The p -harmonic approximation and the regularity of p -harmonic maps

DOI: 10.1007/s00526-003-0233-x

Calc. Var. 20, 235–256 (2004) Calculus of Variations

Frank Duzaar · Giuseppe Mingione

The p-harmonic approximation and the regularityof p-harmonic maps

Received: 2 November 2002 / Accepted: 10 July 2003 /Published online: 4 September 2003 – c© Springer-Verlag 2003

Abstract. We extend to the degenerate case p = 2, Simon’s approach to the classicalregularity theory of harmonic maps of Schoen & Uhlenbeck, by proving a “p-HarmonicApproximation Lemma". This allows to approximate functions with p-harmonic functions inthe same way as the classical harmonic approximation lemma (going back to De Giorgi) doesvia harmonic functions. Finally, we show how to combine this tool with suitable regularityestimates for solutions to degenerate elliptic systems with a critical growth right hand side,in order to obtain partial C1,α-regularity of p-harmonic maps.

Mathematics Subject Classification (2000): 35J70, 49N60, 49Q60

1. Introduction

The aim of this paper is the study of regularity properties of (local) minimizers –energy minimizing p-harmonic maps – of the p-energy functional

Dp(u;Ω) =∫

Ω

|Du|p dx

whereΩ is an open subset of Rn,n ≥ 2, p > 1,M is a compact smooth Riemannian

manifold of dimensionm ≥ 2 which is isometrically embedded in some Euclideanspace R

m+k, k ≥ 1, and u ∈ W 1,ploc (Ω,M). Du stands for the n× (m+ k) matrix

with entriesDαuj (uj the components of u), and |Du|2 =

∑nα=1

∑m+kj=1 (Dαu

j)2.The regularity theory for energy minimizing harmonic maps, i.e. the case p = 2,

was initiated in the pioneering work of Schoen and Uhlenbeck [14], where theinterior regularity is proved at interior points with small excess. The notion ofexcess refers to the quantity

E(u;B) =1n

∫B

|u− u|2 dx

F. Duzaar: Mathematisches Institut der Friedrich-Alexander-Universitat zuNurnberg-Erlangen, Bismarckstr. 1 1/2, 91054 Erlangen, Germany(e-mail: [email protected])

G. Mingione: Dipartimento di Matematica, Universita, via M. D’Azeglio 85/a, 43100, Parma,Italy (e-mail: [email protected])

236 F. Duzaar, G. Mingione

forB Ω (i.e. the mean square deviation of u onB from its mean value u overB). A new and elegant treatment of the Schoen-Uhlenbeck result was presentedby Simon [17]. This proof avoids technique of smoothing the harmonic map withsmoothing radius varying with respect to the independent variable x. It insteaduses a direct PDE argument which can roughly be described as follows: In a firststep, using the minimality of the harmonic map, in particular the monotonicityformula, and a Lemma of Luckhaus, one proves a Caccioppoli type inequalitywhich states that the scaled energy (/2)2−n

∫B/2

|Du|2 dx is estimated by the

excess E(u;B) for balls B Ω. As it is well known from the regularity theoryfor nonlinear elliptic systems a Caccioppoli inequality is sufficient to prove an “ε-regularity theorem" for solutions of nonlinear elliptic systems with right hand sideof critical growth (the usual smallness assumption on the solution being used onlyto derive the Caccioppoli inequality itself). In the case of an energy minimizingharmonic map the Euler-Lagrange equation takes the form

∆u = Au(Du,Du) on Ω

where Az : TzM × TzM → (TzM)⊥ stands for the second fundamental form M .Clearly, this equation is an elliptic system of second order (of diagonal structure)with right hand side of critical growth, i.e. |Au(Du,Du)| ≤ c|Du|2 for someconstant c. In [17] it was then shown in the so called “technical lemma” that a weaksolution of the elliptic system

∆u = F on Ω

with right hand side F of critical growth, i.e. |F | ≤ c|Du|2 on Ω, satisfying aCaccioppoli inequality is of class C1,α, 0 < α < 1 on a ball B/4 provided theexcess E(u;B) over B Ω is sufficiently small. We note here that the ideaof approximating minimizers to variational problems by harmonic functions goesback to DeGiorgi’s work [4] on the regularity of minimal surfaces.

Under the assumption of p-growth with p = 2 the regularity theory for min-imizing p-harmonic maps was sucessfully carried out in [8,11], and [13]. TheEuler-Lagrange equation for energy minimizing p-harmonic maps takes the form

div(|Du|p−2Du) = |Du|p−2Au(Du,Du) on Ω

which is a degenerate elliptic system with right hand side of critical growth, i.e.|Du|p−2|Au(Du,Du)| ≤ c|Du|p. The applied methods of proof are either anadaptation of the original Schoen-Uhlenbeck method to the case of p-growth [8]or of indirect nature [11,13]. In the latter case this is done by considering a blow-up sequence of p-energy minimizers ui on the unit ball B which do not exhibit acertain energy decay, but which have total energy

∫B

|Dui|p dx = λpi converging 0

as i → ∞. The weak limit h of the rescaled functions λ−1i (ui −(ui)1) then satisfies

the blow-up system div(|Dh|p−2Dh) = 0. Solutions of the blow-up system satisfycertain regularity estimates (see [18] for the case 1 < p < 2 and [19] for the casep > 2), and these estimates can be transferred back to the blow-up sequence to givethe desired contradiction. This approach is close to the original ideas of DeGiorgi

p-harmonic approximation 237

[4]. However it has the disadvantage that the constants in the regularity theoremsare not explictly computable.

Up to now in the case of energy minimizing p-harmonic maps (or for degenerateelliptic systems) there is no direct analogue of Simon’s approach to the regularityproblem. The main missing tool is a “p-harmonic approximation lemma", which isfundamental, in this approach, to the problem of regularity for energy minimizingharmonic maps. The precise statement in the degenerate case and one of the mainresults of the present paper is

Lemma 1 (p-harmonic approximation lemma). Let B be the unit ball in Rn.

