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New Tools for Nonlocal Elliptic ProblemsMini-Course @ Winter School 2014, St. Etienne de Tinee
Enno Lenzmann1
Department of MathematicsUniversity of Basel
February 4, 2014
1Joint work with Rupert Frank (CalTech) and Luis Silvestre (Chicago)
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Overview
Class of Problems
Lu + f (x , u) = 0 in Rn
L is nonlocal differntial operator, e. g. L = (−4)s with s ∈ (0, 1).
f is nonlinearity
Contents of Mini-Course:
1 Motivation & History
2 New Results
3 Tools & Methods
4 Applications & Outlook
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Overview
Class of Problems
Lu + f (x , u) = 0 in Rn
L is nonlocal differntial operator, e. g. L = (−4)s with s ∈ (0, 1).
f is nonlinearity
Contents of Mini-Course:
1 Motivation & History→ Physics & Maths
2 Results→ Symmetry, Uniqueness, Oscillations
3 Tools & Methods→ s-Harmonic Extension, Topological Bounds, Hamiltonian Estimates
4 Applications & Outlook→ Stability and Blowup for Nonlinear Dispersive PDE
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Overview
Class of Problems
Lu + f (x , u) = 0 in Rn
L is nonlocal differntial operator, e. g. L = (−4)s with s ∈ (0, 1).
f is nonlinearity
Contents of Mini-Course:
1 Motivation & History→ Physics & Maths
2 Results→ Symmetry, Uniqueness, Oscillations
3 Tools & Methods→ s-Harmonic Extension, Topological Bounds, Hamiltonian Estimates
4 Applications & Outlook→ Stability and Blowup for Nonlinear Dispersive PDE
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Overview
Class of Problems
Lu + f (x , u) = 0 in Rn
L is nonlocal differntial operator, e. g. L = (−4)s with s ∈ (0, 1).
f is nonlinearity
Contents of Mini-Course:
1 Motivation & History→ Physics & Maths
2 Results→ Symmetry, Uniqueness, Oscillations
3 Tools & Methods→ s-Harmonic Extension, Topological Bounds, Hamiltonian Estimates
4 Applications & Outlook→ Stability and Blowup for Nonlinear Dispersive PDE
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Overview
Class of Problems
Lu + f (x , u) = 0 in Rn
L is nonlocal differntial operator, e. g. L = (−4)s with s ∈ (0, 1).
f is nonlinearity
Contents of Mini-Course:
1 Motivation & History→ Physics & Maths
2 Results→ Symmetry, Uniqueness, Oscillations
3 Tools & Methods→ s-Harmonic Extension, Topological Bounds, Hamiltonian Estimates
4 Applications & Outlook→ Stability and Blowup for Nonlinear Dispersive PDE
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Nonlocality of Type I
Question: Where do nonlocal problems arise?
Example: Let Ω ⊂ Rn and L local (elliptic) differential operator, e. g.,Laplacian L = −4
Classical Boundary-Value ProblemLu = 0 in Ωu = g on ∂Ω
Ω
∂Ω
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Nonlocality of Type I
Question: Where do nonlocal phenomena arise?
Example: Let Ω ⊂ Rn and L local (elliptic) differential operator, e. g.,Laplacian L = −4
Classical Boundary-Value ProblemLu = 0 in Ωu = g on ∂Ω
Ω
∂Ω
rpNonlocality due to change of g
g = g+“Pertubation around p ∈ ∂Ω”
=⇒ u(x) 6= u(x) ∀x ∈ Ω
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Boundary-Value Problem from Skandinavia...
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Boundary-Value Problem from Skandinavia...
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Elliptic & Parabolic vs. Hyperbolic
Deeper reason for nonlocal phenomena lies in the class of PDE.
Elliptic and Parabolic PDE such as
4u = 0 und ∂tu −4u = 0
exhibit Nonlocality via Boundary-/Initial conditions.
Hyperbolic PDE such as wave equation
1
c2∂ttu −4u = 0
preserve Locality (with c > 0 speed of propagation)
vs.
