use estimate (9.2.14); on its RHS we invokeestimate (9.2.17). This way we obtain

(

1 +1

ε

)

C

∫

Q

e2τϕi |fi|2dQ+ CT

∫

Q

e2τϕi [|∇wi|2 + |wi|2]dQ

+ Cτ3e2τσ∗

∫

Q

|w|2dQ+ Ce2τσ∫ T

0

E(t)dt

+ τCdi,T e−2τδ[Ei(T ) + Ei(0)] +BΣ(wi)

≥ δ0

(

2ρτ − ε

2

)

∫ T

0

∫

Ω

e2τϕi |∇wi|2dQ+ [4ρ(1 − δ0)p2τ3 + O(τ2)]

∫

Qi(σ)

e2τϕi |wi|2dx dt. (9.2.19)

But τ3e2τσ∗ ≤ ce2τσ since σ∗ < σ by (9.1.11): hence, on the LHS of (9.2.19) the third integral term containing

∫

Q |w|2dQ is absorbed by the fourth integral term containing∫ T

0 E(t)dt, E(t) in (1.1.11). Moreover, [Ei(T ) +

Ei(0)] ≤ c[E(T ) + E(0)], since χi ≤ c. This way, (9.2.19) readily yields (9.2.5) and Proposition 9.2.1 isestablished.

Remark 9.2.2. Alternatively (over the use of property (p) = (9.1.13)), one may introduce an additionalcut-off function ψi(x), such that

ψi(x) ≡

1 in Ωi,

0 outside Ωi,

where Ωi is a slightly enlarged set Ωi of Ωi, for which |∇di(x)| ≥ p > 0 on Ωi. This can alwaysbe achieved by (9.1.13), since ∇di is continuous. Then, wi(t, x) and fi(t, x) in (9.2.2) are replaced bywi(t, x) = ψi(x)χi(t, x)w(t, x) = ψi(x)wi(t, x); fi(t, x) = ψi(x)χi(t, x)w(t, x) = ψi(x)fi(t, x). This yields cor-responding wi-problems in place of (9.2.3). Regarding the critical term Bi(τ) in (9.2.7), we distinguish twocases: (i) on Ωi, we proceed as before in (9.2.8a), since |∇di(x)| ≥ p > 0 on Ωi; (ii) in the complementary partΩ/Ωi, we have that wi ≡ 0.

Finally, given x ∈ Ω, then x ∈ Ωi for some i by Ω = Ω1 ∪ Ω2, and so |∇di(x)| ≥ p > 0 for this i. Thus∇d1(x) and ∇d2(x) do not vanish simultaneously, and we can then combine the two cases. 2


9.3 Carleman estimate, first version, for the w-problemBuilding up on Proposition 9.2.1, we obtain the counterpart of Theorem 5.1 (or [20, Theorem 10.4.1]).

Theorem 9.3.1. Let T > 0 and assume the setting of the present Section 9. Let w ∈ C2(Rt ×Rnx ; C) be asolution of Eqn. (1.1.11) [and no B.C.] under the standing assumption (1.1.3) for F (w) and (1.1.4) for f . Then:the following one-parameter family of estimates hold true for all τ > 0 sufficiently large and all ε > 0 small:

BΣ(w) + C∫

Q |f |2dQ ≥[

δ0

(

2ρτ − ε

2

)

−(

1 +1

ε

)

CT

]

e2τσ∫ t1

t0

E(t)dt

− C e2τσ∫ T

0

E(t)dt − τ CT,de−2τδ[E(T ) + E(0)]; (9.3.1a)

BΣ(w) =

2∑

i=1

BΣ(wi), CT,d =

2∑

i=1

CT,di. (9.3.1b)

Proof. Step 1. We recall property (9.1.7): ϕi ≥ σ > 0 in [t0, t1] × Ω; and, moreover, that [t0, t1] × Ωi ⊂Qi(σ) by (9.1.12), where wi ≡ w on Qi(σ), since χi ≡ 1 here by (9.2.1). Thus, we estimate the first integralterms on the RHS of (9.2.5) for i = 1, 2:

2∑

i=1

∫ T

0

∫

Ω

e2τϕi [|∇wi|2 + |wi|2]dΩ dt ≥2∑

i=1

∫ t1

t0

∫

Ωi

e2τσ[|∇wi|2 + |wi|2]dΩidt

= e2τσ2∑

i=1

∫ t1

t0

∫

Ωi

[|∇w|2 + |w|2]dΩidt

≥ e2τσ∫ t1

t0

∫

Ω

[|∇w|2 + |w|2]dΩdt, (9.3.2)

where the last step follows, since the integral terms over Ωi, i = 1, 2, collects also contributions on the non-emptyportion Ω1 ∩ Ω2, as assumed.

Step 2. With τ sufficiently large as to have [4ρ(1− δ0)p2τ3 +O(τ2)] > 0, so that Remark 9.2.1 applies, we

sum up Eqn. (9.2.5) of Proposition 9.2.1 for i = 1, 2, and obtain, by virtue of (9.3.2):

(

1 +1

ε

)

C

2∑

i=1

∫

Q

e2τϕi |fi|2dQ+ Ce2τσ∫ T

0

E(t)dt

+

2∑

i=1

BΣ(wi)

≥[

δ0

(

2ρτ − ε

2

)

−(

1 +1

ε

)

CT

]

e2τσ∫ t1

t0

∫

Ω

|∇w|2 + |w|2dΩ dt− τ

(

2∑

i=1

Cdi,T

)

e−2τδ[E(T ) + E(0)],

(9.3.3)

and (9.3.1) is established, via (1.1.11). 2

Final Remark. The remaining Sections 10.4–10.7 of [20] admit a faithful counterpart mutatis mutandisfrom second-order hyperbolic equations to Schrodinger equations and will not be repeated. This therefore leadsto Theorem 9.1.1. 2

10. LOWER-LEVEL (L2)-CARLEMAN ESTIMATES FOR (1.1.1) WITH l.o.t.: THEOREM2.6.1

Orientation. Generally, lower-level energy estimates are obtained from known higher-level energy estimatesby ‘lifting the topology.’ More precisely, the desired lower-level topology is lifted to a suitable higher-level


topology (“natural” energy level of the equations—which is more appropriate for energy methods), wherea preliminary energy estimate is already available in the first place. Such technique of lifting the topology(via pseudo-differential operators) is standard in the PDE theory of regularity [4], [42], etc. However, in thecontext of observability estimates, the issue is more delicate, due to the very nature of the inverse, rather thanforward, problem. A few variants of this lifting technique were used in the context of observability inequalitieswhich arise from the uniform stabilization problem of originally conservative systems in: [17] (wave equationin L2(Ω) × H−1(Ω), see also prior references therein), [15] (Schrodinger equation in H−1(Ω)), and variousplate-equations by the same authors. The technical approach was successful as the stabilization problem (dueto its dissipative nature) does not involve energy-level terms in the equation. By contrast, in the case of afully general equation with energy level terms such as (1.1.1), “standard” lifting techniques produce intractablecommutators. It was a device constructed by D. Tataru [37] that allows one to overcome this additional seriousdifficulty in the general case. The main idea is the construction of a very special localization of the problem,by virtue of a clever cut-off function z (see Eqn. (10.2.1) below), which controls the sign of the aforementionedcommutator (see (10.2.29a) and (10.2.20) below). In the present section, we follow Tataru’s idea. See alsoRemark 10.2.1 below. 2

10.1 Background from [44] H1(Ω)-level estimates for w with l.o.t.

The present section aims at obtaining Carleman estimates at a lower level of energy: L2(Ω), rather thanH1(Ω) as in previous sections. Thus, the issue now is not to avoid l.o.t. in the estimates. Indeed, the finalestimates will contain l.o.t. Accordingly, we may start as well by recalling from [44] the method that pro-duces H1(Ω)-level estimates containing l.o.t. This method is based on the main multipliers eτϕ∇ϕ · ∇w andw div(e2τϕh) [44], where ϕ is the function defined in (1.1.8a), where for simplicity we take now d(x) = ‖x−x0‖2,so that ∇ϕ = h = 2(x − x0). Our starting point is the following result contained in [44, Theorems 2.2.1 and2.1.1]. We could also use the Riemannian version in [45].

Theorem 10.1.1. Assume (1.1.3) on F and (1.1.4) on f . Let w be a solution of (1.1.1): Pw = F (w) + fin the following class:

w ∈ C([0, T ];H1(Ω));

|∇w|Γ| ∈ L2(0, T ;L2(Γ)), wt|Γ ∈ L2(0, T ;H−1(Γ)).(10.1.1)

Let ϕ be the function defined in (1.1.8a) with d(x) = ‖x − x0‖2, hence ∇ϕ = h = 2(x − x0). Then, thefollowing results hold true:

(i)

Bw(Σ) = 2

∫

Q

e2τϕ|∇w|2dQ+ τ

∫

Q

e2τϕ|h · ∇w|2dQ

+ Re

∫

Q

[F (w) + f ]e2τϕh · ∇w dQ

− i

2

∫

Q

wd(e2τϕ)

dth · ∇w dQ+

1

2

∫

Q

w∇w · ∇(div(e2τϕh))dQ

+1

2

∫

Q

[F (w) + f ]w div(e2τϕh)dQ+i

2

[∫

Ω

we2τϕh · ∇w dΩ]T

0

, (10.1.2)

where the boundary terms BΣ(w) are explicitly given by

BΣ(w) ≡ 1

2

∫

Σ

∂w

∂νw div(e2τϕh)dΣ +

i

2

∫

Σ

wwte2τϕh · ν dΣ

+ Re

∫

Σ

e2τϕ∂w

∂νh · ∇w dΣ

− 1

2

∫

Σ

e2τϕ|∇w|2h · ν dΣ. (10.1.3)


If, moreover, we assume that

w|Σ0 ≡ 0, h · ν ≤ 0 on Γ0, hence

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

= |∇w|2 on Γ0, (10.1.4)

then (10.1.3) specializes to

BΣ(w) =1

2

∫

Σ1

∂w

∂νw div(e2τϕh)dΣ1 +

i

2

∫

Σ1

wwte2τϕh · ν dΣ1 + Re

∫

Σ1

e2τϕ∂w

∂νh · ∇w dΣ1

− 1

2

∫

Σ1

e2τϕ|∇w|2h · ν dΣ1 −1

2

∫

Σ0

e2τϕ∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

h · ν dΣ0, (10.1.5)

where div(e2τϕh) = 2τe2τϕ|h|2 + e2τϕdiv h, ∇ϕ = h. Using h · ν ≤ 0 on Γ0:

BΣ(w) ≤ Cϕ

∫

Σ1

e2τϕ[|∇w|2 + |w|2]dΣ1 +1

2

∣

∣

∣

∣

∫

Σ1

wwte2τϕh · ν dΣ1

∣

∣

∣

∣

. (10.1.6)

(ii) For all τ sufficiently large, the following estimate holds true:

BΣ(w) +2

τ

∫

Q

|f |2e2τϕdQ+ Cϕ‖eτϕw‖2L2(Q)

≥(

2 − CTτ

− 1

τ

)∫

Q

e2τϕ|∇w|2dQ− e−2δτ

τ[E(T ) +E(0)] (10.1.7)

where E(t) is defined in (1.1.11).

Proof. (i) This is identity (2.2.2) of Theorem 2.2.1 in [44] with some changes of sign since in [44] dealswith iwt = ∆w + F (w) + f rather than (1.1.1): iwt + ∆w = F (w) + f .

(ii) See [44, Theorem 2.2.4, inequality (2.2.25)]. However, for purposes of Section 10.2 when dealing withthe localized problem, we need to explicitly reproduce here a key portion of the argument given in [44] whichleads to inequality (10.1.7) starting with identity (10.1.2). Our focus will be on the first three terms on theRHS of identity (10.1.2) (see [44, p. 477]). We begin with

Re

∫

Q


≥ − ε

2

∫

Q

|F (w) + f |2e2τϕdQ− 1

2ε

∫

Q


≥ −εCT∫

Q

[|∇w|2]e2τϕdQ− ε

∫

Q

|f |2e2τϕdQ− 1

2ε

∫

Q

e2τϕ|h · ∇w|2dQ, (10.1.8)

where CT is the constant in (1.1.3) and ε > 0 is arbitrarily preassigned. Thus, regarding the first three termson the RHS of identity (10.1.2), we estimate by means of (10.1.8), with τ sufficiently large:

2

∫

Q

e2τϕ|∇w|2dQ+ τ

∫

Q

e2τϕ|h · ∇w|2dQ+ Re

∫

Q


≥ (2 − εCT )

∫

Q

e2τϕ|∇w|2dQ+

(

τ − 1

2ε

)∫

Q


− ε

∫

Q

|f |2e2τϕdQ− εCT

∫

Q

e2τϕ|w|2dQ (10.1.9)

≥ (2 − εCT )

∫

Q

e2τϕ|∇w|2dQ− ε

∫

Q

|f |2e2τϕdQ− εCT

∫

Q

e2τϕ|w|2dQ, (10.1.10)


where, in the last step, with ε > 0 preassigned, we have selected τ sufficiently large as to have τ − 12ε > 0, so

that the corresponding integral term∫

Q is dropped. The passage from (10.1.9) to (10.1.10) shows the virtue

of the factor e2τϕ with free parameter τ , over the ‘classical’ multiplier h · ∇w used in [15]. 2

10.2 H1(Ω)-estimates for the time localized problem (zw)

Definition of a special cut-off function z. To begin with, following [37], let ψ ∈ C1(R) be a functionsatisfying the following properties:

(i)ψ(s) ≡ 0, for −∞ < s ≤ 0; ψ(s) > 0 for s > 0; (10.2.1)

(ii)

ψ′(s) ≥ 0, for s ∈ R;

[

[ψ′(s)]2

ψ(s)

]

≤ C, for 0 < s ≤ 1. (10.2.2)

A typical example for such function is given by

ψ(s) = (s+)4 =

s4 for s ≥ 0;

0 for s < 0.(10.2.3)

With ψ as in (10.2.1), (10.2.2), we next introduce the new variable z(x, t) by setting

z(x, t) = ψ(ϕ(x, t)), (10.2.4)

where ϕ(x, t) is the function defined in (1.1.8a). We justify below that:

ψ(ϕ(x, t)) ≡ 0 for all t ≤ 0 and all t ≥ T, ∀ x ∈ Ω; that is,

supp z(x, t) = supp ψ(ϕ(x, t)) ⊂ (0, T ), uniformly in Ω.(10.2.5)

Thus, z is a cut-off function that cuts off in the t-direction, uniformly in x ∈ Ω. Indeed, by the definition(1.1.8a) and subsequent property (1.1.9) for ϕ, we obtain:

for t ≤ 0 and for t ≥ T : ϕ(x, t) ≤ ϕ(x, 0) = ϕ(x, T ) ≤ −δ, uniformly in x ∈ Ω, (10.2.6)

and property (10.2.1)(i) above for ψ yields (10.2.5).

Remark 10.2.1. Although there are many ways of constructing cut-off functions, some of the most naturalchoices would not work for the present purposes. This justifies the introduction of the above special cut-offfunction z [37]. Indeed, subsequent estimates are very sensitive to perturbations at the H1-energy level. Onthe hand, a commutator of such energy level H1 is produced every time one considers a commutator of the type[P , mult. op.], where P = i∂t + ∆ is the principal part of the differential operator (of anisotropic order 1) andmult. op. is a multiplication operator (pseudo-differential operator of order 0). Unless such commutator has avery special structure, it cannot be absorbed in the calculations. More specifically, the special cut-off function zin (10.2.4) will have the advantage of controlling the sign of the relevant commutator, as quantitatively assertedby Eqn. (10.2.19), (10.2.20) below. 2

A new variable w. Finally, using (10.2.4), we introduce a new variable

w = zw, (10.2.7)

where w is a regular solution of Eqn. (1.1.1), rewritten now as

Pw = f on Q = (0, T ] × Ω; P = i∂t + ∆ − F = P − F. (10.2.8)

Applying z to such equation, we obtain

Pw = zf + [P, z]w ≡ fnew, (10.2.9)


where [P, z] = Pz−zP denotes the commutator between P and z. The main result of this section is the followingcounterpart of Theorem 2.1.1, Eqn. (2.1.2), as applied, however, to the new Eqn. (10.2.9) in the new variablew.

Theorem 10.2.1. Assume (1.1.3) on F , (1.1.4) on f , T > 0 preassigned. Let w be a solution of Eqn. (1.1.1)on (0, T ] × Ω [with no boundary conditions imposed] within the class (10.1.1), so that w, defined by (10.2.7),is a corresponding solution of Eqn. (10.2.9). Assume, moreover, hypothesis (10.1.4) on Γ0. Then, for allτ > 0 sufficiently large, there exist constants C > 0, CT > 0, such that the following one-parameter family ofestimates in w = zw holds true:

∫ T

0

∫

Ω

|∇w|2e2τϕdΩ dt+ ‖weτϕ‖2

H12 (0,T ;L2(Ω))

≤ CT

∫ T

0

∫

Γ1

|∇w|2e2τϕdΓ1dt+ ‖weτϕ‖2

H12 (0,T ;L2(Γ1))

+C

τ

∫ T

0

∫

Ω

e2τϕ|(zf)|2dQ+ CT,τ l.o.t.(z1w), (10.2.10)

where z1 is a smooth function such that

z1 ≡ 1 on supp z ⊂ supp z1, (10.2.11)

while

l.o.t.(w) = O(

‖w‖2L2(Q) + ‖w‖2

L2(Σ)

)

. (10.2.12)

In light of the notation to be introduced in Section 10.3 below, we can rewrite estimate (10.2.10) for w = zwas

‖weτϕ‖2H1

a(Q) ≤ CT ‖weτϕ‖2H1

a(Σ1)) +C

τ‖(zf)eτϕ‖2

L2(Q) + CT,τ l.o.t.(z1w), (10.2.13)

where H1a(Q) = L2(0, T ;H1(Ω)) ∩H 1

2 (0, T ;L2(Ω)); similarly for H1a(Σ).

Proof. Step 1. First, we show that the commutator term in (10.2.9) satisfies

[P, z]w = 2ψ′(ϕ)∇ϕ · ∇w + l.o.t.(z1w). (10.2.14)

Indeed, with P = i∂t + ∆ − F as in (10.2.8), we have

[P, z]w = Pzw − zPw = [(i∂t(zw) − iz∂tw]

+[∆(zw) − z∆w] + [zFw − F (zw)], (10.2.15)

where direct computations using (10.2.4) yield after a cancellation

[i∂t(zw) − iz∂tw] = iψ′(ϕ)ϕtw; (10.2.16)

[∆(zw) − z∆w] = 2ψ′(ϕ)∇ϕ · ∇w + ψ′′(ϕ)|∇ϕ|2w + ψ′(ϕ)∆ϕw, (10.2.17)

where all terms vanish outside supp z. Substituting (10.2.16), (10.2.17) into (10.2.15) and using that [z, F ]is a 0 + 1 − 1 = 0-order commutator on supp z, yields (10.2.14), as desired. In fact, all lower-order termsin the RHS of (10.2.15) containing w vanish outside supp z; they are of the form O(w|supp z). But, in turn,w|supp z = O(z1w), where z1 is defined in (10.2.11).