For each ε > 0 there exists a positive constant δ ∈ (0, 1] depending only onn,N, p and ε such that the following is true: whenever u ∈ W 1,p(B,RN ) with∫

B|Du|pdx ≤ 1 is approximately p-harmonic in the sense that∣∣∣∣∫

B

|Du|p−2Du ·Dϕdx∣∣∣∣ ≤ δ sup

B

|Dϕ|

holds for all ϕ ∈ C1c (B,RN ), then there exists a p-harmonic function h ∈

W 1,p(B,RN ) such that∫B

|Dh|p dx ≤ 1 and∫

B

|h− u|p dx ≤ εp .

When p = 2 the previous result is nothing but the classic harmonic approximationof De Giorgi (see [4,17]). The previous assertion is proved by contradiction. Themain difficulty can roughly be described as follows: If (uk)k∈N is a sequence inW 1,p(B,RN ) converging weakly to u ∈ W 1,p(B,RN ) which does not, for someε > 0, satisfy the assertion of the Lemma then Duk converges pointwise almosteverywhere on B to Du. This would yield strong convergence of Duk to Du inLs(B,RnN ) for any s ∈ [1, p) and, in particular, would imply that the weak limitu is p-harmonic on B. The convergence of Duk to Du almost everywhere in Bwill certainly be true if, for some θ ∈ (0, 1), we could show:

limk→∞

∫B

[(|Duk|p−2Duk − |Du|p−2Du) · (Duk −Du)

]θdx = 0

Therefore, it is natural (apart from the fact that uk − u = 0 on ∂B, which is aminor technical point) to test

∣∣∫B

|Duk|p−2Duk ·Dϕdx∣∣ ≤ 1k supB |Dϕ| with

ϕ = uk − u. However, uk − u is not of class W 1,∞(B,RN ) and therefore notallowed as test function. To overcome this difficulty one has to replace uk − uby a W 1,∞-truncation, i.e. a Lipschitz truncation. This idea has been successfullyapplied in the study of steady flows of fluids with shear dependent viscosity where asimilar problem occurs; see [7]. Using the Lipschitz truncation Lemma [7], Propo-sition 1, we succeed in proving the desired convergence (see Sect. 3). Let us alsoobserve that similar truncation methods i.e. truncating the maximal function to ob-tain convergence assertions, have been introduced in the Calculus of Variations byAcerbi and Fusco in their classic paper [1].

Once the p-harmonic approximation Lemma is established it is not difficult toperform the other steps in Simon’s approach. The Caccioppoli inequality for energy


minimizing p-harmonic maps can be proved exactly along the lines of [17], §2.8,using the monotonicity formula for those maps and the Interpolation Lemma ofLuckhaus [13]; cf. [17], §2.6. Since this is well known to experts in regularity ofharmonic maps we do not carry this out here and assume the validity of Caccioppoliinequality throughout the paper.

The second step is the “degenerate analogue” of the before mentioned technicallemma of Simon. This lemma states, that a weak solution of the degenerate ellipticsystem

div(|Du|p−2Du) = F on Ω

with right hand side F of critical growth (i.e. |F | ≤ c|Du|p on Ω) satisfyingCaccioppoli’s inequality(

2

)p−n∫

B/2

|Du|p dx ≤ c

n

∫B

|u− u|p dx =: Ep(u;Bρ)

on any ball B Ω is of class C1,α for some α ∈ (0, 1) on B/4 provided theexcess E(u;B) is sufficiently small. All ingredients in the proof of the technicalLemma have already been used previously in regularity theory for degenerate ellip-tic problems. We believe nevertheless that the way in which we have combined theingredients in our proof contains new arguments and has conceptual advantages:it gives explicitly computable constants in the regularity estimates; it enables us toderive an explicit modulus of continuity for the derivative; it is very flexible andcan, for example, be applied to a wide class of degenerate problems, as we shallshow in a forthcoming paper, [5].

2. Notation and Preliminaries

Throughout the paperΩ denotes a bounded open subset of Rn,B(y) the open ball

with center y ∈ Rn and radius > 0 in R

n. If u is an integrable function onB(y)then we define

uy, =∫–

B(y)u(x) dx =

1|B(y)|

∫B(y)

u(x) dx ,

where |B(y)| = αnρn is the Lebesgue measure of B(y) and αn = |B1(0)|.

We also adopt the convention of writing B, u instead of B(y), uy,, when thecenter is the origin 0 of R

n, or when the center is clear from the context. Also, cwill denote a generic constant, possibly varying from line to line, while we shallemphasize only the important connections, when occurring. Finally, in the rest ofthe paper, p ∈ (1,+∞) will always be such that p = 2; for the case p = 2 and theapproach we are going to follow, the reader is referred to [17].

From [7] we recall Proposition 1. Here we need only the special case when thedomainΩ is the unit ballB ⊂ R

n. From now, when v ∈ W 1,p0 (B), we shall define

v ∈ W 1,p0 (Rn) as the extension of v to the whole R

n obtained, letting v ≡ 0 outsideB. With a bit of ambiguity of notation we shall go on denoting such extension byv.


Proposition 1. There exists a constant c depending only on n such that wheneverwk → 0 weakly in W 1,p

0 (B), then for any λ > 0 there exists a sequence (wλk )k∈N

of maps wλk ∈ W 1,∞

0 (B) such that

‖wλk‖W 1,∞(B) ≤ cλ . (1)

Moreover, denoting Aλk = x ∈ B : wλ

k (x) = wk(x) then

|Aλk | ≤ c

λp‖Dwk‖p

Lp(B) . (2)

Consequently,

‖Dwλk‖p

Lp(B) ≤ c‖Dwk‖pLp(B) ≤ c sup

k∈N

‖Dwk‖pLp(B) < ∞ . (3)

Moreover Aλk ⊂ Rλ

k = Fλk ∪Gλ

k ∪Hλk , where Fλ

k , Gλk and Hλ

k satisfy

|Fλk | ≤ c

λp‖Dwk‖Lp(B) , (4)

|Gλk | ≤ c

λ2p‖Dwk‖Lp(B) , (5)

|Hλk | = 0 .