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Nonlocality of Type II
Nonlocality in equations involving Pseudo-Differentialoperators
EquationLu = f in Ω ⊂ Rn
with Pseudo-Differentialoperator L.
E. g. fractional Laplacian L = (−4)s with s > 0 in Rn defined via Fouriertransform as
((−4)s f )(ξ) := |ξ|2s f (ξ)
For s ∈ (0, 1) we have integral formula by Aronszajn-Smith:
((−4)s f )(x) = cn,s
∫Rn
f (x)− f (y)
|x − y |n+2sdy
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Harmonic Extension
Problem
Let f : Rn → R be bounded. Find F : Rn+1+ → R bounded such that
4F = 0 in Rn+1+
F = f on ∂Rn+1+
-
6
Rn
t
F = f
4F = 0
Derivative on boundary ∂Rn+1+
∂tF (·, t)∣∣t=0
= −(−4)1/2f
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Harmonic Extension
Problem
Let f : Rn → R be bounded. Find F : Rn+1+ → R bounded such that
4F = 0 in Rn+1+
F = f on ∂Rn+1+
-
6
Rn
t
F = f
4F = 0
Derivative on boundary ∂Rn+1+
∂tF (·, t)∣∣t=0
= −(−4)1/2f
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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s-Harmonic Extension: (−4)s as Dirichlet-Neumann map
Observation by Caffarelli-Silvestre ’06 (see also Graham-Zworski ’04,Ostrovskii-Molchanov ’69)
Let s ∈ (0, 1) und f : Rn → R bounded. Let F = F (x , t) be solution to∇ · (t1−2s∇F ) = 0 in Rn+1
+
F = f on ∂Rn+1+
-
6
Rn
t
F = f
∇ · (t1−2s∇F ) = 0
(Weighted) Boundary Derivative on ∂Rn+1+
t1−2s∂tF (·, t)∣∣t=0
= −cs(−4)s f
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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The Five-Fold Way to (−4)s
Fractional Laplacian (−4)s in Rn with s ∈ (0, 1) can be represented as:
1 Multiplication by |ξ|2s in Fourier space.
2 Singular integral operator∫ f (x)−f (y)
|x−y|n+2s dy .
3 Generator of the (heat) semigroup e−t(−4)st≥0.
4 By spectral calculus,
(−4)s =sin(πs)
π
∫ ∞0
µs−1 −4−4+ µ
dµ
5 Dirchlet–Neumann operator for Ls = ∇ · (t1−2s∇·) on Rn+1+ .
Remarks:
Hence many different situations, where (−4)s arises.
Large toolbox due to diverse defintions.
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Extension Principle in Retrospective
For many interesting pseudo-differentialoperators L it holds:
Extension Principle
Lu = f in Rn ⇐⇒
LlocalU = 0 in Rn+1+
∂U∂n
= f auf ∂Rn+1+
Examples: L = (−4)s , (−4+ 1)s with s ∈ (0, 1), L = LILW (water waves)etc.
Manifolds and domains: Hn, Tn, and Ω ⊂ Rn etc.
Regularity, Existence/Nonexistence, Harnack inequalities, uniquecontinuation for L etc.
Oscillation bounds for H = (−4)s + V .
Montonicity formulae (Hamiltonian estimates) for radial solutions.
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Why study Nonlocal Equations?
Concrete Examples from Physics & Maths
Stable Levy Processes & Anomalous Diffusion:
pt = e−t(−4)s , ∂tu = (−4)su + f (u)
Song-Wu ’99, Bogdan et al. ’99, Banuelos et al. ’04, Caffarelli & Vazquez ’11, ...
Relativistic Schrodinger Operators:
H = (−4)s + V
Herbst ’77, Carmona-Master-Simon ’90, ...
Relativsitic Quantum Mechanics, Gravitatonal Collapse, Long-RangeSystems:
i∂tu = (−4)su + f (u) (fNLS)
Fefferman-de la Llave ’86, Lieb-Yau ’87, Elgart-Schlein ’05, Frohlich-Lenzmann
’07, ...
