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PARTIAL REGULARITY FOR LOCAL MINIMIZERS OF VARIATIONAL

INTEGRALS WITH LOWER ORDER TERMS

JUDITH CAMPOS CORDERO

ABSTRACT. We consider functionals of the form

F(u) :=

∫Ω

F (x, u,∇u) dx,

where Ω ⊆ Rn is open and bounded. The integrand F : Ω × R

N × RN×n → R is assumed

to satisfy the classical assumptions of a power p-growth and the corresponding strong quasicon-

vexity. In addition, F is Holder continuous with exponent 2β ∈ (0, 1) in its first two variables

uniformly with respect to the third variable, and bounded below by a quasiconvex function de-

pending only on z ∈ RN×n. We establish that strong local minimizers of F are of class C1,β in

an open subset Ω0 ⊆ Ω with Ln(Ω \ Ω0) = 0. This partial regularity also holds for a certain

class of weak local minimizers at which the second variation is strongly positive and satisfying

a BMO-smallness condition. This extends the partial regularity result for local minimizers by

Kristensen and Taheri (2003) to the case where the integrand depends also on u. Furthermore, we

provide a direct strategy for this result, in contrast to the blow-up argument used for the case of

homogeneous integrands.

Key words: Strong local minimizers, regularity, quasiconvexity

MSC (2010): 35B65, 35J50, 35J60, 49N99

DATE: November 4, 2021

1. INTRODUCTION

We investigate the regularity properties of W1,q-local minimizers of variational integrals of

the form

(1.1) F(u) :=

∫

ΩF (x, u,∇u) dx.

Here, Ω ⊆ Rn is an open and bounded set and u ∈ W1,p(Ω,RN ) for a fixed p ≥ 2. We

recall at this point that, for a given q ∈ [1,∞], a map u ∈ W1,p(Ω,RN ) is said to be a

W1,q-local minimizer if there exists a δ > 0 such that, for every ϕ ∈ W1,q0 (Ω,RN ) satisfy-

ing ‖∇ϕ‖Lq(Ω,RN×n) < δ, it holds that F(u) ≤ F(u + ϕ). When q ∈ [1,∞), we say that u is a

strong local minimizer, whereas if q = ∞, we call u a weak local minimizer.

Regarding the integrand F : Ω× RN × R

N×n → R, we assume that it satisfies the following

standard conditions:

(H0) F = F (x, u, z) is a continuous function, of class C2 in the variable z ∈ RN×n, and such

that Fzz is continuous in Ω× RN × R

N×n.

(H1) F satisfies a natural growth condition: for a fixed p ≥ 2 and for a constant L > 0 it holds

that, for every (x, u, z) ∈ Ω× RN × R

N×n,

|F (x, u, z)| ≤ L(1 + |z|p).1

http://arxiv.org/abs/2108.08869v2

2 J. CAMPOS CORDERO

(H2) F is strongly quasiconvex: there is a constant ℓ > 0 such that, for every (x0, u0, z0) ∈

Ω× RN × R

N×n and every ϕ ∈ W1,p0 (Ω,RN ),

ℓ

∫

Ω|V (∇ϕ)|2 dx ≤

∫

Ω(F (x0, u0, z0 +∇ϕ)− F (x0, u0, z0)) dx,

where the auxiliary function V : RN×n → R is defined by

V (z) := (|z|2 + |z|p)12 .

(H3) There exists a function G : RN×n → R such that, for every (x, u, z) ∈ Ω×RN ×R

N×n,

G(z) ≤ F (x, u, z).

Furthermore, G is assumed to be strongly quasiconvex, in the sense that G is con-

tinuous and, for a constant ℓG > 0, it holds that for every z0 ∈ RN×n and every

ϕ ∈ W1,p0 (Ω,RN ),

ℓG

∫

Ω|∇ϕ|p dx ≤

∫

Ω(G(z0 +∇ϕ)−G(z0)) dx.