The sets Fλk and Gλ

k are actually given by

Fλk = x ∈ B : λ < M(Dwk)(x) ≤ λ2 ,Gλ

k = x ∈ B : M(Dwk)(x) > λ2 ,(6)

where

M(v)(x) = sup>0

∫–

B(x)|v(z)| dz

denotes the Hardy-Littlewood maximal function of a function v.Now, if wk = uk − u ∈ W 1,p

0 (B) for k ∈ N and uk → u weakly in W 1,p(B)we can apply Proposition 1. Moreover, with regard to the uniform bound

supk∈N

∫B

|Duk|p dx+∫

B

|Du|p dx ≤ K < ∞ ,

Proposition 2 in [7] states:

Proposition 2. Fixed ε > 0, there exists a subsequence (uk)k∈K, K ⊂ N, andλ ≥ 1

ε independent of k ∈ K such that for any k ∈ K∫F λ

k

|Duk|p + |Du|p dx ≤ ε (7)

where Fλk = x ∈ B : λ < M(D(uk − u))(x) ≤ λ2.

The following selection result is due to Eisen, [6].


Lemma 2. Let G ⊂ B be a measurable set. Assume that Bk is a sequence ofmeasurable subsets of G such that, for some ε > 0, the following estimate holds:

|Bk| ≥ ε ∀ k ∈ N.

Then there exists a subsequence Bkh such that⋂

h∈N

Bkh= ∅ .

We shall widely use the function V = Vp : RnN → R

nN defined by

V (A) = |A| p−22 A (8)

forA ∈ RnN and for any p > 1. We will need a number of elementary estimates and

facts concerning V ; they are all readily available in the literature (see e.g. [2,10])and we shall highlight here those ones we are mainly interested in. The followinginequality holds for any A,B ∈ R

nN , where c ≡ c(n,N, p) ≥ 1:

c−1(|A|2 + |B|2) p−22 |A−B|2 ≤ |V (A) − V (B)|2

≤ c(|A|2 + |B|2) p−22 |A−B|2 . (9)

From the basic inequality(|A|p−2A− |B|p−2B) · (A−B) ≥ c(p)(|A| + |B|)p−2|A−B|2 (10)

with c(p) = p− 1 in the case 1 < p < 2 (cf. [12], Lemma 2) and c(p) = 2−2p−2

in the case p > 2 (cf. [9], Lemma 2.2) we deduce, via (9):(|A|p−2A− |B|p−2B) · (A−B) ≥ c−1 |V (A) − V (B)|2 , (11)

for A,B ∈ RnN and c ≡ c(n,N, p) ≥ 1. Another consequence of (9) is the

elementary:

Lemma 3. If f : Bs → RnN is such that V (f) is a Holder continuous function

with exponent β ∈ (0, 1), then the function f itself is Holder continuous inBs withexponent α := min1, (2/p)β.

Proof.. If V (f) is Holder continuous, then it is also bounded and therefore so f is;say |f | ≤ M . Using (9), if p ≥ 2 and x, y ∈ Bs, then:

|f(x) − f(y)|p ≤ c(|f(x)|2 + |f(y)|2) p−22 |f(x) − f(y)|2

≤ cMp−22 |V (f(x)) − V (f(y))|2

≤ cMp−22 [V (f)]20,β |x− y|2β

and the assertion follows in this case. If 1 < p < 2 then, again by (9):

|f(x) − f(y)|2 ≤ c(|f(x)|2 + |f(y)|2) 2−p2 |V (f(x)) − V (f(y))|2

≤ cM2−p2 |V (f(x)) − V (f(y))|2

≤ cM2−p2 [V (f)]20,β |x− y|2β

and also this case is covered.


Finally we state a refined version of the fundamental regularity result for solutionsto the p-Laplacian system, due to K. Uhlenbeck ([19]).

Proposition 3. There exist constants γ ∈ (0, 1) and c with the following property:Whenever h ∈ W 1,p(BR,R

N ) is a solution of∫BR

|Dh|p−2Dh ·Dϕdx = 0 for all ϕ ∈ C1c (BR,R

N ),

then for any 0 < r ≤ R

Φ(h; r) ≤ c( rR

)2γ

Φ(h;R) , (12)

where

Φ(h; r) =∫–

Br

|V (Dh) − (V (Dh))r|2 dx

and the function V is defined as in (8).

The case p > 2 was proved in [10], Theorem 3.1; in this case a similar result– using a different excess – was also proved in [9]. Both papers make strong useof the fundamental regularity result [19]. The case 1 < p < 2 was treated in[2], Proposition 2.11. In that case we have γ > 1

2 . Lower bounds for the Holdercontinuity exponent γ can be retrieved looking at the proofs of the above mentionedpapers.

3. The p-harmonic approximation lemma

The main result of this section is the p-harmonic approximation lemma whichgeneralizes the harmonic approximation lemma from [16], §1, [17], §2 to the caseof the p-Laplacian system.

Lemma 4. For any ε > 0 there exists a positive constant δ ∈ (0, 1], dependingonly onn,N, p and ε, such that the following is true: Whenever u ∈ W 1,p(B,R

N )with n−p

∫B

|Du|pdx ≤ 1 is approximately p-harmonic in the sense that∣∣∣∣∣p−n

∫B

|Du|p−2Du ·Dϕdx∣∣∣∣∣ ≤ δ sup

B

|Dϕ|

holds for all ϕ ∈ C1c (B,R

N ), then there exists a p-harmonic function h ∈W 1,p(B,R

N ) such that

p−n

∫B

|Dh|p dx ≤ 1 and −n

∫B

|h− u|p dx ≤ εp .


Proof.. We may assume = 1, the general case follows by scaling y → u(y).Supposing the lemma to be false we have for some ε > 0 the existence of a sequenceuk ∈ W 1,p(B,RN ), with ∫

B

|Duk|p dx ≤ 1 (13)

and ∣∣∣∣∫B

|Duk|p−2Duk ·Dϕdx∣∣∣∣ ≤ 1

ksupB

|Dϕ| (14)

for all ϕ ∈ C1c (B,RN ) such that the inequality∫

B

|uk − h|p dx > εp (15)

holds for each k and each p-harmonic function h ∈ W 1,p(B,RN ) with∫B

|Dh|p dx ≤ 1 .