Fluid Dynamics:
∂tu + ∂x(−4)su − |u|p−1∂xu = 0 (gBO)
Weinstein ’87, Amick-Toland ’91, Kenig-Martel-Robbiano ’11, ...
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Why study Nonlocal Equations?
Concrete Examples from Physics & Maths
Stable Levy Processes & Anomalous Diffusion:
pt = e−t(−4)s , ∂tu = (−4)su + f (u)
Song-Wu ’99, Bogdan et al. ’99, Banuelos et al. ’04, Caffarelli & Vazquez ’11, ...
Relativistic Schrodinger Operators:
H = (−4)s + V
Herbst ’77, Carmona-Master-Simon ’90, ...
Relativsitic Quantum Mechanics, Gravitatonal Collapse, Long-RangeSystems:
i∂tu = (−4)su + f (u) (fNLS)
Fefferman-de la Llave ’86, Lieb-Yau ’87, Elgart-Schlein ’05, Frohlich-Lenzmann
’07, ...
Fluid Dynamics:
∂tu + ∂x(−4)su − |u|p−1∂xu = 0 (gBO)
Weinstein ’87, Amick-Toland ’91, Kenig-Martel-Robbiano ’11, ...
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Why study Nonlocal Equations?
More Examples
Fractional Obstacle and Free Boundary Problems:
min((−4)su, u − ϕ) = 0
Silvestre ’05, Caffarelli-Salsa-Silvestre ’08, ...
Layer Solutions (Peirls-Nabarro) Phase Transitions:
(−4)su = f (u) in Rn with uxn > 0
Toland ’97, Cabre-Sola-Morales ’05, Cabre-Sire ’12, ...
Conformal Geometry: Fractional Yamabe Problem, Paneitz operators,etc.Graham-Zworski ’03, Chang-Mar ’11, Mar-Mazzeo-Sire ’12, ...
Nonlocal ‘Minimal’ Surfaces:
L(A,B) =
∫ ∫χA(x)χB(y)
|x − y |n+2sdx dy .
Caffarelli-Roquejoffre-Savin ’10, Davila-del Pino-Wei ’13, ...
n/2-Fractional Harmonic Maps:
(−4)n/2u ∧ u = 0 for u : Rn → Sn−1
da Lio-Riviere ’11, Schikorra ’12, ...
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Why study Nonlocal Equations?
More Examples
Fractional Obstacle and Free Boundary Problems:
min((−4)su, u − ϕ) = 0
Silvestre ’05, Caffarelli-Salsa-Silvestre ’08, ...
Layer Solutions (Peirls-Nabarro) Phase Transitions:
(−4)su = f (u) in Rn with uxn > 0
Toland ’97, Cabre-Sola-Morales ’05, Cabre-Sire ’12, ...
Conformal Geometry: Fractional Yamabe Problem, Paneitz operators,etc.Graham-Zworski ’03, Chang-Mar ’11, Mar-Mazzeo-Sire ’12, ...
Nonlocal ‘Minimal’ Surfaces:
L(A,B) =
∫ ∫χA(x)χB(y)
|x − y |n+2sdx dy .
Caffarelli-Roquejoffre-Savin ’10, Davila-del Pino-Wei ’13, ...
n/2-Fractional Harmonic Maps:
(−4)n/2u ∧ u = 0 for u : Rn → Sn−1
da Lio-Riviere ’11, Schikorra ’12, ...
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Scope of this Mini-Course
We will study nonlocal elliptic problems with the fractional Laplacian (−4)s
given by((−4)su)(ξ) = |ξ|2s u(ξ)
Type of Problem
(−4)su + f (x , u) = 0 in Rn
We’ll focus on s ∈ (0, 1) with f (x , u) either linear or nonlinear.
Symmetry, uniqueness and non-degeneracy of ground states u(x) > 0.
Applications to time-dependent problems.