(H4) There is an increasing function ρ(s) ≥ 2L such that, for a constant c > 0 and for every

(x, u, z), (y, v, z) ∈ Ω× RN × R

N×n,

|F (x, u, z) − F (y, v, z)| ≤ c ϑ(|v|, |x − y|2 + |u− v|2)(1 + |z|p),

with ϑ(s, t) = min2L, ρ(s)tβ, β ∈ (0, 12).

On the other hand, when dealing with weak local minimizers, we will replace assumption (H0)

by the following regularity condition on F :

(H0w) F = F (x, u, z) is a continuous function, of class C2 in the variables (u, z) ∈ RN ×

RN×n, and such that Fuu and Fzz are continuous in Ω× R

N × RN×n.

For the case of weak local minimizers we will also require the following growth condition on Fu:

(H1w) For a constant K > 0 it holds that, for every (x, u, z) ∈ Ω× RN × R

N×n,

|Fu(x, u, z)| ≤ K(1 + |z|p).

Remark 1.1. Assumption (H3) is a coercivity condition on F that is uniform in (x, u) ∈ Ω×RN ,

see [14]. In this regard, we emphasize that the value of the constants ℓ and ℓG does not play an

important role in our context and, furthermore, we can normalize the integrand F by minℓ, ℓGwhile the set of local minimizers remains unchanged. Whereby, for the rest of the paper we shall

assume, without loss of generality, that ℓ = ℓG = 1.

We note that (H3) plays a fundamental role in obtaining a preliminary higher integrability

on the local minimizers under investigation. On the other hand, assumption (H4) is a Holder

continuity condition on the (x, u) variables and uniform in z.

We also record that assumptions (H3) and (H4) enable us to establish suitable estimates for

the case under study, in which F depends on the lower order terms x ∈ Ω and u ∈ RN , in

addition to its dependence on z ∈ RN×n. A corresponding assumption of the form of (H4) is

also necessary when F depends only on x and z, see [7], while condition (H3) is only required

when the dependence on u is present. However, we recall that all the hypotheses (H0)-(H4) are

standard when proving partial regularity of global minimizers if the integrand depends also on

(x, u), see, for example, [1, 5, 30, 31, 36, 40].

The study of local minimizers in the calculus of variations has been broadly motivated by

models arising in materials science. In the same spirit, a fundamental example of John in [39]

PARTIAL REGULARITY FOR LOCAL MINIMIZERS OF VARIATIONAL INTEGRALS WITH LOWER ORDER TERMS3

establishes non-uniqueness of equilibrium points in nonlinear hyperelasticity. The annular shape

of the domain in this example and the symmetric properties of the possible solutions to the

equilibrium equations provided strong evidence that the topology of Ω plays a central role in the

number of local minimizers admitted by the functional.

The question of existence of local minimizers was addressed by Post and Sivaloganathan for

low dimensions in [45], see also [50]. Kristensen and Taheri further extended the previous exam-

ples to obtain W1,p-local minimizers for homogeneous integrands satisfying a natural p-growth

condition. Furthermore, it was shown by Taheri in [51] that, if the domain is star-shaped, then

strong local minimizers are actually global minimizers if they are subject to linear boundary con-

ditions. The problem of existence was finally settled in great generality by Taheri in [52], where

a lower bound for the number of local minimizers of functionals of the form (1.1) is given in

terms of certain topological properties of the domain.

The foundations of regularity theory for solutions to elliptic systems rely on the astounding

works of De Giorgi [17] and Almgren [3, 4], that were followed by those of Giusti and Miranda

[32], Morrey [43], and others. Later on, Giaquinta and Giusti [28] established partial regularity

for minimizers of functionals as in (1.1) under the assumption of quadratic growth and strong

convexity in the z variable. Many other works followed in the case of convexity in the z-variable,

see [25, 31].

On the other hand, in the quasiconvex setting it wasn’t until the work of Evans [20] that partial

regularity of class C1,α was established for minimizers, under the assumption that the integrand

satisfies a controlled quadratic growth. Several important generalizations of his work have been

established, including [1, 2, 12, 13, 15, 18, 19, 21, 22, 23, 26, 33, 36, 37, 40, 47]. Of particular

relevance to us is the direct approach to establish partial regularity of minimizers of functionals

of the form (1.1), with F satisfying a p-growth as in (H1) and a strong quasiconvexity condition

of the form (H2). The foundations of this method go back to the early works of [25]-[30].