Clearly we may assume∫–

Bukdx = 0 (otherwise we replace uk by uk − ∫–

Bukdx)

and apply Poincare’s inequality to obtain∫

B|uk|pdx ≤ c

∫B

|Duk|pdx ≤ c so thatsupk≤1 ‖uk‖W 1,p(B,RN ) ≤ 1+c < ∞.Applying Rellich’s theorem we obtain (afterpassing to a subsequence, if necessary, and using (13) with lower semicontinuity)

uk → u weakly in W 1,p(B,RN ) , (16)

uk → u strongly in Lp(B,RN ) , (17)

uk → u a.e. in B , (18)∫B

|Du|p dx ≤ 1 . (19)

Next we want to show that there exists a subsequence and hence an infinite setK2 ⊂ N, such that

Duk(x) →k∈K2 Du(x) for a.e. x ∈ B . (20)

For this we fix θ ∈ (0, 1) and consider

lim supk→∞

∫B


]θdx .

Note that the previous integrand is well defined in view of (10).We first prove (20) under the additional assumption that

wk = uk − u ∈ W 1,p0 (B,RN ) (21)


for each k ∈ N; we shall subsequently remove (21). In this case we can apply Propo-sition 1 to conclude: For any λ > 0 there exists a sequence wλ

k ∈ W 1,∞0 (B,RN )

such that

‖wλk‖W 1,∞(B,RN ) ≤ c(n)λ , (22)

|Aλk | ≤ c(n)

λp‖Dwk‖p

Lp(B,RnN ) , (23)

where Aλk = x ∈ B : wλ

k (x) = wk(x). Moreover, again up to not relabelledsubsequences, we have

wλk → w ∈ W 1,s

0 (B,RnN ) weakly in W 1,s0 (B,RnN ) (24)

for any s ∈ [1,∞) and since the functions wλk are uniformly Lipschitz then

wλk (x) → w(x) ∀ x ∈ B . (25)

Furthermore, Aλk ⊂ Rλ

k = Fλk ∪Gλ

k ∪Hλk , Hλ

k being a negligible set, where

|Fλk | ≤ c(n)

λp‖Dwk‖p

Lp(B,RnN ) , (26)

|Gλk | ≤ c(n)

λ2p‖Dwk‖p

Lp(B,RnN ) . (27)

We now estimate

Ik =∫

B


]θdx

=∫

B\Aλk

. . . dx+∫

Aλk

. . . dx = I ′k + I ′′

k . (28)

Using Holder’s inequality and (23) the second term I ′′k of the right-hand side of

(28) is estimated as follows:

|I ′′k | ≤ 3θ

∫Aλ

k

(|Duk|p + |Du|p)θ dx

≤ 3θ

(∫Aλ

k

|Duk|p + |Du|p dx)θ

|Aλk |1−θ

≤ 3θ( c

λp‖Dwk‖p

Lp(B,RnN )

)1−θ(∫

B

|Duk|p + |Du|p dx)θ

≤ cλ−p(1−θ) supk≥1

∫B

|Duk|p + |Du|p dx

= cKpλ−p(1−θ) . (29)

Here we have abbreviated

Kp = supk≥1

∫B

|Duk|p + |Du|p dx . (30)


Using again Holder’s inequality and the fact that Dwλk = Dwk = Duk −Du on

B\Aλk we get for the first term I ′

k of the right-hand side of (28)

|I ′k| ≤ |B \Aλ

k |1−θ

(∫B\Aλ

k

(|Duk|p−2Duk − |Du|p−2Du) ·Dwλ

k dx

)θ

= |B \Aλk

∣∣1−θ

(∫B

. . . dx−∫

Aλk

. . . dx

)θ

= |B \Aλk |1−θ(J ′

k − J ′′k )θ . (31)

For J ′k we conclude using (14), (24) and (22):

lim supk→∞

|J ′k|

≤ lim supk→∞

(∣∣∣∣∫B

|Duk|p−2Duk ·Dwλk dx

∣∣∣∣+∣∣∣∣∫

B

|Du|p−2Du ·Dwλk dx

∣∣∣∣)≤ lim sup

k→∞

1k

supB

|Dwλk | +

∣∣∣∣∫B

|Du|p−2Du ·Dw dx∣∣∣∣

≤ lim supk→∞

cλ

k+∣∣∣∣∫

B

|Du|p−2Du ·Dw dx∣∣∣∣

=∣∣∣∣∫

B

|Du|p−2Du ·Dw dx∣∣∣∣ . (32)

The second term J ′′k in the right hand side of (31) is estimated as follows:

|J ′′k | =

∣∣∣∣∣∫

Aλk

(|Duk|p−2Duk − |Du|p−2Du) ·Dwλk dx

∣∣∣∣∣≤∫

Aλk

(|Duk|p−1 + |Du|p−1)|Dwλk | dx

=∫

F λk

. . . dx+∫

Gλk

. . . dx . (33)

Now, we use Holder’s inequality, (22) and (27) to conclude∫Gλ

k

. . . dx ≤ c

(∫Gλ

k

|Duk|p + |Du|p dx)1−1/p(∫

Gλk

|Dwλk |p dx

)1/p

≤ c

(∫B

|Duk|p + |Du|pdx)1−1/p

λ|Gλk |1/p

≤ cKp−1 1λ

‖Dwk‖Lp(B,RnN )

≤ cKp

λ. (34)


Similarly, we obtain using Holder’s inequality, (22) and (26)

∫F λ

k

. . . dx ≤ c

(∫F λ

k

|Duk|p + |Du|p dx)1− 1

p(∫

F λk

|Dwλk |p dx

)1/p

≤ c

(∫F λ

k

|Duk|p + |Du|p dx)1−1/p

λ|Fλk |1/p

≤ cK

(∫F λ

k

|Duk|p + |Du|p dx)1−1/p

.