Remarks on bounded domains Ω ⊂ Rn with exterior Dirichlet conditions
(−4)su + f (x , u) = 0 on Ω and u ≡ 0 on Rn \ Ω
First: Brief recap for “classical” local case when s = 1
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Scope of this Mini-Course
We will study nonlocal elliptic problems with the fractional Laplacian (−4)s
given by((−4)su)(ξ) = |ξ|2s u(ξ)
Type of Problem
(−4)su + f (x , u) = 0 in Rn
We’ll focus on s ∈ (0, 1) with f (x , u) either linear or nonlinear.
Symmetry, uniqueness and non-degeneracy of ground states u(x) > 0.
Applications to time-dependent problems.
Remarks on bounded domains Ω ⊂ Rn with exterior Dirichlet conditions
(−4)su + f (x , u) = 0 on Ω and u ≡ 0 on Rn \ Ω
First: Brief recap for “classical” local case when s = 1
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Recap: Classical Local Theory
Elliptic Problem
−∆u + f (x , u) = 0 in Ω ⊂ Rn
In many cases (P) stems from variational problem given by
E(u) =
∫|∇u|2 +
∫F (u, x)
Linear Setting: Theory of Schrodinger Operators H = −∆ + V (x) with
Existence and Regularity of Eigenfunctions Hun = λnun
Bounds on N(V ) (number of eigenvalues)
Estimates on λn (e. g. spectral gaps)
Nodal Properties of un.
Unique continuation, absence of embedded eigenvalues...
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Recap: Classical Local Theory
Elliptic Problem
−∆u + f (x , u) = 0 in Ω ⊂ Rn
In many cases (P) stems from variational problem given by
E(u) =
∫|∇u|2 +
∫F (x , u)
Nonlinear Setting:
Existence of Ground States u(x) > 0 and Excited States u(x) 6≥ 0
Regularity
Symmetry (Gidas-Ni-Nirenberg ’79): If Ω = BR(0) or Ω = Rn then
u(x) > 0 ⇒ u = u(|x − x0|) > 0
for large class of local nonlinearities f = f (u) (e. g. locally Lipschitz).
Uniqueness and Nondegeneracy of Ground States u(x) > 0 (by reductionto ODE).
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Recap: Classical Local Theory
Elliptic Problem
−∆u + f (x , u) = 0 in Ω ⊂ Rn
In many cases (P) stems from variational problem given by
E(u) =
∫|∇u|2 +
∫F (x , u)
Nonlinear Setting:
Existence of Ground States u(x) > 0 and Excited States u(x) 6≥ 0
Regularity
Symmetry (Gidas-Ni-Nirenberg ’79): If Ω = BR(0) or Ω = Rn then
u(x) > 0 ⇒ u = u(|x − x0|) > 0
for large class of local nonlinearities f = f (u) (e. g. locally Lipschitz).
Uniqueness and Nondegeneracy of Ground States u(x) > 0 (by reductionto ODE).
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Ground States: ODE Analysis
By moving planes argument, can restrict to radial functions and hence ODE.
Initial-Value Problem
−ϕ′′ − n−1rϕ′ + f (ϕ) = 0 for r > 0
ϕ(r) > 0 and ϕ(r)→ 0 as r →∞
For n = 1 uniqueness of ϕ is very simple.For n ≥ 2 analysis substantially harder.
Coffman ’73: Cubic case f (ϕ) = ϕ− ϕ3 and n = 3
Lieb ’77: Choquard-Pekar f (ϕ) = ϕ− (|x |−1 ∗ |ϕ|2)ϕ and n = 3
McLeod & Serrin ’81: f (ϕ) = ϕ− ϕp for some p and n ≥ 2.
Kwong ’89: General power-case f (ϕ) = ϕ− ϕp with 1 < p < n+2n−2
andn ≥ 2.
Many further results...
All proofs depend on ODE techniques!
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Ground States: ODE Analysis
By moving planes argument, can restrict to radial functions and hence ODE.