In the quasiconvex setting, a full regularity result under a smallness condition on the boundary

datum has been obtained in [9]. Furthermore, the impact of regularity on uniqueness has been

studied by the author and Jan Kristensen in [10], where a uniqueness result for global minimizers

of quasiconvex functionals has been obtained, once again under a natural smallness restriction

on the Dirichlet boundary condition.

Regarding the case of local minimizers, Kristensen and Taheri established in [41] that, under

the classical assumptions of natural growth and strong quasiconvexity of a homogeneous inte-

grand F = F (z), W1,q-local minimizers are of class C1,α in a subset of the domain Ω of full

n-dimensional measure. They observed that an important challenge when dealing with W1,q-

local minimizers, if q > p, is that of obtaining higher integrability properties. An important

reason leading to this obstacle is that, while establishing partial regularity via an indirect blow-

up argument, the Caccioppoli inequality that can be obtained carries on the right hand side a term

of the form

(1.2) θ

∫

B(x0,r)|∇uj −∇aj|

p dx,

where θ ∈ (0, 1), (uj) is the blown-up sequence and (aj) is a sequence of suitable affine maps.

In the case of global minimizers, this term can be made to disappear after an iteration process as

in [20]. However, in the case of local minimizers, such an iteration would require letting j → ∞,

which cannot be done at this stage since the sequence (uj) is only known to converge weakly

in W1,p. On the other hand, Evans and Gariepy had already observed in [21] that a Cacciop-

poli inequality of the first kind (in the sense of [31, Ch. 9]) is not really necessary to establish

4 J. CAMPOS CORDERO

convergence in W1,2 of the blown-up sequence. However, their argument does not resolve the

issue for strong local minimizers. In [41], the obstacle of not being able to iterate the Cacciop-

poli inequality carrying the term in (1.2) is overcome for W1,q-local minimizers by means of a

measure-theoretical argument, thanks to which strong convergence can be established bypassing

the need of the usual estimates that give higher integrability. Other interesting regularity results

concerning local minimizers can be found in [5, 6, 11, 15, 18, 46].

In [7] it was shown that the blow-up strategy can be suitably adapted to obtain partial regularity

of minimizers when the integrand is of the form F (x, z), with (x, z) ∈ Ω × RN×n and the

treatment can even be taken to establish partial regularity up to the boundary of Ω. However,

the case in which the integrand F depends also on u ∈ RN poses extra technical challenges.

Such difficulties arise mainly from the fact that a uniform coercivity assumption of the form (H3)

cannot suitably be blown up and, just as in the case of global minimizers for functionals with

lower order terms, see [30], the strong quasiconvexity condition (H2) also fails to be enough to

establish a pre-Caccioppoli inequality in this case.

On the other hand, and in remarkable contrast with the aforementioned regularity results,

Muller and Sverak [44] built examples of Lipschitz solutions to the weak Euler-Lagrange equa-

tions associated to a quasiconvex variational problem, but that are nowhere of class C1. Kris-

tensen and Taheri [41] provided a modification of these examples so that the second variation

at the Lipschitz solutions to the Euler-Lagrange equations was actually uniformly positive, and

hence the solutions were, in fact, weak local minimizers. Furthermore, the importance of the

regularity properties of strong local minimizers has also been made evident in the sufficiency

result established by Grabovsky and Mengesha in [34], see also [8].

In this setting, the main objective of this work is to establish a partial C1,α-regularity result

for W1,q-local minimizers of variational functionals of the form (1.1), where the integrand also

depends on u. It is worth noting that, just as in the case of homogenous integrands first established

in [41], and then extended to integrands with x dependence in [7], we require the local minimizer

to be a priori a W1,qloc map. It remains unclear whether this assumption is really necessary.