Then, by substituting (10.2.14) in the RHS of (10.2.9), we arrive at

Pw = fnew ≡ zf + 2ψ′(ϕ)∇ϕ · ∇w + l.o.t.(z1w). (10.2.18)


Step 2. Following [37], [44], we next apply the multiplier e2τϕ∇ϕ ·∇ ¯w on both sides of Eqn. (10.2.18), andshow that we obtain the following key decomposition on supp z ⊂ Q ≡ (0, T ]× Ω, given by (10.2.5):

[P, z]w e2τϕ∇ϕ · ∇ ¯w = Π(w) +B(w, w) on supp z

≡ 0 outside supp z

Π(w) = 2ψ′(ϕ)

ψ(ϕ)|∇ϕ · ∇w|2e2τϕ ≥ 0

|B(w, w)| = 2|∇ϕ|2∣

∣

∣

∣

ψ′(ϕ)2

ψ(ϕ)

∣

∣

∣

∣

|l.o.t.(w)∇ϕ · ∇ ¯w|e2τϕ

≤ e2τϕ

εC|w|2 + ε|∇ϕ · ∇ ¯w|2e2τϕ,

(10.2.19a)

(10.2.19b)

(10.2.20)

(10.2.21)

(10.2.22)

for any positive ε > 0. The term Π is ‘positive’ (non-negative) while the term B is ‘benign.’

Proof of (10.2.19a). Indeed, to prove (10.2.19a), we apply the multiplier e2τϕ∇ϕ · ∇ ¯w to both sides of(10.2.14), use

z∇ϕ · ∇w = ∇ϕ · ∇(zw) − w∇ϕ · ∇z; ∇z = ∇(ψ(ϕ)) = ψ′(ϕ)∇ϕ, (10.2.23)

and obtain the following identities, all on supp z, see (10.2.5):

([P, z]w)(e2τϕ∇ϕ · ∇ ¯w) = [2ψ′(ϕ)∇ϕ · ∇w + l.o.t.(w)]e2τϕ∇ϕ · ∇ ¯w (10.2.24)

=

2ψ′(ϕ)

z(z∇ϕ · ∇w)∇ϕ · ∇ ¯w + l.o.t.(w)∇ϕ · ∇ ¯w

e2τϕ (10.2.25)

(by (10.2.23)-(10.2.24)

=

2ψ′(ϕ)

z[∇ϕ · ∇(zw) − wψ′(ϕ)∇ϕ · ∇ϕ]∇ϕ · ∇ ¯w + l.o.t.(w)∇ϕ · ∇ ¯w

e2τϕ (10.2.26)

=

2ψ′(ϕ)

ψ(ϕ)|∇ϕ · ∇w|2 − 2w

[ψ′(ϕ)]2

ψ(ϕ)|∇ϕ|2∇ϕ · ∇ ¯w + l.o.t.(w)∇ϕ · ∇ ¯w

e2τϕ. (10.2.27)

Thus, (10.2.27) establishes the decomposition (10.2.19a) on supp z, where the term Π(w) is (critically) non-

negative by recalling that ψ′

ψ is non-negative by (10.2.1), (10.2.2) on supp z ⊂ (0, T ), as noted in (10.2.20);

while estimate (10.2.22) follows from (10.2.21) via (10.2.2).

Step 3. Integrating (10.2.19a) over Q, noticing that such term vanishes outside supp z by (10.2.19b), weshall obtain that

Re

∫

Q

([P, z]w) e2τϕ∇ϕ · ∇ ¯w dQ

≥ −εC∫

Q∩ supp z

|∇w|2e2τϕdQ− C

ε

∫

Q∩ supp z

|w|2e2τϕdQ.

≥ −εC∫

Q


ε

∫

Q

|z1w|2e2τϕdQ,

(10.2.28a)

(10.2.28b)

where the function z1 is defined in (10.2.11).


Proof of (10.2.28). By (10.2.19) we compute

Re

∫

Q

([P, z]w)(e2τϕ∇ϕ · ∇ ¯w)dQ

= Re

∫

Q∩ supp z

([P, z]w)(e2τϕ∇ϕ · ∇ ¯w)dQ

(10.2.29)

= Re

∫

Q∩ supp z

[Π(w) +B(w, w)]dQ

(10.2.30)

(by (10.2.20)) ≥ Re

∫

Q∩ supp z

B(w, w)dQ

(10.2.31)

(by (10.2.22)) ≥ −ε∫

Q∩ supp z

|∇ϕ · ∇ ¯w|2e2τϕdx dt

− C

ε

∫

Q∩ supp z

|w|2 e2τϕdx dt, (10.2.32)

where (10.2.32) follows from (10.2.22). Thus, (10.2.32) establishes (10.2.28a). Then, (10.2.28b) follows at oncefrom (10.2.28a) since w|supp z = O(z1w) by the definition of z1 in (10.2.11). 2

Step 4. By property (10.2.5) and definition (10.2.7) of w = zw, we have that

Ew(0) = Ew(T ) = 0, Ew(t) =

∫

Ω

[|∇w(t)|2 + |w(t)|2]dΩ, (10.2.33)

in agreement with the notation in (1.1.11).

Step 5. Proposition 10.2.2. Under the setting of Theorem 10.2.1, in particular with Γ0 defined in(10.1.4), the following estimate holds true, for all τ sufficiently large:

∫ T

0

∫

Ω

|∇(zw)|2e2τϕdQ ≤ Cϕ

∫ T

0

∫

Γ1

e2τϕ[|∇(zw)|2 + |(zw)|2]dΣ1

+ C‖(zw)eτϕ‖2

H12 (0,T ;L2(Γ1))

+1

τ

∫ T

0

∫

Ω

|(zf)|2e2τϕdQ+ Cϕ,τ l.o.t.(z1w). (10.2.34)

Proof of Proposition 10.2.2. We now invoke the argument in (10.1.8)–(10.1.10) leading to inequality(10.1.7) [with BΣ subject to estimate (10.1.6)], which was obtained for w solution of Pw = f , P = i∂t+ ∆−F ,and apply it now to w solution of Eqn. (10.2.9) (or (10.2.18)): Pw = fnew, where fnew = (zf) + [P, z]w.Accordingly, recalling (10.1.8) mutatis mutandis as well as (10.2.28b), we estimate

Re

∫

Q

[F (w) + fnew]e2τϕh · ∇wdQ

= Re

∫

Q

[F (w) + (zf)]e2τϕh · ∇ ¯wdQ

+ Re

∫

Q

[P, z]w e2τϕh · ∇ ¯w)dQ

≥ − 1

2ε

∫

Q

e2τϕ|h · ∇ ¯w|2dQ− ε C

∫

Q


ε

∫

Q

|z1w|2e2τϕdQ. (10.2.35)


Hence, the counterpart of (10.1.9)–(10.1.10)—this time for w—is, by use of (10.2.35):

2

∫

Q

e2τϕ|∇w|2dQ+ τ

∫

Q

e2τϕ|h · ∇w|2dQ+ Re

∫

Q

[F (w) + (fnew]e2τϕh · ∇ ¯wdQ

≥ (2 − εCT − εC)

∫

Q

e2τϕ|∇w|2dQ+

(

τ − 1

2ε

)

1∫

Q


− ε

∫

Q

|zf |2e2τϕdQ− εCT

∫

Q

e2τϕ|w|2dQ− C

ε

∫

Q

|z1w|2E2τϕdQ, (10.2.36)

where again we take τ large enough as to have(

τ − 12ε

)

> 0 and, accordingly, we drop the correspondingintegral term. This way we obtain the counterpart of inequality (10.1.7) this time for w, i.e., with energyvanishing at t = 0 and t = T by (10.2.33). We obtain setting ε = 1

2 :

[

2 − CTτ

− C1

τ

]∫

Q

e2τϕ|∇w|2dQ

≤ CTτ

‖eτϕw‖2L2(Q) + τC‖(z1w)eτϕ‖2

L2(Q) +1

τ

∫

Q

|zf |2e2τϕdQ+BΣ(w) (10.2.37)

(by (10.1.6)) ≤ CTτ

‖eτϕw‖2L2(Q) + τC‖(z1w)eτϕ‖2

L2(Q)

+1

τ

∫

Q

|zf |2e2τϕdQ+ Cϕ

∫

Σ1

e2τϕ[|∇w|2 + |w|2]dΣ1 +1

2

∣

∣

∣

∣

∫

Σ1

¯wwte2τϕh · ν dΣ1

∣

∣

∣

∣

, (10.2.38)

where in the last step we have invoked, under assumption (10.1.4), estimate (10.1.6) with w there replaced byw now; moreover, h = ∇ϕ.

Finally, we let the superscript ∼ denote Fourier transform in time t → iσ. Recall that w = zw, and suppz ⊂ (0, T ) from (10.2.4), (10.2.5), by use of

(zw)teτϕ = ((zw)eτϕ)t − (zw)eτϕτϕt,

we then estimate∣

∣

∣

∣

∣

∫ T

0

∫

Γ1

(zw)(zw)t e2τϕh · ν dΓ1dt

∣

∣

∣

∣

∣

≤ C

∫

R1t

∫

Γ1

|((zw)eτϕ)t| |(zw)eτϕ| dΓ1dt+ τ

∫

R1t

∫

Γ1

|(zw)|2e2τϕ|ϕt|dΓ1dσ

(10.2.39)

= C

∫

R1σ

∫

Γ1

∣

∣

∣|σ| 12 ˜((zw)eτϕ)

∣

∣

∣

∣

∣

∣|σ| 12 ˜((zw)eτϕ)

∣

∣

∣dΓ1dσ + τ

∫ T

0

∫

Γ1

|(zw)|2e2τϕ|ϕt|dΓ1dt

(10.2.40)

= C ‖((zw)eτϕ)‖2

H12 (R1

t ;L2(Γ1))+ Cτ l.o.t.(z1w) (10.2.41)

= C ‖(zw)eτϕ‖2

H12 (0,T ;L2(Γ1))

+ Cτ l.o.t.(z1w), (10.2.42)

recalling (10.2.11) and (10.2.12). Substituting estimate (10.2.42) for the last integral term in (10.2.38) yieldsestimate (10.2.34), and Proposition 10.2.2 is proved, with w = (zw) = O(z1w). 2


Step 6. Proposition 10.2.3. Under the setting of Theorem 10.2.1, in particular with Γ0 as in (10.1.4),the following estimate holds true, for all τ

‖(zw)eτϕ‖2

H12 (0,T ;L2(Ω))

≤ C

τ

∫ T

0

∫

Ω

|∇((zw)eτϕ)|2dQ+C

τ

∫

Σ1

∣

∣

∣

∣

∂

∂ν((zw)eτϕ)

∣

∣

∣

∣

2

dΣ1

+1

τ

∫

Q

|zf |2e2τϕdQ+ Cτ l.o.t.(z1w). (10.2.43)

Proof. Define, for simplicity of notation,

W ≡ (zw)eτϕ = weτϕ; W = Fourier transform of W in t : t→ iσ. (10.2.44)

We multiply both sides of Eqn. (1.1.1) by zeτϕ and obtain after standard manipulations

i∂t((zw)eτϕ) + ∆((zw)eτϕ) = F ((zw)eτϕ) + ((zf)eτϕ) + Kw; (10.2.45)

Kw ≡ i[∂t, zeτϕ]w + [∆, zeτϕ]w − [F, zeτϕ]w, (10.2.46)

where [ , ] denotes commutator: the first commutator is of order zero in time/space variables; the secondof order one in the space variable; the third of order zero in time/space variables. In the notation of (10.2.44),we rewrite Eqn. (10.2.45) as

i∂tW + ∆W = F (W ) + (zf)eτϕ + Kw;

−σW + ∆(W ) = F (W ) + ˜((zf)eτϕ) + (Kw),

(10.2.47)

(10.2.48)

where the superscript ∼ again denotes the Fourier transform in t→ iσ. Next, we multiply (10.2.48) by ¯W (thecomplex conjugate of W ) and integrate in Ω: by use of Green’s first theorem, we then obtain via (10.1.4):

σ

∫

Ω

|W |2dΩ = −∫

Ω

|∇W |2dΩ +

∫

Γ1

∂W

∂ν¯WdΓ1 −

∫

Ω

F (W ) ¯WdΩ

−∫

Ω

˜((zf)eτϕ) ¯WdΩ −∫

Ω

(Kw) ¯WdΩ. (10.2.49)

Further integration over R1σ and majorization with ε = 1

τ yields, since both F and K are first-order operatorsin space:

∫

R1σ

∫

Ω

∣

∣

∣|σ| 12 W

∣

∣

∣

2

dΩ dσ ≤ 1

2τ

∫

R1σ

∫

Γ1

∣

∣

∣

∣

∣

∂W

∂ν

∣

∣

∣

∣

∣

2

dΓ1 dσ +τ

2

∫

R1σ

∫

Γ1

|W |2dΓ1 dσ

+C

τ

∫

R1σ

∫

Ω

∣

∣

∣∇W

∣

∣

∣

2

dΩ dσ + Cτ

∫

R1σ

∫

Ω

∣

∣

∣W∣

∣

∣

2

dΩ dσ

+1

τ

∫

R1σ

∫

Ω

∣

∣

∣

˜((zf)eτϕ)∣

∣

∣

2

dΩ dσ +

∫

R1σ

∫

Ω

∣

∣

∣(Kw)

∣

∣

∣

2

dΩ dσ

. (10.2.50)

By Plancherel theorem, since z ≡ 0 outside (0, T ), we rewrite estimate (10.2.50) as

‖W‖2

H12 (0,T ;L2(Ω))

≤ 1

2τ

∫ T

0

∫

Γ1

∣

∣

∣

∣

∂W

∂ν

∣

∣

∣

∣

2

dΓ1dt+τ

2

∫ T

0

∫

Γ1

|W |2dΓ1dt

+C

τ

∫ T

0

∫

Ω

|∇W |2dΩ dt+ τ

∫ T

0

∫

Ω

|W |2dΩ dt+ C

τ

∫ T

0

∫

Ω

|zf |2e2τϕdΩ dt. (10.2.51)


Then (10.2.51), where W = zw eτϕ by (10.2.44), is precisely (10.2.43), recalling (10.2.11), (10.2.12). 2

Step 7. By summing up (10.2.34) of Proposition 10.2.2 with (10.2.43) of Proposition 10.2.3, we obtain(10.2.10) of Theorem 10.2.1, after taking τ large enough to absorb the first term on the RHS of (10.2.43) withcoefficient C

τ by the term on the LHS of (10.2.34) with coefficient 1.

10.3 Lower-level L2-energy estimates for the localized problem (10.2.9) on (zw)

We have seen that the new Theorem 2.1.3 and the old Theorem 10.1.1 (after [44]) both give energy estimatesat the H1(Ω)-level: the first without l.o.t. and the second with l.o.t., respectively. We now seek energy estimatesfor problem (1.1.1) in the original variable w at the lower L2(Ω)-level with l.o.t. To this end, the key step isthe following result, which achieves the desired goal, although—as a first step—only for the localized problem(10.2.9) in the localized variable (zw). The space H1

a was defined below (10.2.13). See also (10.3.6) below.

Theorem 10.3.1. Assume (1.1.3) on F and f ∈ H−1a (Q). Let w be a solution of Eqn. (1.1.1) in the class

(11.1.1), satisfying condition (10.1.4) on Γ0. Then, the following estimate holds true for all τ sufficiently large:

∫

Q

|zw|2e2τϕdQ ≤∫ T

0

∫

Γ1

|zw|2e2τϕdΣ1 +

∥

∥

∥

∥

∂

∂ν(eτϕzw)

∥

∥

∥

∥

2

H−1a (Σ1)

+C

τ‖(zeτϕ)f‖2

(0,−1a) + ‖z1w‖2H−1(Q). (10.3.1)

Proof. Step 1 (Partition of unity, space localization, M-S local change of variables to flatten the boundary).We cover the original bounded domain Ω by a finite number of boundary (collar) subdomains Mj , j = 1, . . . , J ,plus an interior subdomain Ωint: Ω ⊂ ⋃jMj ∪ Ωint. To achieve our goal of proving estimate (10.3.1), we willshow below that it will suffice, as usual, to reduce our analysis to the following two situations, one involving atypical boundary subdomain Mj and one involving the interior subdomain Ωint. Thus, we seek to obtain thefollowing two ingredients:

(1) the required boundary estimates on one such typical boundary subdomain Mj around a fixed boundarypoint ζ ∈ Γ = ∂Ω; and

(2) the required interior estimates on the interior subdomain Ωint.

Construction of Ωint and Mj and corresponding space localization. To begin with, we take aboundary collar neighborhood C of ∂Ω, diffeomorphic to, say, [0, 1

2 ] × ∂Ω, and we call (2C) the larger collarneighborhood of ∂Ω, diffeomorphic to [0, 1]× ∂Ω. We then define Ωint = Ω \ (2C).

Ωint:β0≡0

collar (2C) collar C: β0≡19

Fig. 1: Boundary Collars C and (2C);Interior Subdomain Ωint = Ω \ (2C); Function β0.


C@@R

ζ•

Mj : βj≡1

-M-S

coordinates

66y

1

x1-

−1

34

-β≡1

Ωc

β=0

12

Fig. 2

We introduce a smooth function β0, 0 ≤ β0 ≤ 1, such that: β0 ≡ 1 on the boundary collar C, and β0 ≡ 0 onthe interior subdomain Ωint. Next, we cover, in turn, the collar C by a finite number of boundary subdomainsMj , j = 1, . . . , J , and introduce smooth functions βj , 0 ≤ βj ≤ 1, such that: βj ≡ 1 on Mj and βj ≡ 0

on a slightly larger subdomain, so that β0 ≡ ∑Jj=1 βj , and β1, . . . , βJ , (1 − β0) is a partition of unity over

Ω :∑J

j=1 βj + (1 − β0) ≡ 1.

Flattening Mj locally. As the Laplacian ∆ in problem (1.1.1) over the original domain Ω is a second-order differential operator with real principal symbol −|ξ|2 and with noncharacteristic boundary, then nearlyany point ζ ∈ Γ ≡ ∂Ω, we may choose [34, pp. 597–598] local coordinates, called henceforth (x, y), x scalar,y ∈ Rn−1, centered at ζ, such that Ω is locally given by x ≥ 0 and the Laplacian ∆ is replaced by (may betaken to have) the following form

∆ = D2x + a(x, y)D2

y + l.o.t. in Dy; D2x =

∂2

∂x2; D2

y =

n−1∑

j=1

∂2

∂y2j

, (10.3.2)

with a(x, y) real and smooth. Thus, we may consider the original problem (10.2.9): P(zw) = fnew, as definedon the collar domain

Ωc = 0 ≤ x < 1; |y| < 1,where ∆ may be replaced by ∆ of the form (10.3.2). Such new problem over Ωc may be viewed as correspondingto the original problem (10.2.9), defined, however, only over the boundary (collar) subdomain Mj of Ω andacting on the solution (zw) having compact support in space within a subdomain slightly larger than Mj

and in time within (0, T ) [by (10.2.5)], after implementing the M-S change of variable from the original spacecoordinates on Mj to the new coordinates (x, y) on Ωc. Consequently, the new problem over Ωc with ∆ givenin the form (10.3.2) may be considered for a solution (zw) of (10.2.9) vanishing outside (0, T ) × |y| < 1 foreach value of x, 0 ≤ x < 1, as a parameter.

Final strategy. Let w be a solution of (1.1.1), or (10.2.8), so that (zw) is a solution of (10.2.9). By theabove discussion, we may write

zw = β0(zw) + (1 − β0)(zw) =

J∑

j=1

βj(zw) + (1 − β0)(zw). (10.3.3)

In conclusion, calling β a typical βj , we may confine our analysis to consider two terms of the solution in(10.3.3).