Initial-Value Problem
−ϕ′′ − n−1rϕ′ + f (ϕ) = 0 for r > 0
ϕ(r) > 0 and ϕ(r)→ 0 as r →∞
For n = 1 uniqueness of ϕ is very simple.For n ≥ 2 analysis substantially harder.
Coffman ’73: Cubic case f (ϕ) = ϕ− ϕ3 and n = 3
Lieb ’77: Choquard-Pekar f (ϕ) = ϕ− (|x |−1 ∗ |ϕ|2)ϕ and n = 3
McLeod & Serrin ’81: f (ϕ) = ϕ− ϕp for some p and n ≥ 2.
Kwong ’89: General power-case f (ϕ) = ϕ− ϕp with 1 < p < n+2n−2
andn ≥ 2.
Many further results...
All proofs depend on ODE techniques!
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Warm-Up: One-Dimensional Case
Consider ODE for the ground state problem with power-type nonlinearity.
ODE Problem
−ϕ′′ + ϕ− |ϕ|p−1ϕ = 0 on R+
ϕ(0) = ϕ0 > 0, ϕ′(0) = 0, and ϕ(x)→ 0 as |x | → ∞
Define Energy by
E(r) =1
2|ϕ′(r)|2 + V (ϕ(r)) with V (ϕ) = − 1
2|ϕ|2 + 1
p+1|ϕ|p+1
Simple calculation shows dE/dr = 0 and hence E(r) ≡ const. Combinedwith limr→∞ E(r) = 0 (by decay of ϕ) this gives
E(r) =1
2|ϕ′(r)|2 + V (ϕ(r)) ≡ 0
Since ϕ′(0) = 0, must have V (ϕ(0)) = 0 and hence ϕ(0) = ( p+12
)1
p−1 .
By-Product: No sign-changing ϕ solution with ϕ(∞) = 0 exists!
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Warm-Up: One-Dimensional Case
Consider ODE for the ground state problem with power-type nonlinearity.
ODE Problem
−ϕ′′ + ϕ− |ϕ|p−1ϕ = 0 on R+
ϕ(0) = ϕ0 > 0, ϕ′(0) = 0, and ϕ(x)→ 0 as |x | → ∞
Define Energy by
E(r) =1
2|ϕ′(r)|2 + V (ϕ(r)) with V (ϕ) = − 1
2|ϕ|2 + 1
p+1|ϕ|p+1
Simple calculation shows dE/dr = 0 and hence E(r) ≡ const. Combinedwith limr→∞ E(r) = 0 (by decay of ϕ) this gives
E(r) =1
2|ϕ′(r)|2 + V (ϕ(r)) ≡ 0
Since ϕ′(0) = 0, must have V (ϕ(0)) = 0 and hence ϕ(0) = ( p+12
)1
p−1 .
By-Product: No sign-changing ϕ solution with ϕ(∞) = 0 exists!
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Next Step: Higher-Dimensional Case
ODE Problem for (n ≥2)
−ϕ′′ − n−1rϕ′ + ϕ− |ϕ|p−1ϕ = 0 on R+
ϕ(0) = ϕ0 > 0, ϕ′(0) = 0, and ϕ(x)→ 0 as |x | → ∞
Again, consider energy given by
E(r) =1
2|ϕ′(r)|2 + V (ϕ(r)) with V (ϕ) = − 1
2|ϕ|2 + 1
p+1|ϕ|p+1
Simple calculation shows monotonicity
dE
dr= −n − 1
r|ϕ′(r)|2 ≤ 0
Intuitive Picture (McLeod, Tao): Motion of particle with position ϕ(r) andvelocity ϕ′(r) at ‘time’ r in potential V (ϕ) subject to friction force n−1
r2 ϕ′.