As usual, our regularity result will rely on a suitable decay of the mean oscillations denoted

by

E(y, r) := −

∫

B(y,r)|V (∇u− (∇u)y,r)|

2 dx,

were B(y, r) ⊆ Ω is a ball and the bar on the integral signifies its mean value, as specified in the

list of notations below.

Our main theorem is the following.

Theorem 1.2. Let F : Ω×RN×R

N×n → R be an integrand of class C2 in the variable in RN×n

and satisfying (H0)-(H4). Let q ∈ [2,∞] and assume that u ∈ W1,p(Ω,RN ) ∩W1,qloc(Ω,R

N ) is

a W1,q-local minimizer of F . Let

Σ0 :=

x ∈ Ω : lim supr→0

|(u)x,r|+ |(∇u)x,r| = ∞ or lim infr→0

E(x, r) > 0

and define Ω0 := Ω \ Σ0. Then, Ω0 is open, Ln(Σ0) = 0 and, if q ∈ [2,∞), then u is locally of

class C1,β in Ω0. On the other hand, if q = ∞, we assume in addition that the second variation

at u is strongly positive, as in (H5), and that assumptions (H0w) and (H1w) also hold. Then,

there exists a δ0 > 0 such that, if δ1 ∈ (0, δ0) and if

(1.3) lim supr→0

‖∇u(y)− (∇u)x,r‖L∞(B(x,r),RN×n) < δ1


locally uniformly in x ∈ Ω, then u is also locally of class C1,β in Ω0.

Note that, by the examples cited above, condition (1.3) is indeed necessary for the case of

W1,∞-local minimizers.

The strategy that we follow is fundamentally inspired by the direct argument of Giaquinta and

Modica [30] to establish partial regularity of global minimizers. The application of this approach

is new in the context of W1,q-local minimizers. An important challenge to overcome is that, just

as for the case without u-dependence, we can only obtain a Caccioppoli of the first kind, which

is stated in Theorem 4.4.

As in the homogeneous case, the restrictions imposed by the fact that we are dealing with local

minimizers do not allow us to perform an iteration to force the term θ −∫

Br|V (∇u − z0)|

2 dx to

vanish. However, a generalized version of Gehring’s Lemma, originally established in [24], was

obtained by Stredulinsky in [49]. This generalized version enables the Caccioppoli inequality

to self-improve into a similar one with higher exponents on all the relevant terms. Whereby, by

combining this with a preliminary higher integrability obtained in a classical way, we can finally

obtain a suitable reverse Holder inequality, which is stated in Theorem 4.5.

The reverse Holder inequality will finally allow us to establish a good decay rate for the mean

oscillations of ∇u. This is achieved by comparing the mean oscillations of the minimizer u with

those of the (regular) minimizer of a second order Taylor polynomial of a frozen integrand of the

form F (x0, u0, z), as we show in Theorem 4.7.

The main body of the paper is organized as follows. In Section 2 we recall classical estimates

for the integrands satisfying our main assumptions. Furthermore, we state Stredulinsky’s version

of Gehring’s Lemma, being one of the main ingredients of our regularity proof. The final part of

Section 2 includes the statements of the decay rate and the Lp-estimates satisfied by A-harmonic

maps, namely, the minimizers of strongly quasiconvex quadratic functionals. The main content

of this work is contained in Sections 3 and 4. In Section 3 we prove that weak local minimizers

at which the second variation is strongly positive are, in fact, W1,BMO-local minimizers, in the

sense that u minimizes the energy under perturbations for which the BMO-seminorm of the

derivative is small. Finally, in Section 4 we establish the proof of Theorem 1.2. Concerning the

different steps of the proof, we remark that the linearization process in Subsection 4.4 makes use

of the aforementioned W1,BMO-minimality result for the case of weak local minimizers, since it

is by these means that we can use the local minimality property when comparing the energy at uwith that of the solution to the linearized problem.

We now introduce the notation that we use in the rest of the paper.