(i) βzw, in the collar Ωc, with

β =

1 in the rectangle 0 ≤ x ≤ 12 ; |y| ≤ 3

4;

0 outisde the rectangle 0 ≤ x ≤ 1; |y| ≥ 1,


with compact support in (0, T ) × |y| < 1, for each 0 ≤ x ≤ 1, so that one can legitimately take the Fouriertransform of (βzw) in t and y, holding x, 0 ≤ x ≤ 1, as a parameter. [In particular, application of thepseudo-differential operator Λtan below on βzw is legitimate.]

(ii)

(1 − β0)zw, vanishing in time outside (0, T ) and vanishingin space in the collar C and outside Ω.

so that we can legitimately take the Fourier transform of (1 − β0)(zw) in t, x, y. [In particular, application ofthe pseudo-differential operator Λ below on (1 − β0)(zw) is legitimate.]

Primal and dual variables, anisotropic orders and spaces. To the variables (t, y, x) = (time, tangen-tial space variable, normal space variable), we associate the dual variables (σ, η, ξn); x, ξn ∈ R1

+; y, η ∈ Rn−1.In order to account for the anisotropy of the Schrodinger operator, see below in this section, we need to defineappropriately the operation of multiplication. Set [37],

ξ = (σ, η, ξn) = (ξ′, ξn), ξ′ = (σ, η), (10.3.4)

and define multiplication by positive scalars α > 0 by

αξ = α(σ, η, ξn) = (α2σ, αη, αξn) (10.3.5a)

in agreement with the fact that, at the derivative level, the single time derivative counts as two space derivatives.Accordingly, the corresponding anisotropic metric is

|ξ| = [σ2 + |η|4 + ξ4n]14 . (10.3.5b)

With the above definitions, the following properties hold true:(i) α|ξ| = |αξ|, α ≥ 0;(ii) p(ξ) = iσ − η2 − ξ2n is a homogeneous polynomial of anisotropic order two;(iii) the sets (x, αξ), α ≥ 0 represent rays in the co-tangent space.The anisotropic Sobolev spaces

Hsa(Q) = L2(0, T ;Hs(Ω)) ∩H s

2 (0, T ;L2(Ω)) (10.3.6)

are defined with respect to the new definition of metric in (10.3.5). Let Λ and Λtan be two pseudo-differentialoperators defined in terms of the variables t, x, y, σ, η, ξn and t, y, σ, η with 0 ≤ x ≤ 1 as a parameter,respectively, in the following classes:

Λ ∈ S1a(Q× Rn+1) with symbol λ ≡ |ξ| = [σ2 + |η|4 + ξ4n]

14 ∈ s1a of anisotropic order 1; (10.3.7)

Λtan(x) = Λtan ∈ S0,1a(Σ × [0, 1]× Rn) with symbol λtan

≡ |ξ|′ = [σ2 + |η|4] 14 ∈ s0,1a of anisotropic order 1, (10.3.8)

with symbols defined away from zero, with x as a parameter 0 ≤ x ≤ 1. The first superindex refers to theorder in the normal direction ξn, while the second superindex refers to the anisotropic order in the tangentialdirection [σ, η]. We shall also write S0,1

a instead of S0,1a. Both symbols are elliptic, and hence we have

Λ−1 ∈ S−1a ; Λ−1

tan ∈ S0,−1a ≡ S0,−1a . (10.3.9)

We finally note that

‖u‖H1a(Σ) = ‖Λtanu‖L2(Σ) = λtan‖u‖L2(Σ) = [σ2 + |η|4] 1

4 ‖u‖L2(Σ), (10.3.10a)


for any u compactly supported in the variable t and y, for each value of the parameter x, 0 ≤ x ≤ 1,

‖u‖H−1a (Σ) = ‖Λ−1

tanu‖L2(Σ) =1

λtan‖u‖L2(Σ) (10.3.10b)

|u|H1a(Q) = |Λu|L2(Q) = λ|u|L2(Q) = [σ2 + |η|4 + ξ4n]

14 |u|L2(Q), (10.3.11a)

for any u compactly supported in the variables t, x, y;

‖u‖H−1a (Q) = ‖Λ−1u‖ =

1

λ‖u‖L2(Q). (10.3.11b)

Remark 10.3.1. (i) The usual rules of ‘homogeneous symbols’ apply in obtaining the anisotropic order (inall variables) of a commutator [A,B], involving the operators Λ, Λ−1, etc. That is, if A ∈ Sma , B ∈ Sra, then[A,B] is of anisotropic order = m+ r − 1.

(ii) By contrast, when dealing with the commutator involving the operators Λtan, Λ−1tan, etc., the above rule

applies only to the sum of the two superindices, but not to the normal or tangential indices separately. SeeAppendix E, Lemma E.1.

The spaces H(m,s)(Rn) [4, Vol. III, p. 477]. Let m, s ∈ R. We denote by H(m,s)(R

n) the set of allu ∈ S ′(Rn) with u ∈ L2

loc and

‖u‖H(m,s)(Rn) =1

(2π)n2

∫

|u(ξ)|2(1 + |ξ|2)m(1 + |ξ′|2)sdξ

12

, (10.3.12)

ξ ∈ Rn, ξ′ = ξ1, ξ2, . . . , ξn−1. Thus, m is the order in the normal direction to the plane xn = 0 (which playsa distinguished role), while (m+ s) is the order in the tangential direction.

Case m = 0. We haveH(0,s)(R

n) = L2(Rxn;Hs(Rn−1)), (10.3.13)

or: v ∈ H(0,s)(Rn) means that v can be regarded as an L2 function of xn with values in Hs(Rn−1).

Case m a positive integer. We have

H(m,s)(Rn) ≡

u : Djnu ∈ H(0,s+m−j)(R

n) ≡ L2(Rxn;Hs+m−j(Rn−1)) for 0 ≤ j ≤ m

. (10.3.14)

Various properties of these spaces including trace theorems are given in [4, Vol. III, Appendix B].

The spaces H(m,s)(Rn+) [4, Vol. III, p. 478]. Let Rn+ denote the open half space of Rn defined by xn > 0,

∂Rn+ = x ∈ Rn : xn = 0.Case m a non-negative integer. We have

H(m,s)(Rn+) = u ∈ S ′(Rn+) : Dj

nu is an L2 function of xn ∈R+ with values in Hs+m−j(Rn−1

+ ) when 0 ≤j ≤ m

(10.3.15)

‖u‖H(m,s)(Rn) equivalent to

m∑

j=0

∫ ∞

0

‖Djnu( · , xn)‖2

Hs+m−jdxn

12

. (10.3.16)

The spaces H(m,s)(C) in a collar neighborhood C of the boundary ∂Ω of a bounded domain Ω[42, Vol. 1, p. 393]. Let Ω be a bounded domain in Rn, with sufficiently smooth boundary ∂Ω. Take a collarneighborhood C of ∂Ω, diffeomorphic to [0, 1]×∂Ω, sitting inside a larger collar neighborhood N , diffeomorphicto [0, 2] × ∂Ω. The space H(m,s)(C) is defined by functions u such that the norms below are finite:

‖u‖(0,s) ≡∫ 1

0

‖u( · , xn)‖2Hs(∂Ω)dxn

12

, (10.3.17)


or more generally (Dxn= Dν = derivative in the normal direction),

‖u‖(m,s) ≡

m∑

j=0

∫ 1

0

‖Djxnu( · , xn)‖2

Hm+s−j(∂Ω)dxn

12

. (10.3.18)

One can similarly define norms on H(m,s)(N ). These spaces depend on the choice of collaring. An intrinsictreatment, with intrinsic spaces based on lacunary symbols have been introduced in [33] and further studiedalso in [41].

The pseudo-differential operator Λ. We start with the symbol λ(σ, η, ξn) = |ξ| = [σ2 + |η|4 + ξ4n]14 in

(10.3.7) of anisotropic order 1, and define the corresponding pseudo-differential operator Λ by

(Λu)(t, x, y) =

∫

λ(σ, η, ξn)u(σ, η, ξn)ei[tσ+y·η+x·ξn]dσ dη dξn; (10.3.19)

u(σ, η, ξn) = (2π)−n∫

u(t, x, y)e−i[tσ+xξn+y·η]dt dx dy, (10.3.20)

for u having, say, compact support in R1t × R1

x × Rn−1y . This is the case for the function u = (1 − β0)zw in

Section 10.3, so that Λ(1 − β0)zw is well-defined.

The pseudo-differential operator Λtan. We start with the symbol λtan(σ, η) = |ξ′| = [σ2 + |η|4] 14 in

(10.3.8) of anisotropic order 1, and define the corresponding pseudo-differential operator Λtan by

(Λtanu)(t, x, y) =

∫

λtan(σ, η)u(σ, x, η)ei[tσ+y·η]dσ dη (10.3.21)

u(σ, x, η) = (2π)−n∫

u(t, x, y)e−i[tσ+y·η]dt dy, (10.3.22)

for u having, say, compact support in R1t × Rn−1

y , for each value of the parameter x, 0 ≤ x ≤ 1. This is thecase for the function u = βzw of Section 10.3, so that Λtanβzw is well-defined.

Anisotropic symbols sm,sa and corresponding operators in Sm,sa .

10.4 Continuation of proof of Theorem 10.3.1

Step 2. (New variables and their equations) Let z be the cut-off function introduced in (10.2.4). We startwith w being a solution of Eqn. (10.2.8): Pw = f . We introduce then two new variables, v1 and v2, and theirtime-localizations

v1 ≡ Λ−1tanβzw; and its time localization v1 = zv1, near Σ, (10.4.1)

v2 ≡ Λ−1(1 − β)zw; and its time localization v2 = zv2, in Q away from Σ, (10.4.2)

where we note that v1 and v2 are well defined, since (βzw) is compactly supported in R1t × Rn−1

y , for each0 ≤ x ≤ 1 as a parameter, while (1 − β)zw is compactly supported in R1

t × R1x × Rn−1

y , see Section 10.3.Beginning with this section, β in (10.4.1), (10.4.2) stands for either βj or else β0 in (10.3.3) of Section 10.3.We further note that the pseudo-local property [42, p. 39] implies that

(i) v1 is arbitrarily smooth outside Ωc where β ≡ 0 and outside (0, T ) where z ≡ 0;(ii) v2 is arbitrarily smooth in the neighborhood collar of the boundary 0 ≤ x < 1

2 , |y| < 34, where β ≡ 1

and outside (0, T ) where z ≡ 0.We shall show below that the new variables v1 and v2 satisfy the following equations

Pv1 = [P, z]v1 + F1(w) + zΛ−1tanβzf in a collar of Σ;

F1(w) ≡ −[Λ−1tanβz,P](zw) − [z, [Λ−1

tanβz,P]]w;

(10.4.3a)

(10.4.3b)


Pv2 = [P, z]v2 + F2(w) + zΛ−1(1 − β)zf in Q away from Σ;

F2(w) ≡ −[Λ−1(1 − β)z,P](zw) − [z, [Λ−1(1 − β)z,P]]w.

(10.4.4a)

(10.4.4b)

We notice that: Pvi contains the sum of [P, z]vi, where the commutator [P, z] has the “right sign,” see (10.2.20),(10.2.21), and a term Fi(w). For i = 2, the term F2(w) consists of a commutator of zero order in all variablesapplied to (zw) and a commutator of anisotropic order −1 in all variables applied to w, see Remark 10.3.1.For i = 1, the same applies to F1(w), but only in the sense of Remark 10.3.1 and above all of Lemma E.1 inAppendix E.

Proof of (10.4.4.) Applying zΛ−1(1− β)z to both sides of Pw = f , and using the corresponding commu-tator yields

P(zΛ−1(1 − β)zw) = zΛ−1(1 − β)zf + [P, zΛ−1(1 − β)z]w, (10.4.5)

where by definition

[P, zΛ−1(1 − β)z]w = PzΛ−1(1 − β)zw − zΛ−1(1 − β)zPw (10.4.6)

= Pz(Λ−1(1 − β)zw) − zPΛ−1(1 − β)zw + [Λ−1(1 − β)z,P]w (10.4.7)

= [P, z]Λ−1(1 − β)zw − z[Λ−1(1 − β)z,P]w. (10.4.8)

Inserting (10.4.8) into (10.4.5) and recalling the definition of v2 and v2 in (10.4.2), we obtain

Pv2 = zΛ−1(1 − β)zf + [P, z]v2 − z[Λ−1(1 − β)z,P]w. (10.4.9)

As to the last term in (10.4.9), we rewrite it as

z[Λ−1(1 − β)z,P]w = [Λ−1(1 − β)z,P](zw) + [z, [Λ−1(1 − βz,P]]w. (10.4.10)

Substituting (10.4.10) for the last term in (10.4.8) yields (10.4.4), as desired.The proof of (10.4.3) is identical, mutatis mutandis.

Step 3. (Localized estimates for new variables) Our next step consists in applying the localized estimates(10.2.13) of Theorem 10.2.1 derived for the w solution of Eqn. (10.2.9) to the new variables v1 and v2, solutionsof (10.4.3a) and (10.4.4a), respectively. We need, of course, to account now for the “right-hand sides” inEqns.(10.4.3a) and (10.4.4a), with respect to Eqn. (10.2.9). We obtain via (10.2.13) the following estimates.

Estimates (in a collar of the boundary Σ) for the variable v1 = (zv1) (arbitrarily smooth outsideΩc):

‖v1eτϕ‖H1a(Q) ≤ CT ‖v1eτϕ‖2

H1a(Σ1) +

C

τ

∫ T

0

∫

Ω

[|zΛ−1tanβzf |2 + |F1(w)|2]e2τϕ dQ+ CT,τ‖z1v1‖2

L2(Q). (10.4.11)

Estimates in the interior Q (away from Σ) for the variable v2 = (zv2) (arbitrarily smooth in acollar of Σ):

‖v2eτϕ‖2H1

a(Q) ≤CTτ

∫ T

0

∫

Ω

[|zΛ−1(1 − β)zf |2 + |F2(w)|2]e2τϕ dQ+ CT,τ‖z1v2‖2L2(Q). (10.4.12)

Since v2 is arbitrarily smooth within the neighborhood boundary collar 0 ≤ x < 12 ; |y| < 3

4 ⊂ Ωc, where1− β0 ≡ 0, then Eqn. (10.4.12) does not have boundary terms, as these have been replaced by l.o.t. in L2(Q).

Our next step is to estimate the LHS of Eqns. (10.4.11) and (10.4.12) from below, as well as the RHS ofEqns. (10.4.11) and (10.4.12) from above, in terms of the original variable w.

Step 4. (Estimate from below of LHS of (10.4.11) and (10.4.12) in terms of w)


Proposition 10.4.1. Let v1, v2, and w = (zw) be related by Eqns. (10.4.1), (10.4.2), respectively. Then,recalling the norms in Section 10.3, we have the following estimates in a collar of Σ:

(i)

‖βzeτϕw‖(1,−1a) ≤ C‖eτϕzv1‖(1,0) + Cτ‖βzw‖(1,−2a)

≤ C‖eτϕzv1‖H1a(Q) + Cτ‖βzw‖(1,−2a). (10.4.13)

(ii)‖(1− β)zeτϕw‖L2(Q) ≤ C‖eτϕzv2‖H1

a(Q) + Cτ‖zw‖H−1a (Q). (10.4.14)

(iii) Consequently, via (i) and (ii) we obtain from zeτϕ(zw) = zeτϕw = βzeτϕw + (1 − β)eτϕw:

‖zeτϕ(zw)‖(1,−1a) ≤ C[

‖eτϕzv1‖H1a(Q) + ‖eτϕzv2‖H1

a(Q)

]

+ Cτ [‖βzw‖(1,−2a) + ‖zw‖H−1a (Q)]. (10.4.15)

Proof. (i) We first establish (10.4.13). With w = zw by (10.2.7), we compute in a collar of Σ, recalling(10.4.1),

βzeτϕw = βzeτϕ(zw) = zeτϕΛtan(Λ−1tanβzw) = zeτϕΛtanv1

= Λtan(zeτϕv1) + [zeτϕ,Λtan]v1. (10.4.16)

From (10.4.16), recalling the norms in Section 10.3, in a collar of Σ, we estimate

‖βzeτϕw‖(1,−1a) ≤ ‖Λtan(zeτϕv1)‖(1,−1a) + ‖[zeτϕ,Λtan]v1‖(1,−1a) (10.4.17)

≤ C‖zeτϕv1‖(1,0) + Cτ‖v1‖(1,−1a), (10.4.18)

since the commutator [zeτϕ,Λtan] ∈ S0,0 (Appendix). Moreover, recalling (10.4.1),

‖v1‖(1,−1a)def= ‖Λ−1

tanβzw‖(1,−1a) ≤ C‖βzw‖(1,−2a). (10.4.19)

Invoking (10.4.19) for the last term in estimate (10.4.18) yields estimate (10.4.13), as desired.(ii) Next, we establish (10.4.14). First, the counterpart of (10.4.16) away from Σ in the interior of Q, is

now via (10.4.2):

(1 − β)zeτϕw = (1 − β)zeτϕ(zw) = zeτϕΛ(Λ−1(1 − β)zw) = zeτϕΛv2 (10.4.20)

= Λ(zeτϕv2) + [zeτϕ,Λ]v2. (10.4.21)

Now, from (10.4.21), recalling the interior norms in Section 10.3, we estimate

‖(1 − β)zeτϕw‖L2(Q) ≤ ‖Λ(zeτϕv2)‖L2(Q) + ‖[zeτϕ,Λ]v2‖L2(Q) (10.4.22)

≤ C‖zeτϕv2‖H1a(Q) + Cτ‖v2‖L2(Q), (10.4.23)

recalling (10.3.10) on the first term and the fact that the commutator [zeτϕ,Λ] is of anisotropic order zero inall variables. Finally, recalling (10.4.2) once more, we estimate

‖v2‖L2(Q) = ‖Λ−1(1 − β)zw‖L2(Q) ≤ C‖(1 − β)zw‖H−1a (Q) ≤ C‖zw‖H−1

a (Q). (10.4.24)

Invoking (10.4.24) for the last term in estimate (10.4.23) yields estimate (10.4.14), as desired.


Part (iii) follows from parts (i) and (ii), as follows. First, since (1 − β) ≡ 0 in a collar of the boundary, seebelow (10.3.3), we have by Section 3

‖(1 − β)zeτϕw‖(1,−1a) = ‖(1 − β)zeτϕw‖H0(Q). (10.4.25)

Next, since zeτϕw = βzeτϕw + (1 − β)eτϕw, we obtain, recalling (10.4.13), (10.4.25), and (10.4.14):

‖zeτϕw‖(1,−1a) ≤ ‖βzeτϕw‖(1,−1a) + ‖(1 − β)zeτϕw‖(1,−1a) (10.4.26)

≤ C‖eτϕzv1‖H1a(Q) + Cτ‖βzw‖(1,−2a)

+ C‖eτϕzv2‖H1a(Q) + Cτ‖zw‖H−1

a (Q), (10.4.27)

and (10.4.15) is established.