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Fractional Case
Fractional Elliptic Problem
(−∆)su + f (x , u) = 0 in Ω ⊂ Rn
Linear Setting: Fractional Schrodinger Operators H = (−∆)s + V (x) with
Existence and Regularity of Eigenfunctions Hun = λnun.Simon 79 via e−t(−4)s on Rn; Serra/Ros-Oton & Grubb 13 on domains
Bounds on N(V ) (number of eigenvalues).Daubechies ’85 Lieb-Thirring estimates for (−4)s
Nodal Properties of un.Frank-Lenzmann-Silvestre ’12 and ’13
Unique continuation, absence of embedded eigenvalues...Fall & Felli ’13, Seo ’13, Ruland ’13
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Fractional Case
Fractional Elliptic Problem
(−∆)su + f (x , u) = 0 in Ω ⊂ Rn
In many cases (P) stems from variational problem given by
E(u) =
∫|(−4)s/2u|2 +
∫F (u, x)
Nonlinear Setting:
Existence of Ground States u(x) > 0 and Excited States u(x) 6≥ 0Weinstein ’84, Albert-Bona-Saut ’97, ...
Symmetry Every positive solution u(x) > 0 solving
(−4)su − un+2sn−2s = 0 in Rn
is radial (mod translation). Proof by moving planes for integral equation.Y.Y. Li ’04, Chen-Li-Ou ’06, Ma-Zhao ’10, ...
Uniqueness and Nondegeneracy of Ground States u(x) > 0. ?? See nextslide
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Fractional Case
Fractional Elliptic Problem
(−∆)su + f (x , u) = 0 in Ω ⊂ Rn
In many cases (P) stems from variational problem given by
E(u) =
∫|(−4)s/2u|2 +
∫F (u, x)
Nonlinear Setting:
Existence of Ground States u(x) > 0 and Excited States u(x) 6≥ 0Weinstein ’84, Albert-Bona-Saut ’97, ...
Symmetry Every positive solution u(x) > 0 solving
(−4)su − un+2sn−2s = 0 in Rn
is radial (mod translation). Proof by moving planes for integral equation.Y.Y. Li ’04, Chen-Li-Ou ’06, Ma-Zhao ’10, ...
Uniqueness and Nondegeneracy of Ground States u(x) > 0. ?? See nextslide
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Uniqueness and Nondegenarcy for (−4)s
Amick-Toland ’91: Uniqueness (mod translations) of Q(x) > 0 solving
(−4)1/2Q + Q − Q2 = 0 in R
Proof by complex analysis. “Magic identities”. Very rigid.
Y.Y. Li ’04, Chen-Li-Ou ’06: Uniqueness (mod translations/scalings) ofQ(x) > 0 solving
(−4)sQ − Qn+2sn−2s = 0 in Rn
Use of conformal symmetry. Very rigid.
Frank-Lenzmann ’11, Frank-Lenzmann-Silvestre ’13: Uniqueness andNondegeneracy of ground states Q(x) > 0 solving
(−4)sQ + Q − Qα+1 = 0 in Rn with n ≥ 1
with 0 < α < α∗(s, n).Proof by estimates. New (robust) methods. Substitutes of ODE results.
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Uniqueness and Nondegenarcy for (−4)s
Amick-Toland ’91: Uniqueness (mod translations) of Q(x) > 0 solving
(−4)1/2Q + Q − Q2 = 0 in R
Proof by complex analysis. “Magic identities”. Very rigid.
Y.Y. Li ’04, Chen-Li-Ou ’06: Uniqueness (mod translations/scalings) ofQ(x) > 0 solving
(−4)sQ − Qn+2sn−2s = 0 in Rn
Use of conformal symmetry. Very rigid.
Frank-Lenzmann ’11, Frank-Lenzmann-Silvestre ’13: Uniqueness andNondegeneracy of ground states Q(x) > 0 solving
(−4)sQ + Q − Qα+1 = 0 in Rn with n ≥ 1
with 0 < α < α∗(s, n).Proof by estimates. New (robust) methods. Substitutes of ODE results.
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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Uniqueness and Nondegenarcy for (−4)s
Amick-Toland ’91: Uniqueness (mod translations) of Q(x) > 0 solving
(−4)1/2Q + Q − Q2 = 0 in R
Proof by complex analysis. “Magic identities”. Very rigid.