1.1. Notation: Throughout this work, we shall use the following set of notational conventions:

• | · | denotes the Euclidean norm in any space Rm. If m = N × n, so that we are in a

space of matrices, it denotes the trace norm, so that |z| := (Tr(zzt))12 ;

• B(x0, r) denotes the open ball centred at x0 ∈ Rn and with radius r > 0;

• B(x0, r) denotes the closure in Rn of B(x0, r);

• if we have fixed a ball B(y, r), we shall often write Br instead of B(y, r);

• if ζ > 0 and we have fixed a ball B(y, r), we shall denote by Bζr the ball still centred at

y and with radius ζr.

• for a given open set ω ⊆ Rn of finite Lebesgue measure Ln(ω) ∈ (0,∞), and for a given

map f : ω → Rm, we denote the average value of f over ω by

(f)ω =1

Ln(ω)

∫

ωf dx = −

∫

ωf dx.

6 J. CAMPOS CORDERO

In the particular case that ω = B(x0, r), we denote (f)x0,r := (f)ω;

• for a function f ∈ C1,γ(ω,Rm), with γ ∈ (0, 1), ‖f‖C1 := ‖f‖C1(ω,Rm) + [∇f ]C0,α ,

where [∇f ]C0,α denotes de α-Holder semin-norm of ∇f ;

• for a function f ∈ W1,s(Ω,RN ), we denote ‖f‖1,s := ‖f‖Ls(Ω,RN ) + ‖∇f‖Ls(Ω,RN×n);

• for 1 ≤ p < ∞, W1,p0 (Ω,RN ) refers to the closure in W1,p of the space of compactly

supported smooth functions C∞c (Ω,RN ).

On the other hand, W1,∞0 (Ω,RN ) := W1,∞(Ω,RN ) ∩W1,1

0 (Ω,RN ).For 1 ≤ p ≤ ∞ and u ∈ W1,p(Ω,RN ), W1,p

u (Ω,RN ) denotes the affine space u +

W1,p0 (Ω,RN );

• the letter c will be used to denote a constant and it may change its specific value from

line to line in a chain of equations.

2. PRELIMINARY RESULTS

It is well known that, under the assumptions (H1) and (H2), the following estimates hold:

there exists a constant c = c(L) > 0 such that, ∀(x, u, z), (x, u,w) ∈ Ω× RN × R

N×n,

(2.1) |F (x, u, z) − F (x, u,w)| ≤ c(1 + |z|p−1 + |w|p−1)|z − w|

and

(2.2) |Fz(x, u, z)| ≤ c(1 + |z|p−1).

See [16, 31] as references for these estimates.

Another classical and useful result concerns the following growth properties for shifted frozen

integrands:

Lemma 2.1. Let m > 0 and let (x0, u0, z0) ∈ RN×n be fixed, with |u0| + |z0| ≤ m. Assume

that F : Ω × RN × R

N×n → R is of class C2 in the z variable and that it satisfies assumptions

(H1) and (H2). We define the shifted frozen integrand

F (z) :=F (x0, u0, z0 + z)− F (x0, u0, z0)− Fz(x0, u0, z0)[z]

=

∫ 1

0(1− t)Fzz(x0, u0, z0 + tz)[z, z] dx.

Then, there exists a constant c = c(L,m,F ′′) > 0 such that, for every z ∈ RN×n, we have that

|F (z)| ≤ c(

|z|2 + |z|p)

|F ′(z)| ≤ c(

|z|+ |z|p−1)

.

Furthermore, for every z, w ∈ RN×n, it holds that

|F (z) − F (w)| ≤ c(

|z|+ |w| + |z|p−1 + |w|p−1)

|z − w|.

Under the assumption that |u0|+ |z0| ≤ m, the proof of this lemma follows from the result of

Acerbi & Fusco in [1, Lemma 2.3].

We now state the following version of Gehring’s Lemma. A proof of this result can be found

in [25, Ch. V. Proposition 1.1]. The following version of the result was proved by Stredulinsky in

[49]. See [42] and [29, Proposition 5.1] for earlier developments generalizing Gehring’s Lemma

in this direction.