Step 5. (Estimates from above on F1(w)eτϕ and F2(w)eτϕ on the RHS of (10.4.11) and (10.4.12) in termsof w)

Proposition 10.4.2. Assume (1.1.3) and (2.1.9a) on F . With reference to the terms F1(w) and F2(w)defined by (10.4.3b) and (10.4.4b) respectively, the following estimates (needed on the RHS of (10.4.11) and(10.4.12), respectively) hold true:

(i)‖eτϕF1(w)‖L2(Q) ≤ C‖eτϕzw‖(1,−1a) + Cτ‖z1w‖(1,−2a); (10.4.28)

(ii)‖eτϕF2(w)‖L2(Q) ≤ C‖eτϕzw‖L2(Q) + Cτ‖z1w‖H−1

a (Q). (10.4.29)

Proof. We begin with (ii), which is simpler. We return to the definition (10.4.4b) for F2(w), multiplyacross by eτϕ, use commutators and obtain

−eτϕF2(w) = [Λ−1(1 − β)z,P](eτϕzw) + [z, [Λ−1(1 − β)z,P]](eτϕw)

+ [eτϕ, [Λ−1(1 − β)z,P]](zw) + [eτϕ, [z, [Λ−1(1 − β)z,P]]]w. (10.4.30)

Now, in (10.4.30), we have (Remark 10.3.1) that the first commutator (on eτϕzw) is of anisotropic order−1+2−1 = 0; the second and third commutators (on (eτϕw) and (zw)) are of anisotropic order 0+0−1 = −1;the last commutator (on w) is of anisotropic order 0− 1− 1 = −2. Moreover, they are all active only on suppz. We then obtain from (10.4.30), with F subject to (1.1.3), (2.1.9a),

‖eτϕF2(w)‖L2(Q) ≤ C[

‖eτϕzw‖L2(Q) + ‖z1eτϕw‖H−1a (Q)

]

+ Cτ

[

‖zw‖H−1a (Q) + ‖z1w‖H−2

a (Q)

]

(10.4.31)

≤ C‖eτϕzw‖L2(Q) + Cτ‖z1w‖H−1a (Q), (10.4.32)

and (10.4.32) proves (10.4.29).(i) We return to the definition (10.4.3b) for F1(w), multiply across by eτϕ, use commutators and obtain

−eτϕF1(w) = [Λ−1tanβz,P](eτϕzw) + [z, [Λ−1

tanβz,P]](eτϕw)

+ [eτϕ, [Λ−1tanβz,P]](zw) + [eτϕ, [z, [Λ−1

tanβz,P]]](w). (10.4.33)

We reference to the commutators present in expansion (10.4.33), the following key properties are proved inAppendix E, Lemma E.1, by using the expansion of symbols:

[Λ−1tanβz,P] ∈ S1,−1

a ; [z, [Λ−1tanβz,P]] ∈ S1,−2

a ; (10.4.34)


[eτϕ, [Λ−1tanβz,P]] ∈ S1,−2

a ; [eτϕ, [z, [Λ−1tanβz,P]] ∈ S1,−3

a . (10.4.35)

Accordingly, using (10.4.34), (10.4.35) sequentially, we estimate with F subject to (1.1.3), (2.1.9a),

‖[Λ−1tanβz,P](eτϕzw)‖L2(Q) ≤ C‖eτϕzw‖(1,−1a); (10.4.36)

‖[z, [Λ−1tanβz,P]](eτϕw)‖L2(Q) ≤ C‖z1eτϕw‖(1,−2a); (10.4.37)

‖[eτϕ, [Λ−1tanβz,P]](zw)‖L2(Q) ≤ Cτ‖zw‖(1,−2a); (10.4.38)

‖[eτϕ, [z, [Λ−1tanβz,P]]](w)‖L2(Q) ≤ Cτ‖z1w‖(1,−3a). (10.4.39)

Returning now to expansion (10.4.33) and using here estimates (10.4.36)–(10.4.39), we readily obtain

‖eτϕF1(w)‖L2(Ω) ≤ C

‖eτϕzw‖(1,−1a) + ‖eτϕw‖(1,−2a)

+ Cτ

‖zw‖(1,−2a) + ‖w‖(1,−3a)

≤ C‖eτϕzw‖(1,−1a) + Cτ‖z1w‖(1,−2a),

which proves (10.4.28). 2

Step 6. (Estimates of the boundary terms on the RHS of (10.4.11)) Our next step consists in estimatingthe boundary terms on the RHS of inequality (10.4.11) in a collar of the boundary.

Proposition 10.4.3. With reference to the boundary terms in estimate (10.4.11), we have

‖v1eτϕ‖2H1

a(Σ1)≡∫ T

0

∫

Γ1

|∇(eτϕzv1)|2dΣ1 + ‖(eτϕzv1)‖2

H12 (0,T ;L2(Γ1))

≤ C

‖eτϕzβzw‖2L2(Σ1)

+

∥

∥

∥

∥

∂

∂ν(eτϕzβzw)

∥

∥

∥

∥

2

H−1a (Σ1)

+ Cτ

‖βzw‖2H−1

a (Σ1)+

∥

∥

∥

∥

∂

∂ν(zβw)

∥

∥

∥

∥

2

H−2a (Σ1)

. (10.4.40)

Proof of Proposition 10.4.3. Using the very definition of Λtan in (10.3.9), we rewrite the LHS of (10.4.40)as follows, recalling also v1 in (10.4.1):

‖(eτϕzv1)‖2H1

a(Σ1)

≡∫ T

0

∫

Γ1

[|∇(eτϕzv1)|2dΣ1 + ‖(eτϕzv1)‖2

H12 (0,T ;L2(Γ1))

=

∫

Σ1

∣

∣

∣

∣

∂(eτϕzv1∂µ

∣

∣

∣

∣

2

dΣ1 +

∫

Σ1

|D12t (eτϕzv1)|2dΣ1 +

∫

Σ1

∣

∣

∣

∣

∂(eτϕzv1∂ν

∣

∣

∣

∣

2

dΣ1 (10.4.41)

(by (10.3.10a))

= ‖Λtan(eτϕzv1)‖2

L2(Σ1) +

∫

Σ1

∣

∣

∣

∣

∂(eτϕzv1)

∂ν

∣

∣

∣

∣

2

dΣ1 (10.4.42)

(by (10.4.1))

= ‖Λtan(eτϕz(Λ−1

tanβzw))‖2L2(Σ1) +

∥

∥

∥

∥

∂

∂ν(eτϕzΛ−1

tanβzw)

∥

∥

∥

∥

2

L2(Σ1)

, (10.4.43)


where ∂∂µ = tangential gradient, where in the last step we have recalled the definition of v1 in (10.4.1). We

rewrite and estimate the operators occurring in (10.4.43) by use of commutators

Λtan(eτϕz(Λ−1tanβzw)) = eτϕzΛtanΛ

−1tanβzw + [Λtan, e

τϕz](Λ−1tanβzw); (10.4.44)

∂

∂ν((eτϕz)(Λ−1

tanβzw)) = Λ−1tan

(

∂

∂νeτϕzβzw

)

+

[(

∂

∂ν

)

(eτϕz),Λ−1tan

]

(βzw). (10.4.45)

Next, we note the following properties of the commutators involved in (10.4.44) and (10.4.45):

[Λtan, eτϕz] ∈ S0,0;

[(

∂

∂ν

)

(eτϕz),Λ−1tan

]

∈ S1,−2a . (10.4.46)

See Appendix E, Lemma E.1, Eqn. (E.6), for the first property and Lemma E.2 for the second property, byuse of symbols expansions. Using properties (10.4.46) in, respectively, (10.4.44) and (10.4.45), we obtain

‖Λtan(eτϕz(Λ−1

tanβzw))‖L2(Σ1) ≤ ‖eτϕzβzw‖L2(Σ1) + Cτ‖βzw‖H−1a (Σ1)

, (10.4.47)

recalling (10.3.10b) for Λ−1tan, and

∥

∥

∥

∥

∂

∂ν(eτϕzΛ−1

tanβzw)

∥

∥

∥

∥

L2(Σ1)

≤∥

∥

∥

∥

Λ−1tan

(

∂

∂νeτϕzβzw

)∥

∥

∥

∥

L2(Σ1)

+ Cτ‖βzw‖1,−2a,Σ1 (10.4.48)

≤ C

∥

∥

∥

∥

∂

∂νeτϕzβzw

∥

∥

∥

∥

2

H−1a (Σ1)

,+Cτ

∥

∥

∥

∥

∂

∂ν(βzw)

∥

∥

∥

∥

H−2a (Σ1)

, (10.4.49)

recalling the last step again (10.3.10b) on Λ−1tan and the norm in Section 10.3. Using estimates (10.4.47) and

(10.4.49) on the RHS of (10.4.43), we obtain

‖(eτϕzv1)‖2H1

a(Σ1) ≤ 2‖eτϕzβzw‖2L2(Σ1) + Cτ‖βzw‖2

H−1a (Σ1)

+ C

∥

∥

∥

∥

∂

∂νeτϕzβzw

∥

∥

∥

∥

2

H−1a (Σ1)

+ Cτ

∥

∥

∥

∥

∂

∂νβzw

∥

∥

∥

∥

2

H−2a (Σ1)

, (10.4.50)

which readily yields (10.4.40), as desired. 2

Step 7. (Collecting term)

Proposition 10.4.4. With reference to v1 and v2 defined by Eqns. (10.4.1), (10.4.2), and under assumption(10.1.4) on Γ0, we have the following estimates

(i)

‖zv1eτϕ‖2H1

a(Q) ≤ CT

C

[

‖eτϕzβzw‖2L2(Σ1) +

∥

∥

∥

∥

∂

∂ν(eτϕzβzw)

∥

∥

∥

∥

2

H−1a (Σ1)

]

+ Cτ

[

‖βzw‖2H−1

a (Σ1)+

∥

∥

∥

∥

∂

∂ν(zβw)

∥

∥

∥

∥

2

H−2a (Σ1)

]

+C

τ

‖eτϕzΛ−1tan(βzf)‖2

L2(Q) + C‖zeτϕzw‖2(1,−1a) + Cτ‖z1w‖2

(1,−2a)

+ CT,τ‖z1v1‖2L2(Q);

(10.4.51)


(ii)

‖zv2eτϕ‖2H1

a(Q) ≤C

τ

‖eτϕzΛ−1(1 − β)zf‖2L2(Q) + C‖eτϕzw‖2

L2(Q) + Cτ‖z1w‖2H−1

a (Q)

+ CTτ‖z1v2‖2L2(Q); (10.4.52)

(iii)

‖zv1eτϕ‖2H1

a(Q) + ‖zv2eτϕ‖2H1

a(Q)

≤ CT

C

[

‖eτϕ(zβzw)‖2L2(Σ1) +

∥

∥

∥

∥

∂

∂ν(eτϕzβzw)

∥

∥

∥

∥

2

H−1a (Σ1)

]

+ Cτ

[

‖βzw‖2H−1

a (Σ1)+

∥

∥

∥

∥

∂

∂ν(zβw)

∥

∥

∥

∥

2

H−2a (Σ1)

]

+C

τ

‖(βzf)eτϕ‖2(0,−1a) + ‖(zf)eτϕ‖2

H−1a (Q)

+ ‖zeτϕzw‖2(1,−1a) + Cτ‖z1w‖2

(1,−2a) + ‖eτϕzw‖2L2(Q)

+ Cτ‖z1w‖2H−1

a (Q)

+ CTτ

[

‖z1v1‖2L2(Q) + ‖z1v2‖2

L2(Q)

]

. (10.4.53)

Proof. (i) We return to inequality (10.4.11) and, on its RHS, we invoke Eqn. (10.4.40) of Proposition10.4.3, as well as Eqn. (10.4.28) of Proposition 10.4.2 on F1(w)eτϕ, thus obtaining estimate (10.4.51).

(ii) We return to inequality (10.4.12) and, on its RHS, we invoke Eqn. (10.4.29) of Proposition 10.4.2 onF2(w)eτϕ, thus obtaining estimate (10.4.52) by further majorizing norms.

(iii) Finally, Eqns. (10.4.51) and (10.4.52) yield readily (10.4.53). 2

Step 8. Lemma 10.4.5. With reference to Proposition 10.4.4, we have

(

1 − C

τ

)

‖zeτϕzw‖2(1,−1a) ≤

C

τ

Cτ‖z1w‖2(1,−2a)

+C

τ‖βzfeτϕ‖2

(0,−1a)

+ CT

‖eτϕβzw‖2L2(Σ1) +

∥

∥

∥

∥

∂

∂ν(eτϕβzw)

∥

∥

∥

∥

2

H−1a (Σ1)

+ Cτ

[

‖βzw‖2H−1

a (Σ1)+

∥

∥

∥

∥

∂

∂ν(βzw)

∥

∥

∥

∥

2

H−2a (Σ1)

. (10.4.54)

Proof. We rewrite inequality (10.4.15) of Proposition 10.4.1, and substitute estimate (10.4.53) on its RHS


to yield

‖zeτϕzw‖2(1,−1a) ≤ C

[

‖zv1eτϕ‖2H1

a(Q) + ‖zv2eτϕ‖2H1

a(Q)

]

+ Cτ‖βzw‖2(1,−2a) + ‖zw‖2

H−1a (Q)

(10.4.55)

(by (10.4.53)) ≤ CTτ

‖zeτϕzw‖2(1,−1a) + Cτ‖z1w‖2

(1,−2a) + ‖eτϕzw‖2L2(Q) + Cτ‖z1w‖2

H−1a (Q)

+C

τ

‖βzfeτϕ‖2(0,−1a) + ‖zfeτϕ‖2

H−1a (Q)

+ CTτ

[

‖z1v1‖2L2(Q) + ‖z1v2‖2

L2(Q)

]

+ CT

‖eτϕzβzw‖2L2(Σ1) +

∥

∥

∥

∥

∂

∂ν(eτϕzβzw)

∥

∥

∥

∥

2

H−1a (Σ1)

+ Cτ

[

‖βzw‖2H−1

a (Σ1)+

∥

∥

∥

∥

∂

∂ν(βzw)

∥

∥

∥

∥

2

H−2a (Σ1)

]

. (10.4.56)

Finally, we see that some terms on the RHS of (10.4.56) are redundant, in view of the definitions of the normsinvolved. Thus, the H−1

a (Q)-norm of [zfeτϕ] is contained in its (0,−1a)-norm; the L2(Q)-norm of [eτϕzw] iscontained in its (1,−1a)-norm; theH−1

a (Q)-norm of (z1w) is contained in its (1,−2a)-norm. Moreover, recalling(10.4.1) and (10.4.2) we have

‖z1v2‖L2(Q) = ‖z1Λ−1(1 − β)zw‖L2(Q) ≤ C‖z1w‖H−1a (Q), (10.4.57)

see (10.3.11b) or (10.4.24), and similarly,

‖z1v1‖L2(Q) = ‖z1Λ−1tan(βzw)‖L2(Q) ≤ C‖z1w‖H−1

a (Q), (10.4.58)

see (10.3.11a) and the terms in (10.4.57), (10.4.58) are then contained in the (1,−2a)-norm of z1w. This way,inequality (10.4.56) reduces to inequality (10.4.54), moving its first term on the RHS to the LHS. 2

Step 9. In inequality (10.4.54), we take τ sufficiently large as to have(

1 − Cτ

)

> 0 on the LHS of (10.4.54).We thus obtain

(

1 − C

τ

)

‖eτϕzw‖2L2(Q) ≤

(

1 − C

τ

)

‖eτϕzw‖2(1,−1a) ≤ CT

‖eτϕzw‖2L2(Σ1) +

∥

∥

∥

∥

∂

∂ν(eτϕzw)

∥

∥

∥

∥

2

H−1a (Σ1)

+ Cτ

[

‖zw‖2H−1

a (Σ1)+

∥

∥

∥

∥

∂

∂ν(eτϕzw)

∥

∥

∥

∥

2

H−2a (Σ1)

]

+C

τ‖zfeτϕ‖2

(0,−1a) +Cττ‖z1w‖2

(1,−2a), (10.4.59)

where the first inequality on the LHS of (10.4.59) is true by definition of the norms involved. To obtaininequality (10.3.1) of Theorem 10.3.1 from inequality (10.4.59) we need one more effort, which is treated in thenext Section 10.5, to majorize the ‖z1w‖(1,−2a)-norm by the ‖z1w‖H−1(Q)-norm.

10.5 Completion of the proof of Theorem 10.3.1: Estimate for ‖βw‖(1,−2a) in a collar of theboundary

To complete the proof of Theorem 10.3.1, it is sufficient to establish the following result.


Theorem 10.5.1. Let f satisfy: Λ−1tanf ∈ H−2(Q), Λ−2

tanf ∈ H−1(Q) [a sufficient condition for which isthat f ∈ H(0,−3a)]. Let w be a solution of Eqn. (10.2.8): Pw = f , satisfying w ∈ H−1(Q). Then, in fact,w ∈ H(1,−2a) continuously; more precisely, the following estimate holds true:

‖w‖(1,−2a) ≤ C‖w‖H−1(Q) + ‖Λ−1tanf‖H−2(Q) + ‖Λ−2

tanf‖H−1(Q)

≤ C‖w‖H−1(Q) + ‖f‖(0,−3a), if f ∈ H(0,−3a).

(10.5.1a)

(10.5.1b)

Proof. Step 1. Under present assumptions, with w ∈ H−1(Q), we shall first show that, in fact,

w ∈ H(0,−1a), equivalently Λ−1tanw ∈ L2(Q). (10.5.2)

By [2, p. 111, p. 196], showing (10.5.2) is equivalent to verifying that

Dα(Λ−1tanw) ∈ H−2(Q), ∀ |α| ≤ 2; ∂t(Λ

−1tanw) ∈ H−2(Q), (10.5.3)

that is: all space derivatives up to order 2, as well as the first-order time-derivative, of (Λ−1tanw) are in H−2(Q).

We do this in two steps.First, we verify (10.5.3) for the double tangential -, time - normal/tangential derivatives:

D2yΛ

−1tanw, ∂tΛ

−1tanw ∈ H−2(Q); DνDyΛ

−1tanw ∈ H−2(Q), (10.5.4)

holds true with w ∈ H−1(Q), hence DyΛ−1tanw ∈ H−1(Q).

Next, we now verify (10.5.3) for the double normal:

D2ν(Λ

−1tanw) ∈ H−2(Q). (10.5.5)

In fact, to this end, we shall use that PΛ−1tanw = Λ−1

tanf + [P,Λ−1tan]w, hence

D2ν(Λ

−1tanw) = Λ−1

tanf + [P,Λ−1tan]w − [i∂t + a(x, y)D2

y − F ](Λ−1tanw) + l.o.t.(w), (10.5.6)

since w satisfies Pw = [i∂t + ∆ − F ]w = [i∂t +D2ν + a(x, y)D2

y + l.o.t. − F ]w, recalling (10.3.2) for ∆. Now,each term on the RHS of (10.5.6) is in H−2(Q): the first, by assumption; the third, by application of (10.5.4);and the second, since [P,Λ−1

tan] ∈ S1,−1a (Appendix E, Eqn. (E.2)). We conclude that (10.5.5) holds true. Then,

all the relations in (10.5.3) have been proved, and thus the equivalent statements (10.5.2) are established.

Step 2. By using the result (10.5.2) of Step 1, we now establish that

Λ−2tanw ∈ H1(Q), hence a-fortiori that w ∈ H(1,−2a) as desired (10.5.7)

by verifying thatDα(Λ−2

tanw) ∈ H−1(Q), ∀ |α| ≤ 2; ∂t(Λ−2tanw) ∈ H−1(Q), (10.5.8)

which is equivalent, again by [2, p. 196] to (10.5.7). Eqn. (10.5.8) says that all space derivatives up to order 2,as well as the first time derivative, of Λ−2

tanw are in H−1(Q). Verification of (10.5.8) for the double tangentialderivative Dα = DyDy, for the time derivative Dα = Dt, and for the normal/tangential derivative Dα = Dνdy,that is

D2yΛ

−2tanw, ∂tΛ

−2tanw ∈ H−1(Q), DνDyΛ

−2tanw ∈ H−1(Q), (10.5.9)

hold true with w ∈ H−1(Q): the first two relations are immediate since D2yΛ

−2tan∂tΛ

−2tan are topologically neutral

by definition of Λ−1tan; the third relation follows, as Λ−1

tanw ∈ L2(Q) by (10.5.2) of Step 1 and DyΛ−1tan, is

topologically neutral.We now verify (10.5.8) for the double normal:

D2ν(Λ

−2tanw) ∈ H−1(Q), (10.5.10)


by using the relation

D2ν(Λ

−2tanw) = Λ−2

tanf + [P,Λ−2tan]w − [i∂t + a(x, y)D2

y + l.o.t.− F ](Λ−2tanw) (10.5.11)

counterpart of (10.5.6). Each term on the RHS of (10.5.11) is in H−1(Q): the first, by assumption; the third,by application of (10.5.9); and the second, since [P,Λ−2

tan] ∈ S0,−1a (Appendix E, Eqn. (E.6)). We conclude that

(10.5.10) holds true. Then, all the relations in (10.5.8) have been proved, and thus (10.5.7) (left) is established.We conclude that w ∈ H(1,−2a), as desired, continuously on the original data, and so inequality (10.5.18a) isproved, from which (10.5.18b) follows at once. 2

10.6 From Theorem 10.3.1 to Theorem 2.6.1In this section we complete the proof of Theorem 2.6.1, starting from the statement of Theorem 10.3.1.