Now let us consider an arbitrary subsequence (uk)k∈K0 of our sequence (uk)k∈N;from (30) we have the uniform bound

∫B

|Duk|p + |Du|pdx ≤ Kp. Therefore,fixed ε > 0, by Proposition 2, there exists λ ≥ 1

ε and a further subsequenceK1 ⊂ K0 ⊂ N (λ being independent of k ∈ K1) and such that∫

F λk

|Duk|p + |Du|p dx ≤ ε for any k ∈ K1 . (35)

Applying the previous reasoning (the one based on Proposition 1) to the subse-quence K1 ⊂ K0 ⊂ N and combining (32)–(35) we arrive at

|J ′′k | ≤ c

Kp

λ+ cε1−1/pK ≤ c(Kpε+Kε1−1/p) (36)

for k ∈ K1. From (31), (32) and (36) we infer that

lim supK1k→∞

I ′k ≤ |B|1−θ

[c(Kpε+Kε1−1/p

)+∣∣∣∫

B

|Du|p−2Du ·Dw dx∣∣∣]θ

= c

[Kpε+Kε1−1/p +

∣∣∣∫B

|Du|p−2Du ·Dw dx∣∣∣]θ

. (37)

It remains to estimate the last term coming from (32). In order to do this we shalluse some tricks from [1], going back to [6]. Let us set

G := x ∈ B : w(x) = 0and

G := G ∩ x ∈ B : wk(x) → 0 .By (18), it follows immediately that |G| = |G|. Observe that we also have, lookingat (23), that

|Aλk | ≤ cKpλ−p (38)

with a constant c = c(n, p) ≥ 1. Now let us prove that actually

|G| ≤ 2cKpλ−p , (39)


where the constant c is from (38). Indeed if this is not the case, we would have that

|G ∩ (Aλk)c| > cKpλ−p ∀ k ∈ N .

Hence, we are in position to apply Lemma 2, in order to conclude that, up to passingto a subsequence (⋂

k∈N

(Aλk)c

)∩ G = ∅ .

If x belongs to this last set then (see (25))

w(x) = limk→∞

wλk (x) = lim

k→∞wk(x) = 0

which contradicts the fact that x ∈ G. Therefore (39) holds and we can estimatethe last piece from (32) as follows:∣∣∣∣∫

B

|Du|p−2Du ·Dw dx∣∣∣∣ ≤

(∫G

|Du|p dx) p−1

p(∫

B

|Dw|p dx) 1

p

≤(∫

G

|Du|p dx) p−1

p(

lim infk→∞

∫B

|Dwλk |p dx

) 1p

≤ c

(∫G

|Du|p dx) p−1

p(

supk

∫B

|Dwk|p dx) 1

p

≤ cK

(∫G

|Du|p dx) p−1

p

where c depends only on n and p. Here we have used in the second last line thedefinition of the set Aλ

k and in the last line the bound (30). Therefore denoting byη a continuous non-decreasing function such that η(0) = 0 and∫

A

|Du|p dx ≤ η(|A|) ∀A ⊂ B, A is measurable,

we conclude using also (39)∣∣∣∣∫B

|Du|p−2Du ·Dw dx∣∣∣∣ ≤ cK

(η(2cKpλ−p)

) p−1p

(40)

Now taking (28), (29), (37), (40) and again 1λ ≤ ε into account we finally have

lim supK1k→∞

∫B


]θdx

≤ c

[Kpεp(1−θ) +

(Kpε+Kε1−1/p +K

(η(2cKpεp)

) p−1p

)θ]. (41)

Now since the quantity on the right hand side of the previous estimate tends to0 when ε → 0 and since ε > 0 can be taken arbitrarily small, we can apply theargument described in (35)-(41) starting from the whole sequence (that is taking


K0 = N) and performing a standard diagonal argument; what we finally come upwith is a further subsequence K2 ⊂ N such that

lim supK2k→∞

∫B


]θdx = 0,

i.e.[(|Duk|p−2Duk − |Duk − |Du|p−2Du

) · (Duk −Du)]θ → 0 in L1(B) as

K2 k → ∞. Therefore again up to passing to a further (this time not relabelled)subsequence, we may assume that (|Duk|p−2Duk−|Du|p−2Du)·(Duk−Du) →0 a.e. on B as K2 k → ∞; in particular, this fact and the monotonicity propertyin (10) imply that Duk → Du a.e. on B as K2 k → ∞. We have thereforeproved (20) under the additional assumption in (21).

We now consider the general case; that is we treat the case where (21) is notavailable and establish (20) again. We start from the subsequence in (16)-(19) andobserve, since the strong convergence in (17), that up to another (also this time notrelabelled) subsequence we can suppose∫

B

kp−1|uk − u|p−1 + kp|uk − u|p dx → 0 as k → ∞. (42)

Next, we define

ψk(x) =

1, on B1−1/k,k(1 − |x|), on B \B1−1/k,

and let

uk : = ψkuk + (1 − ψk)u .

Then, for ϕ ∈ C1c (B,RN ) we have∣∣∣∣∫

B

|Duk|p−2Duk ·Dϕdx∣∣∣∣

≤∣∣∣∣∫

B


+

∣∣∣∣∣∫

B\B1−1/k

(|Duk|p−2Duk − |Duk|p−2Duk

) ·Dϕdx∣∣∣∣∣

≤[

1k

+∫

B\B1−1/k

|Duk|p−1 + |Du|p−1 dx

]supB

|Dϕ|

where we have used (14) to estimate the first term in the second last line. By Holder’sinequality and the uniform bound supk≥1

∫B

|Duk|pdx ≤ Kp we have∫B\B1−1/k

|Duk|p−1 dx ≤ Kp−1|B \B1−1/k|1/p .