Y.Y. Li ’04, Chen-Li-Ou ’06: Uniqueness (mod translations/scalings) ofQ(x) > 0 solving
(−4)sQ − Qn+2sn−2s = 0 in Rn
Use of conformal symmetry. Very rigid.
Frank-Lenzmann ’11, Frank-Lenzmann-Silvestre ’13: Uniqueness andNondegeneracy of ground states Q(x) > 0 solving
(−4)sQ + Q − Qα+1 = 0 in Rn with n ≥ 1
with 0 < α < α∗(s, n).Proof by estimates. New (robust) methods. Substitutes of ODE results.
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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The Three Methods
Nonlocal Problem
(−4)su + f (u, x) = 0 in Rn
1 Topological Bounds: For s-harmonic Extension U(x , t) show that
# Nodal Domains of U ≤ C
Yields bounds on # sign changes of U = u on ∂Rn+1+ .
2 “Continuity Argument”: Take s ∈ (0, 1) as parameter and consider
(−4)sus + fs(us , x) = 0
and try to take limit s → 1.
3 Monotonicity Formula: For radial u(r), obtain monotone quantity
H(r) = cs
∫ ∞0
(∂rU(r , t))2 − (∂tU(r , t))2t1−2sdt − F (u(r), r)
Use the fact that H′(r) ≤ 0. Hamiltonian Estimate. Cabre-Sola-Morales
’05, Cabre-Sire ’12, Frank-Lenzmann-Silvestre ’13
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
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The Three Methods
Nonlocal Problem
(−4)su + f (u, x) = 0 in Rn
1 Topological Bounds: For s-harmonic Extension U(x , t) show that
# Nodal Domains of U ≤ C
Yields bounds on # sign changes of U = u on ∂Rn+1+ .
2 “Continuity Argument”: Take s ∈ (0, 1) as parameter and consider
(−4)sus + fs(us , x) = 0
and try to take limit s → 1.
3 Monotonicity Formula: For radial u(r), obtain monotone quantity
H(r) = cs
∫ ∞0
(∂rU(r , t))2 − (∂tU(r , t))2t1−2sdt − F (u(r), r)
Use the fact that H′(r) ≤ 0. Hamiltonian Estimate. Cabre-Sola-Morales
’05, Cabre-Sire ’12, Frank-Lenzmann-Silvestre ’13
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
Page 40
The Three Methods
Nonlocal Problem
(−4)su + f (u, x) = 0 in Rn
1 Topological Bounds: For s-harmonic Extension U(x , t) show that
# Nodal Domains of U ≤ C
Yields bounds on # sign changes of U = u on ∂Rn+1+ .
2 “Continuity Argument”: Take s ∈ (0, 1) as parameter and consider
(−4)sus + fs(us , x) = 0
and try to take limit s → 1.
3 Monotonicity Formula: For radial u(r), obtain monotone quantity
H(r) = cs
∫ ∞0
(∂rU(r , t))2 − (∂tU(r , t))2t1−2sdt − F (u(r), r)
Use the fact that H′(r) ≤ 0. Hamiltonian Estimate. Cabre-Sola-Morales
’05, Cabre-Sire ’12, Frank-Lenzmann-Silvestre ’13
E. Lenzmann Mini-Course Nonlocal Elliptic Problems
Page 41
New Results in Linear Case
Linear Problem
(−4)su + Vu = 0 in Rn
New Results:
Cauchy-Lipschitz Theorem: For suitable V and radial u(r), we have
u(0) = 0 ⇒ u ≡ 0
Simplicity of all radial eigenvalues of H = (−4)s + V .
Sturm Oscillation: Estimates number of zeros for radial eigenfunctions ukof
(−4)suk + Vuk = λkuk .
Key to nondegeneracy proof for nonlinear ground states.
E. Lenzmann Mini-Course Nonlocal Elliptic Problems