Step 1. Theorem 10.6.1. (Carleman estimate, first version, at the L2(Ω)-level) Assume (1.1.3), (2.1.9a),on F and (1.1.4) on f . Then the following estimate holds true for all τ sufficiently large:

‖w|Γ‖2L2(Σ1) +

∥

∥

∥

∥

∂w

∂ν

∥

∥

∥

∥

2

H−1a (Σ1)

+ ‖f‖2L2(Q) + ‖w‖2

H−1(Q)

≥ Cτ

∫ t1

t0

‖w(t)‖2L2(Ω)dt, (10.6.1)

where the interval 0 < t0 <T2 < t1 < T is defined (as in (1.1.10), now more precisely) by (10.6.3) below. Here

we may writeE−1(t) = ‖w(t)‖2

L2(Ω), (10.6.2)

for the ‘energy’ at the ‘lower’ level, one unit below the level in Theorem 2.1.1, Theorem 2.1.3 in Section 2.

Proof. We return to inequality (10.3.1) of Theorem 10.3.1, where we are presently taking d(x) = ‖x−x0‖2 ≥ρ > 0, for a suitable x0 ∈ Rn, x0 /∈ Ω. Then, recalling (1.1.8a) on ϕ and (10.2.1)–(10.2.4), we obtain

minx∈Ω;t∈[t0,t1]

ϕ(x, t) ≥ ρ

2; hence z(x, t) = ψ(ϕ(x, t)) ≥ c > 0, ∀x ∈ Ω, t ∈ [t0, t1]. (10.6.3)

Then, |zw| = |z| |w| = z|w| ≥ c|w|, for all x ∈ Ω, t ∈ [t0, t1]. We then use this information on the LHS of

(10.3.1), while on its RHS we use that ∂(zw)∂ν = z ∂w∂ν + [ ∂∂ν , z]w, with [ ∂∂ν , z] ∈ S0,0 (same proof by symbol

expansion, as in Lemma E.1) and majorize z and z1 by a constant. This way, we readily obtain (10.6.1). 2

Step 2. Proposition 10.6.2 (Counterpart of Lemma 6.1, at the L2(Ω)-level). Assume (1.1.4) on f ∈L2(Q); r1 real with: |r1|, div r1 ∈ L∞(Q), q0 ∈ L∞(Q) as in (2.1.9b) and (1.1.3). Let Γ0 be defined by (10.1.4).Let w be a solution of (1.1.1) in the class (10.1.1). Then:

(i) (Counterpart of (6.11a)–(6.12a)) the following identity holds true for all 0 ≤ s ≤ t ≤ T :

‖w(t)‖2L2(Ω) − ‖w(s)‖2

L2(Ω) =

∫ t

s

∫

Ω

|w|2div r1dΩ dτ − 2 Re

i

∫ t

s

∫

Ω

q0|w|2dΩ dτ

+ 2 Re

i

∫ t

s

∫

Γ1

∂w

∂νw dΓ1 dτ

−∫ t

s

∫

Γ1

|w|2r1 · ν dΓ1 dτ

− 2 Re

i

∫ t

s

∫

Ω

fwdΩ dτ

. (10.6.4)

(ii) (Counterpart of (6.12b) at the L2(Ω)-level. For all 0 ≤ s ≤ t ≤ T and recalling (10.6.2) for E−1(t), wehave:

|E−1(t) −E−1(s)| ≤ Λ(T ) + c

∫ t

s

E−1(τ)dτ ; (10.6.5)


Λ(T ) = k

∫ T

0

∫

Γ1

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

|w|dΓ1dt+

∫ T

0

∫

Ω

|f |2dΩ dt

. (10.6.6)

(iii) (Counterpart of (6.20). For any 0 ≤ s ≤ t ≤ T , we have:

E−1(t) ≤ [E−1(s) + Λ(T )]ec(t−s); E−1(s) ≤ [E−1(t) + Λ(T )]ec(t−s). (10.6.7)

(iv) (Counterpart of (6.21).

‖w(t)‖2L2(Ω) ≡ E−1(t) ≥

E−1(0) +E−1(T )

2e−cT − Λ(T ), 0 ≤ t ≤ T. (10.6.8)

Proof. (i) We take the L2(Ω)-complex inner product of Eqn. (1.1.1) by iw (a component of the multiplierused int he proof of Lemma 6.1) and take real part of the resulting identity. We obtain

Re(wt, w)L2(Ω) − Re

i

∫

Ω

∆w w dΩ

= −Re

i

∫

Ω

[F (w) + f ]w dΩ

. (10.6.9)

We readily verify that

2Re(w(t), wt(t)) =d

dt‖w(t)‖2

L2(Ω); Re

i

∫

Ω

∆w w dΩ

= Re

i

∫

Γ1

∂w

∂νw dΓ1

, (10.6.10)

using, for the second identity, Green’s first theorem, where Rei∫

Ω |∇w|2dΩ = 0.

Next, we recall from assumption (A.3) = (2.1.9) that w.l.o.g. we can take F (w) = −ir1 · ∇w + q0w, r1real-valued. By the divergence theorem with div(w r1) = w div r1 + ∇w · r1, we obtain

2 Re

∫

Ω

r1 · ∇w w dΩ

=

∫

Ω

r1 · ∇w w dΩ +

∫

Ω

wr1 · ∇ w dΩ

=

∫

Γ1

|w|2r1 · ν dΓ1 −∫

Ω

|w|2div r1 dΩ. (10.6.11)

Thus, by (10.6.11),

Re

i

∫

Ω

F (w)w dΩ

= Re

∫

Ω

(r1 · ∇w)w dΩ

+ Re

i

∫

Ω

q0w w dΩ

=1

2

∫

Γ1

|w|2r1 · ν dΓ1 −1

2

∫

Ω

|w|2div r1 dΩ + Re

i

∫

Ω

q0|w|2dΩ

.

(10.6.12)

Substituting (10.6.10), (10.6.12) into (10.6.9) yields

1

2

d

dt‖w(t)‖2

L2(Ω) − Re

i

∫

Γ1

∂w

∂νw dΓ1

=1

2

∫

Ω

|w|2div r1 dΩ − 1

2

∫

Γ1

|w|2r1 · ν dΓ1

− Re

i

∫

Ω

q0|w|2dΩ

− Re

i

∫

Ω

fw dΩ

. (10.6.13)

Integrating (10.6.13) in time from s to t yields identity (10.6.4).


(ii) From (10.6.4) we readily find with 0 ≤ s ≤ t ≤ T :

∣

∣

∣‖w(t)‖2

L2(Ω) − ‖w(s)‖2L2(Ω)

∣

∣

∣≤ c1

∫ t

s

∫

Ω

|w|2dΩ dτ + c2

∫ t

s

∫

Ω

|f |2dΩ dτ

+ c3

∫ t

s

∫

Γ1

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

|w|dΓ1 dτ, (10.6.14)

from which (10.6.5) readily follows, using (10.6.6) and recalling E−1(t) from (10.6.2).(iii),(iv) These parts readily follow now from (10.6.5) as in Lemma 6.2, using the Gronwall inequality in the

case of (10.6.7). 2

Step 3. We substitute inequality (10.6.8) into the RHS of inequality (10.6.1) and obtain

‖w|Γ‖2L2(Σ1)

+

∥

∥

∥

∥

∂w

∂ν

∥

∥

∥

∥

2

H−1a (Σ1)

+ ‖f‖2L2(Q) + ‖w‖2

H−1(Q) + cτ (t1 − t0)Λ(T )

≥ cτ(t1 − t0)e

−cT

2[E−1(0) +E−1(T )], (10.6.15)

or recalling (10.6.6) for Λ(T ) and (10.6.2) for E−1(t):

‖w|Γ‖2L2(Σ1)

+

∥

∥

∥

∥

∂w

∂ν

∥

∥

∥

∥

2

H−1a (Σ1)

+ cτ (t1 − t0)k

∫ T

0

∫

Γ1

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

|w|dΓ1dt

+

∫ T

0

‖f‖2L2(Ω)dt

+ ‖f‖2H−1(Q) + ‖w‖2

H−1(Q)

≥ cτ(t1 − t0)e

−cT

2e−cT [‖w(0)‖2

L2(Ω) + ‖w(T )‖2L2(Ω)], (10.6.16)

and (2.6.1) is proved. 2

10.7 Estimate for D2ν(e

τϕzv1) in a collar of the boundaryThe following result provides additional information on the regularity of (eτϕzv1) in the normal direction

near the boundary.

Theorem 10.7.1. With reference to v1 = Λ−1tanβzw in (10.4.1), the following estimate holds true:

‖D2ν(e

τϕzv1)‖0,−1a ≤ C‖eτϕzv1‖H1a(Q) + Cτ‖z1w‖1a,−2a

+ C‖eτϕβz2f‖L2(Q) + Cτ‖βzf‖0,−2a. (10.7.1)

Proof. Step 1. We return to identity (10.4.3), multiply through by eτϕ, use commutators, and we getsince P = i∂t + ∆ − F :

∆(eτϕv1) = [−i∂t + F ](eτϕv1) − [eτϕ,P]v1 + eτϕg;

g ≡ [P, z]v1 + F1(w) + zΛ−1tanβzf.

(10.7.2)

(10.7.3)

Next, we shall invoke Melrose-Sosjrand’s representation (in suitable coordinates) [34, p. 597-8] of the Lapla-cian near the boundary and write

∆ = D2x + a(x, y)D2

y + l.o.t. in Dy, (10.7.4)


where a(x, y) is real and smooth.Using this representation (10.7.4) on the LHS of (10.7.2), we obtain

D2x(e

τϕv1) = [−i∂t − a(x, y)D2y + F ](eτϕv1) − [eτϕ,P]v1 + eτϕg. (10.7.5)

Thus, from (10.7.5), we estimate

‖D2x(e

τϕv1)‖0,−1a ≤ ‖[i∂t + a(x, y)D2y](e

τϕv1)‖0,−1a

+ ‖F (eτϕv1)‖0,−1a + ‖[eτϕ,P]v1‖0,−1a + ‖eτϕg‖0,−1a. (10.7.6)

Step 2. With reference to the first three terms on the RHS of (10.7.6), we have:(i)

‖[i∂t + a(x, y)D2y](e

τϕv1)‖0,−1a ≤ C‖(eτϕv1)‖0,1a; (10.7.7)

(ii)

‖F (eτϕv1)‖0,−1a ≤ C

‖eτϕv1‖1a,−1a + ‖eτϕv1‖0,1a

(10.7.8)

≤ C‖eτϕv1‖H1a(Q); (10.7.9)

(iii)

‖[eτϕ,P]v1‖0,−1a ≤ Cτ‖v1‖1a,−1a. (10.7.10)

Indeed: (i) estimate (10.7.7) follows by the very definition of the norms involved, see (10.3.9). Next, (ii)estimate (10.7.8) follows since F is a first-order operator in time and space: on the RHS of (10.7.8), the firstterm accounts for the normal derivative, while the second term accounts for the tangential derivative on Σ. Thevery definition of the norms involved readily causes (10.7.8) to imply (10.7.9). Finally, (iii) to obtain estimate(10.7.10), we compute, recalling P = i∂t +D2

x + a(x, y)D2y − F :

‖[eτϕ,P]v1‖0,−1a ≤ ‖[eτϕ, D2x]v1‖0,−1a

+ ‖[eτϕ, i∂t + a(x, y)D2y]v1‖0,−1a + ‖[eτϕ, F ]v1‖0,−1a (10.7.11)

≤ Cτ

‖v1‖1a,−1a + ‖v1‖L2(Q) + ‖v1‖0,−1a

(10.7.12)

≤ Cτ‖v1‖1a,−1a. (10.7.13)

Estimate (10.7.12) follows from (10.7.11) since the various commutators in (10.7.11) are, in the order in whichthey appear, of order 1 in the normal direction; of order 1 in the tangential direction; and of order zero in allvariables, respectively. Finally, the estimates from (10.7.12) to (10.7.13) follow by the very definition of thenorms involved.

Step 3. The following estimate holds true:

‖eτϕ[P, z]v1‖0,−1a ≤ C‖eτϕv1‖1a,−1a + Cτ‖v1‖0,−1a. (10.7.14)

Proof of (10.7.14). We first write

eτϕ[P, z]v1 = [P, z](eτϕv1) + [eτϕ, [P, z]]v1, (10.7.15)


and since P = i∂t +D2x + a(x, y)D2

y − F , we estimate as in going from (10.7.11) to (10.7.13):

‖[P, z](eτϕv1)‖0,−1a ≤ ‖[D2x, z](e

τϕv1)‖0,−1a + ‖[(i∂t + a(x, y)D2y), z](e

τϕv1)‖0,−1a

+ ‖[F, z](eτϕv1)‖0,−1a (10.7.16)

≤ C

‖eτϕv1‖1a,−1a + ‖eτϕv1‖L2(Q) + ‖eτϕv1‖0,−1a

(10.7.17)

≤ C‖eτϕv1‖1a,−1a. (10.7.18)

The definition P = i∂t + ∆ − F implies that [P, z] is of order 1 in the normal direction, and of order 1 in thetangential direction Σ, a fact exploited in (10.7.16)–(10.7.18). Hence, the commutator [eτϕ, [P, z]] is of order 0in the normal direction and of order 0 in the tangential direction. Thus

‖[eτϕ, [P, z]]v1‖0,−1a ≤ Cτ‖v1‖0,−1a. (10.7.19)

Using (10.7.18) and (10.7.19) in (10.7.15) yields (10.7.14).

Step 4. In this step, building upon (10.4.28), we shall establish that

‖eτϕF1(w)‖0,−1a = ‖Λ−1tane

τϕF1(w)‖L2(Q) ≤ C‖eτϕzw‖1a,−2a + Cτ‖z1w‖1a,−3a. (10.7.20)

Proof of (10.7.20). We compute

Λ−1tane

τϕF1(w) = eτϕF1(Λ−1tanw) + [Λ−1

tan, eτϕF1]w. (10.7.21)

Recalling (10.4.28) on eτϕF1, we estimate

‖Λ−1tane

τϕF1(w)‖L2(Q) ≤ ‖eτϕF1(Λ−1tanw)‖L2(Q) + ‖[Λ−1

tan, eτϕF1]w‖L2(Q) (10.7.22)

(by (10.4.28)) ≤ C‖τeτϕΛ−1tanw‖1a,−1a + Cτ‖z1Λ−1

tanw‖1a,−2a

+ C‖w‖0,−3a, (10.7.23)

since, by (10.4.28), eτϕF1 is of order 1 in the normal component and of order −1 in the tangential components.Moreover, using commutators again,

‖eτϕzΛ−1tanw‖1a,−1a ≤ ‖Λ−1

taneτϕzw‖1a,−1a + ‖[eτϕz,Λ−1

tan]w‖1a,−1a (10.7.24)

≤ ‖eτϕzw‖1a,−2a + Cτ‖w‖0,−3a, (10.7.25)

and similarly,

‖z1Λ−1tanw‖1a,−2a ≤ ‖Λ−1

tanz1w‖1a,−2a + ‖[z1,Λ−1tan]w‖1a,−2a (10.7.26)

≤ C‖z1w‖1a,−3a + ‖w‖0,−4a. (10.7.27)

Substituting (10.7.25) and (10.7.27) on the RHS of (10.7.23) yields (10.7.20), as desired.

Step 5. Returning to the definition of g in (10.7.3), we multiply across by eτϕ, use here (10.7.14) and alsoinvoke (10.7.20) for eτϕF1(w), to obtain

‖eτϕg‖0,−1a ≤ ‖eτϕ[P, z]v1‖0,−1a + ‖eτϕF1(w)‖0,−1a

+ ‖eτϕzΛ−1tanβzf‖0,−1a (10.7.28)

[by (10.7.14), (10.7.20)] ≤ C‖eτϕv1‖1a,−1a + Cτ‖v1‖0,−1a

+ C‖eτϕzw‖1a,−2a + C‖z1w‖1a,−3a

+ ‖eτϕzΛ−1tanβzf‖0,−1a. (10.7.29)


Step 6. Using estimates (10.7.7), (10.7.9), (10.7.10), (10.7.29) on the RHS of estimate (10.7.6), yields

‖D2x(e

τϕv1)‖0,−1a ≤ C‖eτϕv1‖0,1a + C‖eτϕv1‖H1a(Q) + Cτ‖v1‖1a,−1a

+ C‖eτϕv1‖1a,−1a + Cτ‖v1‖0,−1a + C‖eτϕzw‖1a,−2a

+ C‖z1w‖1a,−3a + ‖‖eτϕzΛ−1tanβzf‖0,−1a. (10.7.30)

Recalling now v1 = Λ−1tanβzw from (10.4.1), we obtain

‖v1‖0,−1a = ‖Λ−1tanβzw‖0,−1a ≤ C‖βzw‖0,−2a; ‖eτϕv1‖1a,−1a ≤ Cτ‖v1‖1a,−1a ≤ Cτ‖βzw‖1a,−2a, (10.7.31)

recalling (10.4.19). Finally,

eτϕzΛ−1tanβzf = Λ−1

taneτϕβz2f + [eτϕz,Λ−1

tan](βzf);

‖eτϕzΛ−1tanβzf‖0,−1a ≤ ‖Λ−1

taneτϕβz2f‖0,−1a + ‖[eτϕz,Λ−1

tan](βzf)‖0,−1a (10.7.32)

≤ ‖eτϕβz2f‖L2(Q) + Cτ‖βzf‖0,−2a. (10.7.33)

Using (10.7.31) and (10.7.33) in (10.7.30) yields finally

‖D2x(e

τϕv1)‖0,−1a ≤ C‖eτϕv1‖H1a(Q) + Cτ‖v1‖1a,−1a

+ Cτ‖βzw‖1a,−2a + C‖eτϕzw‖1a,−2a

+ Cτ‖z1w‖1a,−3a + ‖eτϕβz2f‖L2(Q) + Cτ‖βzf‖0,−2a. (10.7.34)

Then (10.7.34) readily yields (10.7.1). The proof of Theorem 10.7.1 is complete.

11. APPLICATION OF L2-ESTIMATES IN SECTION 10: UNIFORM BOUNDARYFEEDBACK STABILIZATION IN L2(Ω)

In this section we shall establish a new uniform boundary feedback stabilization result for the pureSchrodinger equation at the L2(Ω)-level of energy. This problem has the advantage of having a boundaryfeedback physically more attractive and more easily implementable than the corresponding results in the liter-ature, see Remark 11.3.1 below.