Similarly, we see that∫B\B1−1/k

|Duk|p−1 dx

≤ c

[∫B\B1−1/k

|∇ψk ⊗ (uk − u)|p−1 dx

+∫

B\B1−1/k

|ψkDuk + (1 − ψk)Du|p−1 dx

]

≤ c

[kp−1

∫B

|u− uk|p−1 dx+Kp−1|B \B1−1/k|1/p

]. (43)

Combining the previous estimates we arrive at∣∣∣∣∫B


≤ c

[1k

+Kp−1|B\B1−1/k|1/p +∫

B

kp−1|uk − u|p−1 dx

]supB

|Dϕ|=: δk sup

B|Dϕ| ,

where by (42) δk → 0 and, similarly to (43), also at∫B

|Duk|p dx ≤ c

[∫B

kp|u− uk|p dx+Kp

]≤: Kp .

This implies that (uk)k satisfies (14) with δk instead of 1k and the bound in (30)

with K replacing K; moreover, by definition, uk − u = ψk(uk − u) ∈ W 1,p0 (B)

for any k. It is easy to show that uk → u weakly in W 1,p(B,RN ) as k → ∞.Hence, we can apply the reasoning from the first case to conclude that there existsa subsequence K2 ⊂ N such that Duk → Du a.e. on B as K2 k → ∞. Takinginto account that ψk(x) = 1 for x ∈ B and k 1 we deduce that Duk → Dua.e. on B as K2 k → ∞. Now, Vitali’s theorem yields that Duk → Du stronglyin Ls(B,RnN ) as k → ∞ for any [1, p). This implies in particular that we canpass to the limit k → ∞ in (14) to conclude that u is weakly p-harmonic on B.This gives the desired contradiction since by (16)

∫B

|uk − u|pdx → 0 as k → ∞contradicting (15), since now one can choose h ≡ u by (19).

4. Regularity for p-Laplacian systems with critical growth

In this section the main result is a technical regularity lemma (the degenerate analogof Simon’s “technical lemma" from [17]) that is going to allow us to give a simplePDE proof of the ε-regularity theorem for energy minimizing p-harmonic maps.(We warn the reader here: in the following lemma, to avoid confusion with Lemma4, the letter ε has been replaced by δ). We recall that in this section we are going toassume the validity of Caccioppoli inequality for p-harmonic maps.


Lemma 5. Suppose α ∈ (0, 1) and cF , cCac ≥ 1 are given. Then there existsδ = δ(n,N, p, cF , cCac, α) > 0 such that the following holds: Whenever u ∈W 1,p(BR,R

N ) satisfies the equation∫BR

|Du|p−2Du ·Dϕdx =∫

BR

F · ϕdx (44)

for all ϕ ∈ C1c (BR,R

N ) where F ∈ L1(BR,RN ) satisfies for a.e. x ∈ BR

|F (x)| ≤ cF |Du(x)|p , (45)

and if u satisfies the Caccioppoli inequality(2

)p−n∫

B/2(y)|Du|p dx ≤ cCac

n

∫B(y)

|u− uy,|p dx (46)

for all B(y) BR, and if

E0 = R−n

∫BR

|u− uR|p dx ≤ δp (47)

then u ∈ C0,α(BR/2,RN ). Moreover, for all x, y ∈ BR/2 we have the estimate

|u(x) − u(y)| ≤ cE1/p0

( |x− y|R/2

)α

,

where the constant c depends only on n,N, p, cF , cCac, and α.

Proof. Let B(y) BR be an arbitrary ball. Using (44) with a test functionϕ ∈ C1

c (B/2(y),RN ) together with the hypotheses (45) and (46) we get∣∣∣∣∣(2)p−n∫

B/2(y)|Du|p−2Du ·Dϕdx

∣∣∣∣∣=

∣∣∣∣∣(2)p−n∫

B/2(y)F · ϕdx

∣∣∣∣∣≤ cF

(ρ2

)p−n∫

B/2(y)|Du|p dx sup

B/2(y)|ϕ|

≤ cF cCac−n

∫B(y)

|u− uy,ρ|p dx supB/2(y)

|Dϕ|

= 2cF cCacE(u; y, ) · ρ2

supB/2(y)

|Dϕ| ,

where we have abbreviated

E(u; y, ) ≡ Ep(u; y, ) = −n

∫B(y)

|u− uy,|p dx .


Therefore, if we introduce

v =u

βwith β = (2cF cCac)

1p−1E(u; y, )1/p

we see that v is approximatively p-harmonic in the following sense: for any ϕ ∈C1

c (B/2(y),RN ) we have∣∣∣∣∣(2)p−n∫

B/2(y)|Dv|p−2Dv ·Dϕdx

∣∣∣∣∣ ≤ E(u; y, )1/p

2sup

B/2(y)|D| . (48)

Using the Caccioppoli inequality once again we also come up with(2

)p−n∫

B/2(y)|Dv|p dx ≤ cCac

(2cF cCac)p

p−1< 1 . (49)

We now apply Lemma 4 to obtain, for a given ε > 0 a p-harmonic function h ∈W 1,p(B/2(y),RN ) such that(

2

)p−n∫

B/2(y)|Dh|p dx ≤ 1 , (50)

and (2

)−n∫

B/2(y)|v − h|p dx ≤ εp , (51)

provided

−n

∫B(y)

|u− uy,|p dx = E(u; y, ) ≤ δp , (52)

where δ = δ(n,N, p, ε) ∈ (0, 1] is the function from the p-harmonic approximationlemma 4.

Now we take θ ∈ (0, 14 ] (to be specified later). Then we have with a constant

c = c(p)

(θ)−n

∫Bθ(y)

|v − h(y)|p dx (53)

≤ c

[(θ)−n

∫Bθ(y)

|v − h|p dx+ (θ)−n

∫Bθ(y)

|h− h(y)|p dx].