11.1 Conservative open-loop and dissipative closed-loop problems in L2(Ω). Well-posednessand stabilization

Let Ω ⊂ Rn be an open bounded domain with sufficiently smooth boundary. In Ω we consider the followingtwo Schrodinger problems:

iyt + ∆y = 0;

y(0, · ) = y0;

y|Γ0 ≡ 0,∂y

∂ν|Γ1 = u;

iwt + ∆w = 0 in Q = (0, T ] × Ω;

w(0, · ) = w0 in Ω;

w|Γ0 ≡ 0,∂w

∂ν|Γ1 = iw in Σi = (0, T ] × Γi,

(11.1.1a)

(11.1.1b)

(11.1.1c)

where the w-problem can be viewed as a closed-loop version of the y-Neumann problem with boundary controlu on Γ1 in the feedback form u = iw. For u ≡ 0, the y-problem is conservative (‘energy’ preserving). It will beestablished below that, in contrast, the w-problem is dissipative.


Theorem 11.1.1. (Well-posedness and strong stabilization) With reference to the w-problem in (11.1.1),we have:

(i) the map w0 → w(t) defines a s.c. contraction semigroup on L2(Ω) : w(t) = eAF tw0 ∈ C([0, T ];L2(Ω))where AF is a maximal dissipative operator (to be explicitly defined below in (11.2.6)).

(ii) Setting

E(t;w0) = E(t) ≡∥

∥eAF tw0

∥

∥

2

L2(Ω)= ‖w(t)‖2 =

∫

Ω

|w(t)|2dΩ, (11.1.2)

we have the dissipativity relations:

dE(t)

dt= −2

∫

Γ1

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

dΓ ≤ 0; E(t) + 2

∫ t

0

∫

Γ1

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

dΓ1dτ = E(0); (11.1.3a)

∫ ∞

0

∫

Γ1

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

dΓ1dt ≤1

2E(0). (11.1.3b)

(iii) The resolvent R(λ,AF ) of AF is well-defined at least for all λ with Re λ ≥ 0, and is compact on L2(Ω).(iv) (Strong stabilization) For w0 ∈ L2(Ω), we have

eAF tw0 → 0 in L2(Ω), as t→ ∞. (11.1.4)

Theorem 11.1.2. (Uniform stabilization) Assume that there exists a coercive vector field h(x) ∈ (C2(Ω))n

[i.e., with Jacobian matrix J satisfying Jv · v ≥ ρ|v|2; in particular, h(x) = ∇d(x) for a strictly convex functiond(x), the case h(x) = x− x0, for some x0 ∈ Rn, being the canonical case], such that

h · ν ≤ 0 on Γ0. (11.1.5)

Then, there exist constants M ≥ 1, δ > 0 such that∥

∥eAF t∥

∥

L(L2(Ω))≤Me−δt, t ≥ 0; equivalently, by (11.1.2): E(t) ≤Me−δtE(0). (11.1.6)

11.2 Abstract model and well-posedness

Abstract model. [15], [18] The abstract model of problems in (11.1.1) is

iyt = A(y − Nu); wt = −iAw −ANN∗Aw, on [D(A)]′, (11.2.1)

where A is the positive self-adjoint operator: L2(Ω) ⊃ D(A) → L2(Ω),

Aψ = −∆ψ, D(A) =

f ∈ H2(Ω) : f |Γ0 = 0,∂f

∂ν

∣

∣

∣

∣

Γ1

= 0

, (11.2.2)

and N is the Neumann map [15]

v = Ng ⇐⇒

∆v = 0 in Ω; v|Γ0 = 0,∂v

∂ν

∣

∣

∣

∣

Γ1

= g

; (11.2.3)

N : Hs(Γ) → Hs+ 32 (Ω), s ∈ R;

N : L2(Γ) → H32 (Ω) ⊂ H

32−2ε(Ω) ≡ D(A

34−ε), ∀ ε > 0;

(11.2.4a)

(11.2.4b)

N∗A∗ϕ = N∗Aϕ =

0 on Γ0

−ϕ on Γ1

ϕ ∈ D(A) [18]. (11.2.5)


[To obtain (11.2.1) (left), for the y-problem, we use (11.2.3) in (11.1.1a) and obtain iyt + ∆(y − Nu) = 0;[y − Nu]Γ0 = 0; ∂

∂ν [y − Nu]Γ1 = 0, hence iyt − A(y − Nu) = 0 by (11.2.2). Thus, (11.2.1) for y follows.Then, (11.2.1) (right) for the w-problem follows from (11.2.1) (left) for y, by using that, in (11.1.1c) for thew-problem, we have u = −i ∂w∂ν = iD∗Aw on Γ1 and u = 0 on Γ0.] The generator of the w-problem (11.2.1) is[15], [18]:

AFw = [A −BB∗]w, w ∈ D(AF ) = w ∈ Y : [A −BB∗]w ∈ Y ; (11.2.6)

A = −iA = −A∗; B = AN ; A− 12B ≡ Q ∈ L(U ;Y ); U ≡ L2(Γ); Y ≡ L2(Ω). (11.2.7)

Well-posedness. The well-posedness of the w-problem (11.1.1)—or its abstract version (11.2.1)—follows,as usual, via the Lumer-Phillips theorem. Let x ∈ D(AF ). Then we can write

AFx = [A −BB∗]x = A12 [I − (A− 1

2B)(B∗A− 12 )]A

12x

= A12 [I + iQQ∗]A

12x = f ∈ Y = L2(Ω), (11.2.8)

using Q = A− 12B ∈ L(U ;Y ), Q∗ = B∗A∗− 1

2 = −iBA− 12 , since skew-adjointness A∗ = −A yields A∗ 1

2 = iA12 ,

or A∗− 12 = −iA− 1

2 . It is clear that the operator [I + iQQ∗] is boundedly invertible on Y = L2(Ω), whereQ∗Q ∈ L(Y ). Thus (11.2.8) yields

x = A−1F f = A− 1

2 [I + iQQ∗]−1A− 12 f ∈ D)AF ), f ∈ Y ; (11.2.9)

A−1F ∈ L(Y ), with D(AF ) ⊂ D(A

12 ) ⊂ D(B∗). (11.2.10)

We now show that AF is dissipative. Let x ∈ D(AF ). Thus, x ∈ D(A12 ) = D(A∗ 1

2 ) ⊂ D(B∗) by (11.2.10).Hence, we can write in the Y -inner product, Y = L2(Ω):

Re(AFx, x) = Re([A −BB∗]x, x) = Re(Ax, x) − |B∗x|2

≤ −|B∗x|2 ≤ 0, ∀ x ∈ D(AF ), (11.2.11)

since Re(Ax, x) = Re−i|A 12 x|2 = 0, where each term in (11.2.11) is well-defined. Thus, AF is dissipative.

Finally, since A−1F ∈ L(Y ) by (11.2.10), then (λ0 −AF )−1 ∈ L(Y ) as well, by a suitably small λ0 > 0, and

the range condition: range (λ0 − AF ) = Y is satisfied, so that AF is maximal dissipative. By Lumer-Phillipstheorem [35, p. 14], AF is the generator of a s.c. contraction semigroup on Y = L2(Ω) Theorem 11.1.1(i) isestablished.

(iii) Moreover, from the expression of A−1F in (11.2.9), it is clear that A−1

F is compact as well on Y = L2(Ω),

since so is A− 12 .

Dissipativity relations. Once the well-posedness part is established for the w-problem in (11.1.1), wedifferentiate E(t) in (11.1.2) at first for smooth data, use wt = i∆w and obtain in the ( , )L2(Ω):

dE(t)

dt= (wt, w) + (w,wt) = 2Re(w,wt) = −2Rei(w,∆w), (11.2.12)

where by Green’s second theorem and recalling the B.C. in (11.1.1c), we obtain

Rei(w,∆w) = Re

[

i

∫

Γ1

w∂w

∂νdΓ1

]

=

∫

Γ1

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

dΓ1. (11.2.13)

Substituting (11.2.13) in (11.2.12) yields (11.1.3a), as desired. One can also multiply the w-problem by w andintegrate by parts to obtain (11.1.3a). Theorem 11.1.1(ii) is proved. 2


Strong stability. The strong stability (11.1.4) follows, as usual, [15] by use of Stone’s theorem andNagy-Foias-Foguel’s theorem. Theorem 11.1.1(iv) is proved.

11.3 Proof of uniform stabilization: Theorem 11.1.2

Step 1. The key step of the proof, by far, is the lower-level (L2(Ω))-energy estimate established in Section10 and as stated in Theorem 2.6.1. We recall the result. Let w be a smooth solution of Eqn. (11.1.1a) andassume hypothesis (11.1.5): h · ν ≤ 0 on Γ0, where w|Γ0 ≡ 0 according to (11.1.1c) [the B.C. on Γ1 in (11.1.1c)is not imposed]. Then the following estimate holds true

E(T ) +E(0) ≤ CT

‖w‖2L2(Σ1) +

∫ T

0

∫

Γ1

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

|w|dΓ1dt+

∥

∥

∥

∥

∂w

∂ν

∥

∥

∥

∥

2

H−1a (Σ1)

+ ‖w‖2H−1(Q)

, (11.3.1)

recalling (11.1.2). We now impose the B.C. w|Γ1 = −i ∂w∂ν in (11.1.1c). Then (11.3.1) becomes

E(T ) +E(0) ≤ CT

∫ T

0

∫

Γ1

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

dΓ1dt+ ‖w‖2H−1(Q)

, (11.3.2)

since the H−1a (Σ1)-norm is dominated by the L2(Σ1)-norm.

Step 2. We want to establish that the last lower-order term on the RHS of inequality (11.3.2) can beabsorbed: i.e., that

‖w‖2H−1(Q) ≤ CT

∥

∥

∥

∥

∂w

∂ν

∥

∥

∥

∥

2

L2(Σ1)

(11.3.3)

for a suitable constant CT > 0. To this end, one seeks to employ a by now standard compactness/uniquenesscontradiction argument. To be carried out, this argument requires the following global uniqueness theorem

iψt + ∆ψ ≡ 0,

ψ|Σ ≡ 0,∂ψ

∂ν

∣

∣

∣

∣

Σ1

≡ 0,on [0, T ] ⇒ ψ ≡ 0, (11.3.4)

which is true by Holmgren’s uniqueness theorem.Once this is done, use of (11.3.3) in (11.3.2) yields

E(T ) +E(0) ≤ CT

∫ T

0

∫

Γ1

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

dΓ1dt. (11.3.5)

Step 3. Inequality (11.3.5) [with E(0) dropped] is of the usual type: “energy at T is dominated by theL2(0, T ; · )-dissipation.” A standard semigroup argument then allows us to establish that ‖eAFT ‖L2(Ω) < 1,and hence [1] that the uniform estimate (11.1.6) holds true. Theorem 11.1.2 is proved. 2

Remark 11.3.1. (Comparison with other uniform stabilization results) First, reference [15] gives a uniformstabilization result on the optimal space of regularity H−1(Ω) with a non-local feedback in the Dirichlet B.C.

w|Σ1 = i ∂(A−1w)∂ν on Σ1, instead of (11.1.1c). Next, reference [32] gives a uniform stabilization result in the

space H1(Ω) [which in this Neumann case is not the space of optimal regularity] by means of a feedback controlin the Neumann B.C.: w|Σ1 = −wt on Σ1.

As a comparison, using for the feedback controller the solution w rather than its velocity wt is more desirable.Moreover, (11.1.1c) is pointwise, rather than non-local. Finally, L2(Ω) is the space of optimal regularity in thiscase. 2

Remark 11.3.2. Consider the second-order differential operator

Aw = −n∑

i,j=1

∂

∂xi

(

aij(x)∂w

∂xj

)

, x = [x1, . . . , xn] (11.3.6a)


with real coefficients aij = aji of class C3, satisfying the uniform ellipticity condition

n∑

i,j=1

aij(x)ξiξj ≥ a

n∑

i=1

ξ2i , x ∈ Ω, (11.3.6b)

for some positive constant a > 0. Then, the main Theorem 11.1.2 of the present section is expected to holdtrue through the following line of argument. First, as noted in Remark 2.6.2, Theorem 2.6.1 holds true also ona Riemannian manifold M, g with Laplacian ∆ in (1.1.1) being replaced by the Riemannian Laplacian ∆g :one just starts from [45] rather than [44] in Section 10.1. This way one obtains inequality (11.3.2) for (11.1.1a)with ∆ replaced by A in (11.3.6a). In this case, the Riemannian manifold is Rn, g, with Riemannian metric gderived from the positive symmetric n× n matrix gij(x) = aij(x)−1. Next, a strategy as the one outlinedin Remark 2.6.1 would yield the required uniqueness result in (11.3.4).

Appendix A: The Most General Form Allowed for the Coefficient q1(t, x) of the Gradient Term∇w in (1.1.1), through Changes of Variables, and Time and Space Rescaling

Regarding the Carleman estimates stated in Section 2.1, we have seen that Theorem 2.1.1 (first version) wasobtained under the ‘minimal’ regularity assumption (1.1.3) on the coefficients (in L∞(Q)) of the differentialoperator F (w) in (1.1.2). In particular, no structural assumption was required on the coefficient q1(t, x) infront of the energy level term ∇w. By contrast, beginning with Theorem 2.1.2 (second version), we have seenthat additional assumptions on q1(t, x) are required, both on smoothness and, above all, structural, such asassumption (A.3) = (2.1.4). The purpose of this Appendix A is two-fold: (i) to give the most general formallowed here for the coefficient q1(t, x); (ii) to ascertain what version we can assume in the text without loss ofgenerality. The approach of this Appendix A is based on elementary computations within standard strategies:suitable changes of variables and time-space rescaling. The structural result on q1(t, x) here achieved withoutloss of generality will agree with results on Schrodinger operators with magnetic potential [36, Thm. X.22,p. 173].

One-dimensional case: dim Ω = 1. Even though the emphasis of this paper is on the multidimensionalcase, dim Ω ≥ 2, the following result in the case dim Ω = 1 may be explicitly noted.

Theorem A.1. Let dim Ω = 1. The 1-dimensional Schrodinger equation

iwt + wxx = F (w) = q1(t, x)wx + q0(t, x)w, (A.1)

under the assumptions thatq0, q1, q1t, q1x ∈ L∞(Q) (A.2)

is transformed into the new form

iwt + wxx = q0(t, x)w; q0 ≡ q0 −i

2πt +

1

4π2x −

1

2πxx ∈ L∞(Q), (A.3)

with no first-order term, by means of the transformation

w(t, x) = e−12π(t,x)w(t, x), π(t, x) ≡

∫ x

0

q1(t, ξ)dξ, (A.4)

so that πx = q1, πxx = q1x. A-fortiori, assumption (A.3) is then satisfied.

Proof. Direct verification with

iwt = ie−π2

[

wt −1

2πtw

]

; wxx = e−12π

[

wxx − πxwx −1

2πxxw +

(

1

2

)2

π2xw

]

, (A.5)

where then πx = q1 is responsible for the cancellation of the first-order term wx. 2

Multidimensional case: dim Ω ≥ 2.


Theorem A.2. Let dim Ω ≥ 2. Consider the Schrodinger equation (1.1.1), as specialized with coefficientq1(t, x) = ∇π(t, x) − ir1(t, x):

iwt + ∆w = (∇π(t, x) − ir1(t, x)) · ∇w + q0(t, x)w, (A.6)

π and r1 being real-valued functions subject to the assumptions

πt, |∇π|,∆π, |r1|, q0 ∈ L∞(Q). (A.7)

Then, modulo the successive changes of variables (A.11), (A.16) defined below, as well as the time-spacerescaling in (A.25) in Propositions A.3 and A.6 below, we can always assume without loss of generality that(A.6) has the following features:

∇π ≡ 0 in Q; and either |r1 · µ|2 ≤ r1 · ν < 1 on Γ; or else r1 · ν ≡ 0 on Γ1, (A.8)

where µ is a unit tangent vector and ν is the unit outward normal at Γ. 2

Proof. The proof of Theorem A.2 is obtained by combining Propositions A.3 through A.6 together. 2

Proposition A.3. Let dim Ω ≥ 2. (i) Equation. The Schrodinger equation of the form (A.6), with π andr1 real-valued functions subject to assumptions (A.7), is transformed into the new form

iwt + ∆w = −ir1(t, x) · ∇w + q0(t, x)w; (A.9)

q0 =

[

q0 −i

2πt +

1

4|∇π|2 − 1

2∆π − i

2r1 · ∇π

]

∈ L∞(Q), (A.10)

that is, with purely imaginary coefficient of the energy term ∇w, by means of the transformation

w(t, x) = e−12π(t,x)w(t, x). (A.11)

(ii) Boundary conditions. Moreover, the change of variable w → w in (A.11) changes the boundary condi-tions

∂w

∂ν+ βw = 0; resp.

∂w

∂ν+ wt = g on Σ, (A.12)

for w, into the boundary conditions

∂w

∂ν+

(

1

2

∂π

∂ν+ β

)

w = 0; resp.∂w

∂ν+ wt +

1

2

(

∂π

∂ν+ πt

)

w = e−π2 g (A.13)

for w.

Proof. Direct verification, with iwt given by the same expression (A.5) (left), and with

wxjxj= e−

12πwxjwj

− e−12ππxjwj

+ e−12π

(

1

4π2xj

− 1

2πxjxj

)

w, (A.14)

see (A.5) (right), hence

∆w = e−12π∆w − e−

12π∇π · ∇w + e−

12π

(

1

4|∇π|2 − 1

2∆π

)

w, (A.15)

along with final use of (A.11) and of

e−12πr1 · ∇w = r1 · ∇w +

1

2(r1 · ∇π)w. 2 (A.16)


Remark A.1. The case of a purely imaginary coefficient q1 = −ir1, r1 real, of the energy level term ∇wcorresponds to the well-known case of Schrodinger operator with magnetic potential; see [36, Theorem X.22,p. 173].

Proposition A.4. Let dim Ω ≥ 2. Let p(t, x) be a real-valued scalar function, subject to the assumptions

pt, |∇p|, ∆p ∈ L∞(Q). (A.17)

(i) Equation. Then, the change of variable

w(t, x) = e−i2p(t,x)w(t, x) (A.18)

transforms the Schrodinger equation

iwt + ∆w = −ir1(t, x) · ∇w + q0(t, x)w, (A.19)

with r1, q0 ∈ L∞(Q) into the new form

iwt + ∆w = −i(r1(t, x) + ∇p(t, x)) · ∇w + q0(t, x)w; (A.20)

q0 =

[

q0 +1

2pt −

i

2∆p− 1

4|∇p|2 +

1

2(r1 + ∇p) · ∇p

]

∈ L∞(Q), (A.21)

still with purely imaginary coefficient of the energy term ∇w. That is, the purely imaginary coefficient −ir1 ofthe energy level term ∇w in (A.17) is transformed into the still purely imaginary coefficient −ir1 ≡ −i(r1+∇p)of the energy term ∇w.

(ii) Boundary conditions. Moreover, the change of variable w → w in (A.16) changes the boundary condi-tions

∂w

∂ν+ βw = 0; resp.

∂w

∂ν+ wt = g on Σ, (A.22)

for w, into the boundary conditions

∂w

∂ν+

(

i

2

∂p

∂ν+ β

)

w = 0; resp.∂w

∂ν+ wt +

i

2

(

∂p

∂ν+ pt

)

w = e−i2pg (A.23)

for w.

Proof. Direct verification, with

wt = e−i2p

[

wt −i

2ptw

]

; (A.24)

wxjxj= e−

i2pwxjxj

− ie−i2ppxj

wxj− e−

i2p

(

1

4p2xj

+i

2pxjxj

)

w (A.25)

(compare with (A.5) (left) and (A.12)), hence

∆w = e−i2 p∆w − ie−

i2 p∇p · ∇w − e−

i2p

(

1

4|∇p|2 +

i

2∆p

)

w, (A.26)

along with final use of (A.16) and of

e−i2 p(−ir1) · ∇w = (−ir1) · ∇w +

i

2w(−ir1) · ∇p; −ir1 = −i(r1 + ∇p). 2 (A.27)


Proposition A.5. With reference to Proposition A.4, given the original real coefficient r1, it is alwayspossible to select, in infinitely many ways, a smooth real function p such that

either infΓ

[(r1 + ∇p) · ν] = infΓ

[

r1 · ν +∂p

∂ν

]

> 0, or else r1 · ν +∂p

∂ν= 0, on Γ0. (A.28)

by means of the inverse trace theorem. 2

Proposition A.6. Let a be a positive parameter.(i) Equation. The time/space rescaling

t′ = a2t, x = ax′, (A.29)

transforms the original Schrodinger equation,

iwt + ∆xw = −ir1 · ∇xw + q0w, (A.30)

into the new equationiWt′ + ∆x′W = −i(ar1) · ∇x′W + a2q0W ; (A.31)

that is, the original coefficient r1 is transformed into (ar1). A-fortiori, at the price of invoking Proposition A.5—that is, at the price of replacing (ar1) with [(ar1)+∇p)] for some suitable p as to make infΓ[(ar1)+∇p]·ν > 0—we can always achieve, in addition, the following condition on Γ:

on Γ : |(ar1) · µ|2 ≤ (ar1) · ν < 1 (A.32)

by choosing a > 0 sufficiently small. [Choosing a > 0 sufficiently small corresponds to suitably expanding theoriginal time t and original space variable x (hence original domain Ω) into the new time scale t′ and the newspace scale x′ (hence new domain Ω′).

(ii) Boundary conditions. Moreover, the rescaling t → t′ and x → x′ in (A.24) transforms the boundaryconditions

∇xw · ν + βw = 0; resp. ∇xw · ν + wt = g on Σ, (A.33)

for w(t, x), into the boundary conditions

∇x′W · ν + aβW = 0; resp. ∇x′W · ν +1

aWt′ = ag on Σ (A.34)

for W (t′, x′).

Proof. We setw(t, x) = w(a2t′, ax′) ≡W (t′, x′), (A.35)

so thatWt′(t

′, x′) = a2wt(t, x); Wx′

j(t′, x′) = awxj

(t, x); Wx′

jx′

j(t′, x′) = a2wxjxj

(t, x); (A.36)

∇x′W (t′, x′) = a∇xw(t, x); ∆x′W (t′, x′) = a2∆xw(t, x). (A.37)

Multiplying (A.26) across by a2 yields, by (A.30), (A.31),

ia2wt + a2∆xw = iWt′ + ∆x′W = −i(ar1) · a∇xw + a2q0w = −i(ar1) · ∇x′W + a2q0W, (A.38)

and (A.31) is established. 2

Remark A.2. Schrodinger equation. We have emphasized in the statement of Proposition A.6, thata small coefficient (ar1) of the energy level term ∇x′W comes at the price of correspondingly expanding theoriginal domain Ω by a factor “a” via x = ax′. Thus, if we attempt to use the classical multiplier [13], say,(x′ − x′0) · ∇x′W to Eqn. (A.27) with small coefficient (ar1) (i.e., with small parameter a), we would obtain

[(ar1) · ∇x′W ](x′ − x′0) · ∇x′W = [(ar1) · ∇x′ ]W(x− x0)

a· ∇x′W, (A.39)


and the achieved benefit of a small coefficient (ar1) is wiped out by a corresponding large vector field x−x0

a , sothat the classical multiplier—which is “perfect” for the canonical Schrodinger equation [15], with no energy levelterms—is not appropriate any longer in the presence of energy level terms, for the purpose of seeking continuousobservability/uniform stabilization inequalities. Thus, more sophisticated multipliers [37-39], [44], are called forin the presence of energy level terms. The same phenomenon occurs in the case of, say, second-order hyperbolicequations [3], [19], [37–39], [46].

Suppose that we start with the original second-order hyperbolic equation

wtt − ∆xw = q1 · ∇xw + q2wt (A.40)

with energy level terms ∇xw and wt. We apply the rescaling

t = at′; x = ax′, (A.41)

with the same parameter a > 0, and set

w(t, x) = w(at′, ax′) ≡W (t′, x′). (A.42)

Then, computations as in (A.29), (A.30) lead to the equation for W (t′, x′) in the form

Wt′t′ − ∆x′W = (aq1) · ∇x′W + (aq2)Wt′ , (A.43)

which has arbitrarily small coefficients (aq1) and (aq2) of the energy level terms for W by taking the parametera > 0 sufficiently small. As a consequence, however, the original domain Ω results correspondingly expandedby a factor “a” and a phenomenon such as the one described in (A.33) (now, with no conjugate sign) occurs:the small coefficients of the energy level terms are compensated by the correspondingly large vector fieldx′ −x′0 = x−x0

a , so that the classical multiplier (x′ −x′0) · ∇x′W is no longer suitable for the purpose of seekingcontinuous observability/uniform stabilization inequalities of wave equations with energy level terms. Hereagain more sophisticated multipliers are called for [37], [38], [39], [3], [19], [22], or approaches [1].

Appendix B: Proof of Identities (6.15) and (6.11a)

Identity (6.15). Let q1 = [q11, . . . , q1n], iq1 = r1 = [r11, . . . , r1n]. Direct computations yield

∇w · ∇(q1 · ∇w) = J trq1∇w · ∇w +

n∑

i=1

q1i∇wxi· ∇w, (B.1)

where J trq1 is defined as in (6.8b). Then (B.1) implies

Rei∇w · ∇(q1 · ∇w) = ReJ trr1∇w · ∇w +

n∑

i=1

r1i Re∇xi · ∇w, (B.2)

where we compute

2 Re∇wxi· ∇w =

n∑

j=1

2 Rewxixjwxj

=

n∑

j=1

∂

∂xi|wxj

|2 =∂

∂xi|∇w|2. (B.3)

Substituting (B.3) in (B.2) yields

Rei∇w · ∇(q1 · ∇w) = ReJ trr1∇w · ∇w +1

2

∑

i=1

r1i∂

∂xi|∇w|2 = ReJ trr1∇w · ∇w +

1

2r · ∇(|∇w|2), (B.4)

and identity (6.15) is proved.


Proof of Identity (6.11). By (6.5),

E(t) − E(s) = 2 Re

∫ t

s

∫

Γ

∂w

∂νwtdΓ dσ

+ 2 Re

i

∫ t

s

∫

Γ

∂w

∂νwdΓ dσ

+ 2 Re

∫ t

s

∫

Ω

[i∆wF (w)dΩ dσ

− 2 Re

∫ t

s

∫

Ω

iw F (w)dΩ dσ

+ 2 Re

∫ t

s

∫

Ω

i∆wf dΩ dσ

− 2 Re

∫ t

s

∫

Ω

iw f dΩ dσ

.

(B.5)

Substituting (6.9) for the third integral term and (6.10) for the fifth term, we obtain

E(t) − E(s) = 2 Re

∫ t

s

∫

Γ

[

∂w

∂νwt + i

∂w

∂νw

]

dΓ dσ

+ 2

[∫ t

s

∫

Γ

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

r1 · ν dΓ dσ + Re

∫ t

s

∫

Γ

∂w

∂ν

∂w


− 1

2

∫ t

s

∫

Γ

|∇w|2r1 · ν dΓ dσ

+ Re

i

∫ t

s

∫

Γ

∂w

∂νq0w dΓ dσ

− Re

∫ t

s

∫

Ω

J trr1∇w · ∇w dΩ dσ

− 1

2

∫ t

s

∫

Ω

|∇w|2div r1 dΩ dσ

− Re

i

∫ t

s

∫

Ω

q0|∇w|2dΩdσ

− Re

i

∫ t

s

∫

Ω

w∇w · ∇q0dΩdσ]

− 2 Re

i

∫ t

s

∫

Γ

∂w

∂νf dΓ dσ

+ 2 Re

i

∫ t

s

∫

Ω

∇w · ∇f dΩ dσ

− 2 Re

∫ t

s

∫

Ω

iw[F (w) + f ]dΩ dσ

; (B.6)

E(t) − E(s) = 2 Re

∫ t

s

∫

Γ

∂w

∂ν[wt + iq0w − if ]dΓ dσ + 2 Re

i

∫ t

s

∫

Γ

∂w

∂νw dΓ dσ

+ (1); (B.7)

(1) =

∫ t

s

∫

Γ

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

r1 · ν dΓ dσ + 2 Re

∫ t

s

∫

Γ

∂w

∂ν

∂w


−∫ t

s

∫

Γ

∣

∣

∣

∣

∂w

∂µ

∣

∣

∣

∣

2

r1 · ν dΓ dσ

− 2 Re

∫ t

s

∫

Ω

[J trr1∇w · ∇w + iq0|∇w|2 + iw∇w · ∇q0

− i∇w · ∇f + iw[F (w) + f ] − 1

2|∇w|2div r1]dΩ dσ

. (B.8)

Then (B.8) proves (6.11a) as desired.

Appendix C: Generation of S.C. Semigroups with Classical Robin or Dissipative B.C.In this appendix, for simplicity, we assume hypothesis (A.4) = (2.2.2) that:

the coefficients r1 and q0 of F are time-independent. (C.0)

This assumption is only for convenience, as it will allow us to work within semigroup theory and avoid the time-dependent theory of evolution operators. Our goal is to collect here relevant results stating that Eqn. (1.1.1),under either classical Robin or dissipative B.C., generates appropriate s.c. semigroups, if moreover, (C.0) isassumed.

C.1 ‘Classical’ B.C. of Robin/Neumann type


Here we consider Eqn. (1.1.1), at first under Robin B.C.:

wt = i∆w − r1 · ∇w − iq0w in (0, T ]× Ω = Q;

w(0, · ) = w0 in Ω;

∂w

∂ν+ βw = 0 in (0, T ]× Γ = Σ.

(C.1a)

(C.1b)

(C.1c)

where β is an L∞-function, generally complex, on Γ, not identically zero, say β 6= 0 on some portion Γ1 6= ∅ ofΓ of positive measure. Neglecting the l.o.t. q0w, we define accordingly the operators

Aw = i∆w − r1 · ∇w ARw = −∆w;

D(A) =

w ∈ H2(Ω) :∂w

∂ν+ βw|Γ = 0

= D(AR),

(C.2a)

(C.2b)

and we note thatH1(Ω) ≡ D(A

12

R) (equivalent norms). (C.3)

Theorem C.1. (1) In addition to the standing assumptions (1.3.3) and (A.3) = (2.1.9) (and (C.0)) on F ,assume further that

|r1 · µ|2 ≤ r1 · ν on Γ, (C.4)

(a subset of (2.1.9c)). Then: (i) the translated operator (A − k2I) is dissipative on D(A12

R), indeed, maximaldissipative, for a suitably large translation constant k2 which depends on |β| on r1.

(ii) Thus, (A− k2I) generates a s.c. contraction semigroup on D(A12

R). In particular, problem (C.1) defines

a s.c. semigroup w0 → w(t) = eAtw0 on D(A12

R), satisfying ‖eAt‖ ≤ ek2t, t ≥ 0.

(2) Assumption (C.4) is redundant: more precisely, under the sole assumptions (1.3.3), (A.3) = (2.1.9) and(C.0) [but without assumption (C.4)], the following stronger conclusion holds true: problem (C.1) in the more

general version with (C.1a) replaced by wt = i∆w− i∇π− r1 ·∇w− iq0w generates a s.c. semigroup on D(A12

R)(non-necessarily contraction). 2

Proof. (1) (i) Dissipativity. We must show that there exists a positive constant k2 sufficiently large(depending on r1 and |β|), such that

Re((A− k2I)w,w)D(A

12R

)≤ 0, ∀ w ∈ D(A). (C.5)

To this end, we take w ∈ D(A) = D(AR) and compute preliminarily from (C.2a):

Re(Aw,w)D(A

12R

)= Rei(∆w,

*ARw)L2(Ω) − Re(r1 · ∇w,ARw)L2(Ω), (C.6)

where the first term vanishes, since ∆w = −ARw for w ∈ D(A) = D(AR). Furthermore, still for such w, wecompute, recalling inequality (6.8a) with q1 = −ir1, as well as

∣

∣

∂w∂ν

∣

∣ = |β| |w| from (C.1c):

−Re(r1 · ∇w,ARw)L2(Ω) = Re

∫

Ω

∆w r1 · ∇w dΩ)

≤∫

Γ

|β|2|w|2dΓ − 1

2

∫

Γ

∣

∣

∣

∣

∂w

∂µ

∣

∣

∣

∣

2

[r1 · ν − |r1 · µ|2]dΓ

− Re

∫

Ω


− 1

2

∫

Ω

|∇w|2 div r1 dΩ. (C.7)

Using (C.7a) in (C.6) yields

Re(Aw,w)D(A

12R

)= − 1

2

∫

Γ

∣

∣

∣

∣

∂w

∂µ

∣

∣

∣

∣

2

[r1 · ν − |r1 · µ|2]dΓ + Oβ,r1

(

‖w‖2H1(Ω)

)

, (C.8)


with constant in O depending on |β| and r1, where in going from (C.7) to (C.8), we have used trace theory onthe first integral over Γ in (C.7). Thus, by (C.8), we see that there exists a constant k2, depending on r1 and|β|, such that

Re((A− k2I)w,w)D(A

12R

)≤ −1

2

∫

Γ

∣

∣

∣

∣

∂w

∂µ

∣

∣

∣

∣

2

[r1 · ν − |r1 · µ|2]dΓ ≤ 0, (C.9)

by invoking assumption (C.4), and thus (C.5) holds true, as desired.

Maximal dissipativity. We must next show the range condition: that for k2 sufficiently large, for λ > 0,and g ∈ H1(Ω), there exists a unique f ∈ D(A) such that [λ− (A− k2I)]f = g. This yields, via (C.2) for A, acorresponding elliptic problem:

−i∆f + r1 · ∇f + (λ + k2)f = g on Ω;

∂f

∂ν+ βf = 0 on Γ,

(C.10a)

(C.10b)

and elliptic theory yields the desired conclusion: given g ∈ H1(Ω), there is a (unique) f ∈ D(A) such thatproblem (C.10) holds true.

(2) Let now F (w) = (∇π − ir1) · ∇w + q0w, as in the general case of assumption (A.3) = (2.1.9). Then,smooth changes of variables given in (A.11) [to obtain ∇π ≡ 0] and in (A.18), (A.29) [to obtain property(C.4) (see the stronger property (A.32) = (2.1.9c)), have the innocuous boundary effect of simply transforming

β into[(

12∂π∂ν + i ∂p∂ν

)

+ β]

. Then, part (1) can be applied to the transformed problem initially for q0 ≡ 0.

But addition of the bounded perturbation a2q0w to the generator, where the time/space scaling constant a isdefined in (A.29), q0 ∈ L∞(Ω), preserves the property of semigroup generation. Finally, the original general

problem defines a s.c. semigroup on D(A12

R).

Remark C.1. Let β ≡ 0 in (C.1c). Then, we must consider the state space H = H1(Ω)/const. given byH1(Ω) quotient the constant functions, topologized by the gradient norm. For w ∈ D(A), we then computewith reference to (C.2) with β ≡ 0:

Re(Aw,w)H = Re(i∆w − r1 · ∇w,w)H = Re

i

∫

Ω

∇∆w · ∇w dΩ

(C.11)

− Re

∫

Ω

∇(r1 · ∇w) · ∇w dΩ

, w ∈ D(A). (C.12)

By using the B.C. (C.2b) with β ≡ 0, we find by Green’s theorem,

0 = Re

i

∫

Ω

∆w∆w dΩ

= −Re

i

∫

Ω

∇w · ∇∆w dΩ

, (C.13)

for the first term in (C.12), while for the second term in (C.12) we invoke identity (6.15). We thus find by(C.13) and (6.15) with iq1 = −r1 followed by (6.18):

Re(Aw,w)H = −Re

∫

Ω

∇(r1 · ∇w) · ∇w dΩ

(C.14)

= −Re

∫

Ω


− 1

2

∫

Γ

|∇w|2r1 · ν dΓ

+1

2

∫

Ω

|∇w|2div r1 dΩ, ∀ w ∈ D(A) (C.15)

= − 1

2

∫

Γ

|∇w|2r1 · ν dΓ + O(∫

Ω

|∇w|2dΩ)

, (C.16)


where in the last step from (C.15) to (C.16) we have used again, that the interior terms in (C.15) are allgradient terms: Thus, there exists a sufficiently large constant k2, depending on r1, such that for w ∈ D(A)we have

Re([A− k2I ]w,w)H ≤ −1

2

∫

Γ

|∇w|2 r1 · ν dΓ ≤ 0, w ∈ D(A), (C.17)

for r1 · ν ≥ 0 on Γ. We have thus shown that dissipativity of (A − k2I) again holds true, this time on H andby virtue only of assumption: r1 · ν ≥ 0 on Γ contained in (C.4).

Maximal dissipativity. This is proved as in the case of Theorem C.1. 2

C.2 Dissipative B.C.Here, we consider now Eqn. (1.1.1) with dissipative B.C.

wt = i∆w − r1 · ∇w in (0, T ]× Ω ≡ Q;

w(0, · ) = w0 in Ω;

∂w

∂ν+ wt = 0 in (0, T ]× Γ ≡ Σ,

(C.18a)

(C.18b)

(C.18c)

where we have deliberately neglected the l.o.t. iq0w in (C.18a). We notice that, using (C.18a), we can rewritethe B.C. (C.18c) as

∂w

∂ν+ i∆w − r1 · ∇w = 0 on Γ, (C.19)

just in terms of w. Accordingly, we define the operator

Aw=i∆w − r1 · ∇w;

D(A)=

w ∈ H2(Ω) : ∆w|Γ ∈ L2(Γ) and

[

∂w

∂ν+ i∆w − r1 · ∇w

]

Γ

= 0

.