The integrand of the second integral of the right-hand side of (53) can be estimatedas follows:

supBθ(y)

|h− h(y)|p ≤(θ sup

Bθ(y)|Dh|

)p

≤ (θ)p supB/4(y)

|Dh|p

≤ c(θ)p

∫–

B/2(y)|Dh|p dx ≤ cθp , (54)


where the constant c depends on n,N , and p. Here we have used (50) in the last in-equality and estimate (with c = c(n,N, p) being independent of the approximatingharmonic function h)

supB/4(y)

|Dh|p ≤ c

∫–

B/2(y)|Dh|p dx ,

in the second last line (see [19] for the case p > 2 and [2] when 1 < p < 2).Combining (53) with (51) and (54) we obtain

(θ)−n

∫Bθ(y)

|v − h(y)|p dx ≤ c(θ−nεp + θp) ,

where c = c(n,N, p). Recalling v = uβ with β = (2cF cCac)

1p−1E(u; y, )1/p we

see that

(θ)−n

∫Bθ(y)

|u− ξ|p dx ≤ c(2cF cCac)p

p−1 (θ−nεp + θp)E(u; y, ) ,

where ξ = βh(y) is a fixed vector in RN . Therefore

infξ∈RN

(θ)−n

∫Bθ(y)

|u− ξ|p dx ≤ c(θ−nεp + θp)E(u; y, ) ,

where c = c(n,N, p, cF , cCac). Note that the infimum on the left-hand side isachieved by a unique ξ0 ∈ R

N . Therefore we get, via Jensen inequality

|uy,θ − ξ0|p ≤∫–

Bθ(y)|u− ξ0|p dx ,

so that

(θ)−n

∫Bθ(y)

|u− uy,θ|p dx ≤ c(θ)−n

∫Bθ(y)

|u− ξ0|p dx

+αn|uy,θ − ξ0|p

≤ c(θ)−n

∫Bθ(y)

|u− ξ0|p dx

≤ c(θ−nεp + θp)E(u; y, ) ,

where c = c(n,N, p, cF , cCac).We now specify θ and ε. First we select θ ∈ (0, 1

4 ] such that cθp < 12θ

pα; notethat θ depends only on n,N, p, cF , cCac and α. Having chosen θ we fix ε > 0such that cθ−nεp < 1

2θpα; this determines ε = ε(n,N, p, cF , cCac, α) and, conse-

quently, δ from the p-harmonic approximation lemma, i.e. δ = δ(n,N, p, ε) =δ(n,N, p, cF , cCac, α). With these specifications we have shown: There existsδ0 = δ0(n,N, p, cF , cCac, α) > 0 such that if

−n

∫B(y)

|u− uy,|p dx ≤ δp0 (55)


for some B(y) BR then

(θ)−n∫

Bθ(y)|u− uy,θ|p dx ≤ θpα−n

∫B(y)

|u− uy,|ρ dx (56)

where θ = θ(n,N, p, cF , cCac, α) ∈ (0, 14 ].

Next, we let E0 = R−n∫

BR

|u− uR|pdx. Then for any y ∈ BR/2 we have

(R/2)−n∫

BR/2(y)|u− uy,R/2|p dx ≤ 2p+nE0 . (57)

If we impose the smallness condition E0 < 2−p−nδp0 ≡ δp and thereby choosing

δ in the statement and in particular in (47), then with = R2 , y ∈ BR/2 we have

B(y) BR and ρ−n∫

B(y) |u − uy,|pdx < δp0 , hence the starting hypothesis

(55) is satisfied on any ball B(y) with = R2 , y ∈ BR/2 and therefore (56)

holds. But this implies that (θ)−n∫

Bθ(y) |u − uy,θ|pdx < δp0 , i.e. the starting

assumption is also satisfied on Bθ(y). Hence (56) holds with θ instead of . Byinduction we easily deduce that

(θjR/2

)−n∫

BθjR/2(y)|u− uy,θjR/2|p dx

≤ θjpα (R/2)−n∫

BR/2(y)|u− uy,R/2|p dx ≤ θjpα2p+nE0 < δp

0 (58)

for each j = 0, 1, 2, . . . , and y ∈ BR/2. Now if σ ∈ (0, R2 ] there is j ∈ N0 such

that θj+1 R2 < σ ≤ θj R

2 . Then, using (57) and (58) we obtain

σ−n

∫Bσ(y)

|u− uy,σ|pdy ≤ 2pσ−n

∫Bσ(y)

|u− uy,θjR/2|p dx

≤ 2p(θj+1R/2

)−n∫

BθjR/2(y)|u− uy,θjR/2|p dx

≤ 2pθ−nθjpα (R/2)−n∫

BR/2(y)|u− uy,R/2|p dx

≤ 2pθ−n−pα

(σ

R/2

)pα

2p+nE0

= 22p+n+pαθ−n−pα( σR

)pα

E0

= cpE0

( σR

)pα

(59)

for any y ∈ BR/2, 0 < σ ≤ R2 , provided we initially assume E0 < 2−p−nδp

0 .Observe that in the previous inequality c ≡ c(θ) depends in the way prescribed in


the statement since so does θ. Then, by Campanato’s Theorem [3] we conclude thatu ∈ C0,α

(BR/2,R

N)

with the estimate

|u(x) − u(y)| ≤ cE1/p0

( |x− y|R/2

)α

(60)

for any x, y ∈ BR/2 where c is the constant from (59). An immediate consequence of the previous lemma is the possibility to derive aMorrey type decay estimate. Indeed, to estimate the left hand side in (59) frombelow we can use the Caccioppoli inequality (46) and obtain the Morrey-spacetype estimate:

σp−n

∫Bσ(y)

|Du|p dx ≤ cpE0

( σR

)pα

(61)

for any 0 < σ ≤ R/4 and y ∈ BR/2 where c exhibits the same dependence on theparameters of the one in (59).We now turn our attention to the question of higher regularity, i.e. C1,α-regularityfor some Holder-exponent 0 < α < 1, proving that Lemma 5 implies such ahigher regularity once α has been chosen sufficiently close to 1. Therefore, fixed α,this reduces the problem of regularity of Du to determine those points where thesmallness assumption (47) is satisfied and this, in turn, leads to Partial Regularitywith estimates on the size of the singular set. This final procedure is standard andwe refer to [14,15,11,8,13].