(C.20a)

(C.20b)

As in the case of Remark C.1, we shall consider the space

H = H1(Ω)/constants, with gradient inner product ( , )H. (C.21)

Theorem C.2. (1) In addition to the standing assumptions (1.3.3) and (A.3) = (2.1.9) (and (C.0)) on F ,assume further that

|r1 · µ|2 ≤ r1 · ν ≤ 1 on Γ. (C.22)

Then: (i) the translated operator (A− k2I) is dissipative on H, indeed, maximal dissipative, for a suitablylarge translation constant k2 depending on r1. (ii) Thus, (A− k2I) generates a s.c. contraction semigroup on

H. In particular, problem (C.18) defines a s.c. semigroup w0 → w(t) = eAtw0 on H, satisfying ‖eAt‖ ≤ ek2t,

t ≥ 0.(2) Assumption (C.22) is redundant: more precisely, under the sole assumption (1.3.3), (A.3) = (2.1.9) and

(C.0) [but without assumption (C.22)], the following stronger conclusion holds true: problem (C.18) in themore general version with (C.1a) replaced by wt = i∆w− i∇π − r1 · ∇w− iq0w, generates a s.c. semigroup onH (non-necessarily contraction. 2

Proof. (1) (i) Our starting point is Eqn. (C.12), this time, however, for the operator A defined in (C.20):

Re(Aw,w)H = Re(i∆w − r1 · ∇w,w)H (C.23)

= Re

i

∫

Ω

∇∆w · ∇w dΩ

− Re

∫

Ω

∇(r1 · ∇w) · ∇w dΩ

, w ∈ D(A). (C.24)


Using this time the B.C. in (C.19) or (C.20b) to extract

on Γ : i∆w = − ∂w

∂ν+ r1 · ∇w = − ∂w

∂ν+∂w

∂ν(r1 · ν) +

∂w

∂µ(r! · µ), (C.25)

using r1 = (r1 · ν)ν + (r1 · µ)µ on Γ, we find now

0 = Re

i

∫

Ω

∆w∆w dΩ

= Re

i

∫

Γ

∂w

∂ν∆w dΓ

− Re

i

∫

Ω

∇∆w · ∇w dΩ

=

∫

Γ

[r1 · ν − 1]

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

dΓ + Re

∫

Γ

∂w

∂ν

∂w

∂µr1 · µ dΓ

− Re

i

∫

Ω

∇∆w · ∇w dΩ

, (C.26)

from which we derive the identity

Re

i

∫

Ω

∇∆w · ∇w dΩ

=

∫

Γ

[r1 · ν − 1]

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

dΓ + Re

∫

Γ

∂w

∂ν

∂w

∂µr1 · µ dΓ

. (C.27)

We return to (C.24) and substitute here (C.27) for its first term, and (C.14)–(C.16) for its second term.We thus obtain for w ∈ D(A):

Re(Aw,w)H =

∫

Γ

[r1 · ν − 1]

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

dΓ − 1

2

∫

Γ

|∇w|2r1 · ν dΓ

+ Re

∫

Γ

∂w

∂ν

∂w

∂µr1 · µ dΓ

+ O(∫

Ω

|∇w|2dΩ)

. (C.28)

But using |∇w|2 =∣

∣

∂w∂ν

∣

∣

2+∣

∣

∣

∂w∂µ

∣

∣

∣

2

in the second term on the RHS of (C.28) as well as the following estimate:

1

2

∣

∣

∣

∣

Re

∫

Γ

2∂w

∂ν

∂w

∂µr1 · µ dΓ

∣

∣

∣

∣

≤ 1

2

∫

Γ

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

+1

2

∫

Γ

∣

∣

∣

∣

∂w

∂µ

∣

∣

∣

∣

2

|r1 · µ|2dΓ, (C.29)

in the third term on the RHS of (C.28), we then obtain for w ∈ D(A):

Re(Aw,w)H =1

2

∫

Γ

[r1 · ν − 1]

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

dΓ

+1

2

∫

Γ

∣

∣

∣

∣

∂w

∂µ

∣

∣

∣

∣

2

[|r1 · µ|2 − r1 · ν]dΓ + O∫

Ω

|∇w|2dΩ

. (C.30)

Thus, there exists a sufficiently large constant k2, depending on r1, such that for w ∈ D(A) we have

Re([A− k2I ]w,w)H ≤ 1

2

∫

Γ

[r1 · ν − 1]

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

dΓ

+1

2

∫

Γ

∣

∣

∣

∣

∂w

∂µ

∣

∣

∣

∣

2

[|r1 · µ|2 − r1 · ν]dΓ ≤ 0, w ∈ D(A), (C.31)

where we have used assumption (C.22) to claim negativity:

Re([A− k2I ]w,w)H ≤ 0, ∀ x ∈ D(A), (C.32)


and dissipativity of [A− k2I ] on H, k2 sufficiently large, is proved.

Maximal dissipativity. We must show the range condition: that for k2 sufficiently large, given a λ > 0,any g ∈ H, there exists a unique solution f ∈ D(A) such that [λ− (A− k2I)]f = g. This yields, via (C.20) forA, a corresponding elliptic problem

−i∆f + r1 · ∇f + (λ+ k2)f = g in Ω;

∂f

∂ν+ i∆f − r1 · ∇f = 0 in Γ,

(C.33a)

(C.33b)

or using (C.33a) in (C.33b):

−i∆f − r1 · ∇f + (λ+ k2)f = g in Ω;

∂f

∂ν+ (λ+ k2)f |Γ = g|Γ in Γ,

(C.34a)

(C.34b)

and elliptic theory yields the desired conclusion: given g ∈ H1(Ω), there exists a (unique) f ∈ D(A) such thatproblem (C.34) holds true.

(2) Same as the proof of part (2) of Theorem C.1, mutatis mutandi. Let now F (w) = (∇π−ir1)·∇w+q0w, asin the general case of assumption (A.3) = (2.1.9). Then, smooth changes of variables given in (A.11) [to obtain∇π ≡ 0] and in (A.18), (A.29) [to obtain property (C.22)] have the boundary effect of simply transforming theboundary condition (C.18c) into

∂w

∂ν+ wt +

[

1

2

(

∂π

∂ν+ πt

)

+i

2

(

∂p

∂ν+ pt

)]

w = 0, (C.35)

in the transformed new variable w. But this new B.C. introduces a lower-order term. Thus, for the problemwithout this l.o.t., part (1) applies. Then the full problem in w with B.C. (C.35) generates a s.c. semigroup onH. 2

Appendix D: (i) Integral Identities and (ii) Estimate in L2(Σ) of the Tangential Trace ∇tanwfor H2,2(Q)-Solutions w of Eqn. (1.1.1) Needed in Section 7.2

In this Appendix D, we collect from the literature an integral identity satisfied by H2,2(Q)-solutions of theSchrodinger equation (1.1.1). As a consequence, we derive an L2(Σ)-estimate for the tangential gradient ∇tanwof such H2,2(Q)-solutions.

Energy identities. From the literature [15], [18], we obtain

Proposition D.1. Let w ∈ H2,2(Q) be a solution of the Schrodinger Eqn. (1.1.11), with no B.C. imposedsubject to (1.1.3) on F and (1.1.4) on f . Then, the following identities hold true:

(i)

Re

∫

Σ

∂w

∂νh · ∇w dΣ

− 1

2

∫

Σ

|∇w|2h · ν dΣ +i

2

∫

Σ

wtwh · ν dΣ = Re

∫

Q

H∇w · ∇w dQ

− 1

2

∫

Q

|∇w|2div h dQ+i

2

∫

Q

wtw div h dQ+i

2

[∫

Ω

wh · ∇w dΩ]T

0

+ Re

∫

Q

[F (w) + f ]h · ∇w dQ

,

(D.1)

where h(x) = [h1(x), . . . , hn(x)] is any real vector field in [C1(Ω)]n and where

H(x) =

∂h1

∂x1, · · · , ∂h1

∂xn...

...

∂hn∂x1

, · · · , ∂hn∂xn

, ν(x) = outward unit normal vector at x ∈ Γ. (D.2)


(ii) Let now h ∈ [C2(Ω)]n. Then

Re

∫

Σ

∂w

∂νh · ∇w dΣ

− 1

2

∫

Σ

|∇w|2h · ν dΣ +i

2

∫

Σ

wtw h · ν dΣ +1

2

∫

Σ

∂w

∂νw div h dΣ

= Re

∫

Q

H∇w · ∇w dQ

+1

2

∫

Q

w∇w · ∇(div h)dQ

+i

2

[∫

Ω

wh · ∇w dΩ]T

0

+ Re

∫

Q

[F (w) + f ]h · ∇w dQ

+1

2

∫

Q

[F (w) + f ]w div h dQ. (D.3)

Proof. (i) Identity (D.1) is given in [15, Section 2.1], [18, Lemma 10.9.10.1, p. 1052], for iwt = ∆w + frather than (1.1.1). It is obtained by multiplying Eqn. (1.1.1) by h ·∇w and integrating by parts. (ii) To obtain(D.3) from (D.1) one further multiplies Eqn. (1.1.1) by w div h. Identity (D.3), as specialized to homogeneousDirichlet B.C. is in [13, Lemma 2.1]. These final identities are derived for H2,2(Q)-solutions, e.g., in the use ofthe identity

Re∇w · ∇(h · ∇w) = Re H∇w · ∇w+h

2· ∇(|∇w|2) (D.4)

(‘Re’ is accidentally missing in the above references which use identity (D.4)).

L2(Σ)-estimate for the tangential gradient ∇tanw. As a corollary of Proposition D.1, we obtain

Proposition D.2. Let w ∈ H2,2(Q) be a solution of the Schrodinger Eqn. (1.1.1), with no B.C. imposed,subject to (1.1.3) on F (w) and (1.1.4) on f . Then, the following estimate on tangential gradient holds true;there is a positive constant Ch—depending on h, where we have specialized h as to achieve h · ν = 1, such that

∫ T

0

∫

Γ

|∇tanw|2dΣ ≤ Ch

∫ T

0

∫

Γ

[

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

+ |w|2]

dΣ +

∫ T

0

‖wt|Γ‖2H−1(Γ)dt+ O(E( · ), f), (D.5)

where we have set for convenience

O(E( · ), f) ≡ O∫ T

0

E(t)dt+ E(T ) + E(0) + ‖f‖2L2(Q)

, (D.6)

with E(t) defined by (1.1.11).

Proof. By (1.1.3), (1.1.4), (D.2), we see at once that the right-hand side of identity (D.3) satisfies by (D.6):

RHS of (D.3) = O(E( · ), f). (D.7)

On the other hand, to handle the left-hand side of identity (D.3), we select preliminarily a vector field h withthe additional property that h · ν ≡ 1 on Γ, write: h = (h · ν)ν + (h · µ)µ = ν + (h · µ)µ, for a unit tangentialvector µ and obtain

h · ∇w =∂w

∂ν+∂w

∂µh · µ; |∇w|2 = |∇tanw|2 +

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

, |∇tanw| =

∣

∣

∣

∣

∂w

∂µ

∣

∣

∣

∣

. (D.8)

Thus, by (D.8) used in the LHS of (D.3), we obtain via (D.7)

LHS of (D.3) =1

2

∫

Σ

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

dΣ + Re

∫

Σ

∂w

∂ν

∂w

∂µh · µ dΣ

+i

2

∫

Σ

wtw dΣ − 1

2

∫

Σ

|∇tanw|2dΣ

(by (D.7)) = RHS of (D.3) = O(E( · ), f). (D.9)


From (D.9), extracting the tangential gradient, we obtain for ε > 0 arbitrarily small:

1

2

∫

Σ

|∇tanw|2dΣ ≤ 1

2

∫

Σ

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

dΣ + ε

∫

Σ

|∇tanw|2dΣ +Chε

∫

Σ

∣

∣

∣

∣

∂w

∂ν

∣

∣

∣

∣

2

dΣ

+ε

4

∫ T

0

|w|2H1(Γ)dt+1

4ε

∫ T

0

|wt|2H−1(Γ)dΓ + O(E( · ), f). (D.10)

But, |w|2H1(Γ) = |∇tanw|2L2(Γ) + |w|2L2(Γ) and hence (D.10) yields estimate (D.5), as desired.

Appendix E: Proof of Properties on Commutators Needed in Section 10

In this appendix we prove some properties of commutators, which are critically needed in Section 10. Theproofs are based on asymptotic expansions of symbol [42]. In short, we shall repeatedly make use of thefollowing result:

Let:

A be a pseudo-differential operator with symbol a(t, x, y;σ, ξn, η)

B be a pseudo-differential operator with symbol b(t, x, y;σ, ξn, η)

Then the Principal symbol (Pr Sym) of the commutator [A,B] is given by

Pr Sym [A,B] =

Dσξnη

a

Dtxy

b

−

Dσξnη

b

Dtxy

a

, (E.1)

where D denotes derivative.

1. We begin by proving the relations in (10.4.34), (10.4.35).

Lemma E.1. With reference to Section 10.4 we have:

(i)

[Λ−1tan,P]; [Λ−1

tanβz,P] ∈ S1,−1a ; (E.2)

(ii)

[z, [Λ−1tanβz,P]] ∈ S1,−2

a ; (E.3)

(iii)

[eτϕ, [Λ−1tanβz,P]] ∈ S1,−2

a ; (E.4)

(iv)

[eτϕ, [z,Λ−1tanβz,P]] ∈ S1,−3

a . (E.5)

(v)

[Λ−2tan,P] ∈ S0,−1

a ; [zeτϕ,Λtan] ∈ S0,0. (E.6)

Proof of (i) = (E.2). It is enough to establish the second claim. We set

i1 = symb[Λ−1tanβz] =

β(x, y)z(t, x, y)

(σ2 + |η|4) 14

i2 = symb[P] = iσ + ξ2n + a(x, y)|η|2 + l.o.t.,

(E.7a)

(E.7b)


recalling the Melrose-Sjostrand’s representation (10.3.2) (in suitable coordinates) of the Laplacian. Using (E.1)for the symbols i1 and i2 in (E.7), we obtain

Pr Sym[Λ−1tanβz,P] =

Dηi1

Dyi2

−

Dσξnη

i2

Dtxy

i2

, (E.8)

= − βz 4|η|34(σ2 + |η|4) 5

4

ay|η|2 − iβzt

(σ2 + |η|4) 14

− 2ξnβxz + βzx

(σ2 + |η|4) 14

− 2a|η| βyz + βzy

(σ2 + |η|4) 14

= ξnβxz + βzx

(σ2 + |η|4) 14

+ l.o.t. ∈ s1,−1a . (E.9)

In (E.8) we have used that the symbol i1 does not depend on ξn, while the symbol i2 does not depend ont. The symbol on (E.9) is of order 1 in the normal direction ξn, and of anisotropic symbol −1 in the tangentialdirection σ, η. Thus, (E.9) proves (E.2).

Proof of (ii) = (E.3). We set: i1 = z(t, x, y), i2 = Pr Sym[Λ−1tanβz,P] given by (E.9). With these

symbols, we then apply (E.1) and obtain since i1 does not depend on (σ, ξn, η):

Pr Sym [z, [Λ−1tanβz,P]] = −

Dσξnη

i2

Dtxy

i1

, (E.10)

= −ξn(βxz + βzx)2σ

(σ2 + |η|4) 54

zt −βxz + βzx

(σ2 + |η|4) 14

zx − ξn(βxz + βzx)4|η|3

(σ2 + |η|4) 54

zy. (E.11)

But the second term in (E.11) is lower order, while, regarding the first and third term, we have

σ

(σ2 + |η|4) 54

∈ s0,−3a ,

|η|3(σ2 + |η|4) 5

4

∈ s0,−2a , (E.12)

since, respectively, we have that

σ

(σ2 + |η|4) 54

[(σ2 + |η|4) 14 ]x =

σ

(σ2 + |η|4) 5−x4

, orσ

σ5−x

2

is bounded for x = 3; (E.13)

|η|3(σ2 + |η|4) 5

4

[(σ2 + |η|4) 14 ]x =

|η|3(σ2 + |η|4) 5−x

4

, or|η|3

|η|5−4is bounded for x = 2. (E.14)

Thus, (E.12), used for the first and third terms in (E.11), says that these are s1,−3a and s1,−2

a respectively.Hence

Pr Sym [z, [Λ−1tanβz,P]] ∈ s1,−2

a , (E.15)

and then (E.3) follows from (E.15), as desired.

Proof of (iii) = (E.4). Same as for (E.3).

Proof of (iv) = (E.5). We take

i1 = z(t, x, y); i2 =ξn

(σ2 + |η|4) 12

, (E.16)


where i2 is justified by (E.15). We apply formula (E.1) and obtain, since i1 does not depend on (σ, ξn, η):

Pr Sym [z, [z,Λ−1tanβz,P]] = −

Dσξnη

i2

Dtxy

i1

, (E.17)

= ξnσzt

(σ2 + |η|4) 32

− zx

(σ2 + |η|4) 12

+ ξn2|η|3zy

(σ2 + |η|4) 32

. (E.18)

The second term in (E.18) is lower order, while the first and third terms satisfy

σ

(σ2 + |η|4) 32

∈ s0,−4a ,

|η|3(σ2 + |η|4) 3

2

∈ s0,−3a , (E.19)

since, respectively,

σ

(σ2 + |η|4) 32

[(σ2 + |η|4) 14 ]x =

σ

(σ2 + |η|4) 12

1

(σ2 + |η|4) 4−x4

is bounded for x = 4; (E.20)

|η|3(σ2 + |η|4) 3

2

[(σ2 + |η|4) 14 ]x =

|η|3(σ2 + |η|4) 3

4

(σ2 + |η|4) x4

(σ2 + |η|4) 34

is bounded for x = 3. (E.21)

Thus, by (E.19) used in (E.18), we conclude that

Pr Sym [z, [z,Λ−1tanβz,P]] ∈ s1,−3

a , (E.22)

and then (E.22) proves (E.5).

Proof of (v). To establish the first claim in (E.6), we let now i2 = symb[P], as given by (E.7b) and

i1 = symb[Λ−2tan] =

1

[σ2 + |η|4] 12

, (E.23)

recalling (10.3.8). We next apply (E.1), with a = i1 , which does not depend on t, x, y, and ξn, and with b = i2 ,which does not depend on t. We obtain

Pr Symb [Λ−2tan,P] = Dη

i1 Dyi2 =

−2|η|3[σ2 + |η|4] 3

2

ay|η|2 ∈ S0,−1a . (E.24)

The symbol in (E.24) is of zero order in the normal component ξn. It is also of anisotropic order −1 in thetangential component, since

|η|5[σ2 + |η|4] 3

2

[(σ2 + |η|4) 14 ]x =

|η|5[σ2 + |η|4] 6−x

4

is bounded for x = 1, (E.25)

and the first claim in (E.6) is proved.

To establish the second claim in (E.6), we let i1 = zeτϕ, i2 = symb [Λtan] = [σ2 + |η|4] 14 . We then apply

(E.1), with a = i1 which does not depend on σ, ξn, η and b = i2 which does not depend on ξn. We obtain

Pr Symb [zeτϕ,Λtan] = −

Dση

i2

Dty

i1

= −1

2

σ

[σ2 + |η|4] 34

i1 t −|η|3

[σ2 + |η|4] 34

. (E.26)


The symbol in (E.26) is of order 0 in the normal direction ξn, and it is also zero in the tangential directionσ, η since it is bounded. Thus the second claim in (E.6) is also established. 2

We next establish the relation in (10.4.46).

Lemma E.2. With reference to Section 10.4, we have

[(

∂

∂ν

)

(eτϕz),Λ−1tan

]

∈ S1,−2a . (E.27)

Proof. We set

i1 = symb

[(

∂

∂ν

)

(eτϕz)

]

= eτϕz ξn; (E.28)

i2 = symb [Λ−1tan] =

1

(σ2 + |η|4) 14

. (E.29)

Applying formula (E.1) for the symbols i1 and i2 yields, since i1 does not depend on (σ, η) and i2 doesnot depend on x and ξn:

Pr Sym

[(

∂

∂ν

)

(eτϕz),Λ−1tan

]

= −

Dση

i2

Dty

i1

, (E.30)

=1

4

2σ

(σ2 + |η|4) 54

(eτϕz)tξn +|η|3

(σ2 + |η|4) 54

ξn(eτϕz)y. (E.31)

Recalling (E.12), we see that the two terms in (E.31) are in s1,−3a and s1,−2

a , respectively. We conclude that

Pr Sym

[(

∂

∂ν

)

(eτϕz),Λ−1tan

]

∈ s1,−2a , (E.32)

and then (E.32) proves (E.27), as desired. 2

Acknowledgement. This paper was completed during the Spring 2002, while the first two authors werevisiting the Scuola Normale Superiore di Pisa, Italy, on a sabbatical leave. They wish to thank the SNS, and inparticular, Professor G. DaPrato, for the warm and stimulating hospitality. The content of the present paperwas presented at a seminar held at the SNS as well as at various conferences. An announcement has been givenin [21].

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Global uniqueness, observability and stabilization of nonconservative Schrödinger equations via pointwise Carleman estimates. Part II: L 2 (Ω)-estimates

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