Lemma 6. Assume that the assumptions of Lemma 5 are satisfied and moreover αis such that α+ pα− p ∈ (0, 1); then Du is Holder continuous in BR/4.

Proof. Let B(y) BR/2 be arbitrary with < R/4. On B(y) we consider theDirichlet-problem

∫B(y)

|Dh|p−2Dh ·Dϕdx = 0 ∀ ϕ ∈ C1c (B(y),RN )

h− u ∈ W 1,p0 (B(y),RN ) .

(62)

Taking the difference of the equations (44) and (62) we obtain∫B(y)

(|Du|p−2Du− |Dh|p−2Dh) ·Dϕdx =

∫B(y)

F · ϕdx

for any ϕ ∈ C1c (B(y),RN ). Since u−h ∈ W 1,p

0 (B(y),RN )∩L∞ (the estimatefor supB(y) |u−h| will be shown later) it is not difficult to check that this equationis also valid for the choice ϕ = u − h; hence using first (11) and then (45), weobtain ∫

B(y)|V (Du) − V (Dh)|2 dx

≤ c

∫B(y)

(|Du|p−2Du− |Dh|p−2Dh) · (Du−Dh) dx

≤ c(n,N, p, cF ) supB(y)

|u− h|∫

B(y)|Du|p dx .


In turn, from Lemma 5 we infer that for a given α ∈ (0, 1) that

|u(x) − u(y)| ≤ 2αcE1/p0

( R

)α

for all x ∈ B(y); i.e. the image u(B(y)) is contained in the ball centered at

u(y) with Radius 2αcE1/p0

(R

)αin R

N , and from the convex-hull property of thefunctional v → ∫

B(y) |Dv|pdx we conclude that also that the image h(B(y)) iscontained in that ball. Therefore, we have∫

B(y)|V (Du) − V (Dh)|2 dx ≤ cE

1/p0

( R

)α∫

B(y)|Du|p dx , (63)

where the constant c depends on n,N, p, cF , cCac, and α. Let 0 < τ ≤ 12 (to be

specified later). Then, by the excess-decay estimate for h onB(y) (cf. Proposition3) and (63) we deduce

Φ(u; y, τ)

=∫–

Bτ(y)|V (Du) − (V (Du))y,τ|2 dx

≤∫–

Bτ(y)|V (Du) − (V (Dh))y,τ|2 dx

≤ 2[Φ(h; y, τ) +

∫–

Bτ(y)|V (Du) − V (Dh)|2 dx

]≤ c[τ2γΦ(h; y, ) + τ−n

∫–


]≤ c[τ2γΦ(u; y, ) + τ−n

∫–


]≤ c[τ2γΦ(u; y, ) + E

1/p0 τ−n

( R

)α∫–

B(y)|Du|p dx

], (64)

where c = c(n,N, p, , cF , cCac, α). The second term of the right-hand side of (64)is estimated by the Morrey-space type estimate (61) yielding

Φ(u; y, τ) ≤ c[τ2γΦ(u; y, ) + τ−nE

1+1/p0 R−α−pαα+pα−p

], (65)

where the constant c again only depends on n,N, p, cF , cCac, and α.We now use a standard iteration argument. According to the assumption on α

we define µ = α + pα − p ∈ (0, 1). We specify 0 < τ ≤ 12 such that cτγ = 1.

Note that τ depends on n,N, p, cF , cCac, α, and γ. With this choice of τ (65) takesthe form

Φ(u; y, τ) ≤ τγΦ(u; y, ) + cE1+1/p0 τ−nR−α−pα µ (66)


for all B(y) BR/2. Iterating (66) k times, k ∈ N, we find

Φ(u; y, τk) ≤ τkγΦ(u; y, ) + cE1+1/p0 τ−nR−α−pαµτ (k−1)µ

k−1∑j=0

τ j(1−µ)

≤ τkγΦ(u; y, ) + cτ−n−µE1+1/p0 R−α−pαµ τkµ

1 − τ1−µ

≤ τkγΦ(u; y, ) + cR−α−pαE1+1/p0 µτkµ ,

where c ≡ c(τ) depends only on the same quantities as before since τ does so.Now, for 0 < σ ≤ there is a unique k ∈ 0, 1, . . . such that τk+1 < σ ≤ τk,and with the usual interpolation argument it is easy to show

Φ(u; y, σ) =∫–

Bσ(y)|V (Du) − (V (Du))y,σ|2 dx

≤ τ−n

∫–

Bτk

(y)|V (Du) − (V (Du))y,τk|2 dx

≤ τ−n[τkγΦ(u; y, ) + cR−α−pαE

1+1/p0 µτkµ

]≤ τ−n

[τ−γ

(σ

)γ

Φ(u; y, ) + τ−µσµcR−α−pαE1+1/p0

]≤ c[(σ

)γ

Φ(u; y, ) +R−α−pαE1+1/p0 σµ

],

for any ball B(y) BR/2, where the constant c depends on the same quantitiesas before. In particular we have

Φ(u; y, σ) ≤ c[( σ

R/4

)γ

Φ(u; y,R/4) +R−α−pαE1+1/p0 σµ

](67)

for all y ∈ BR/4 and 0 < σ ≤ R4 . Noting that Φ(u; y,R/4) ≤ 2nΦ(u;R/2) we

obtain

Φ(u; y, σ) ≤ c[( σ

R/4

)γ

Φ(u;R/2) +R−α−pαE1+1/p0 σµ

](68)

for all y ∈ BR/4 and 0 < σ ≤ R4 . The constant c in (68) depends on the same

parameters as before. This implies that V (Du) is Holder continuous onBR/4 withHolder-exponent β = 1

2 min(γ, µ). Finally, the Holder continuity of Du, withexponent α := min1, (2/p)β, follows from Lemma 3 and the proof is complete.

Acknowledgement. G. Mingione was partially supported by MIUR via the project “Calcolodelle Variazioni" (Cofin 2000). He acknowledges the kind hospitality of the MathematicalInstitute of the Charles University (Prague) in May 2002.


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The p -harmonic approximation and the regularity of p -harmonic maps

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