Theorem 2.2. Let B ⊆ Rn be an open ball and let p > 1. For M ≥ 1, let f, g ∈ Lp(B,RM ),

ζ ∈ (0, 1) and assume that θ ∈ (0, 1) and K > 0 are constants such that, for every Br =B(x0, r) ⊆ B,

−

∫

Bζr

|g|p dx ≤ θ−

∫

Br

|g|p dx+K

(

−

∫

Br

|g|dx

)p

+−

∫

Br

|f |p dx.

Then, there exist ε = ε(K, θ, q, n) > 0 and a constant c = c(K, θ, q, n) > 0 such that, for every

q ∈ [p, p + ε), it holds that g ∈ Lq(12B,RM ) and

(

−

∫

12B

|g|q dx

)1q

≤ c

(

−

∫

B

|g|p dx

) 1p

+ c

(

−

∫

B

|f |q dx

) 1q

.

The following estimate is of standard use in regularity theory and it is an easy consequence

of the convexity of the function V and Jensen’s inequality. We shall make use of it freely for

different domains ω ⊆ Rn.

Lemma 2.3. Let p ≥ 2 and let ω ⊆ Rn be an open set. Then, there exists a constant c = c(p) > 0

such that, for n,M ≥ 1, every f ∈ Lp(ω,RM ) and every ξ ∈ RM ,

(2.3) −

∫

ω|V (f − (f)ω)|

2 dx ≤ c−

∫

ω|V (f − ξ)|2 dx.

We conclude this section by stating the following two results for A-harmonic maps. The

regularity properties satisfied by these is the fundamental cornerstone that enables us to establish

partial regularity in the general nonlinear case.

Theorem 2.4. Let Ω ⊂ Rn be open and bounded. Assume that A : RN×n × RN×n → R is a

symmetric bilinear form satisfying that

(i) for every ξ, η ∈ RN×n, A[ξ, η] ≤ L|ξ||η| and

(ii) A is strongly quasiconvex, i.e., for every ϕ ∈ W1,20 (Ω,RN ),

(2.4) 2

∫

Ω|∇ϕ|2 dx ≤

∫

ΩA[∇ϕ,∇ϕ] dx.

Let u ∈ W1,p(Ω,RN and take BR ⊆ Ω to be an open ball. Then, there exists a unique h ∈W1,2

u (BR,RN ) minimizing the functional

v 7→

∫

BR

A[∇v,∇v] dx,

over W1,2u (BR,R

N ) and, for a constant c = c(n,L) it holds that, for every r ∈ (0, R),

−

∫

Br

|V (∇h− (∇h)r)|2 dx ≤ c

( r

R

)2−

∫

BR

|V (∇h− (∇h)R)|2 dx.

Good references for this classical result are [27], [31, Theorem 10.7].

It is worth recalling at this point that if a function F 0 : RN×n → R is of class C2 and satisfies

the strong quasiconvexity condition that for every z0 ∈ RN×n and every ϕ ∈ W1,2

0 (Ω,RN ) it

holds that∫

Ω|V (∇ϕ)|2 dx ≤

∫

Ω

(

F 0(z0 +∇ϕ)− F 0(z0))

dx,

then for every ϕ ∈ W1,20 (Ω,RN ) the following strong quasiconvexity condition is satisfied:

(2.5) 2

∫

Ω|∇ϕ|2 dx ≤

∫

ΩF 0zz(z0)[∇ϕ,∇ϕ] dx.

8 J. CAMPOS CORDERO

Indeed, for ϕ ∈ C∞0 (Ω,RN ) this follows from the fact that t = 0 minimizes the real valued

function

J (t) :=

∫

Ω

(

F 0(z0 + t∇ϕ)− F 0(z0)− |V (t∇ϕ)|2)

dx

and, hence,

(2.6) 0 ≤ J ′′(0) =

∫

Ω

(

F 0zz(z0)[∇ϕ,∇ϕ] − 2|∇ϕ|2

)

dx.

For ϕ ∈ W1,20 (Ω,RN ), the inequality (2.5) follows then by approximation.

The following result concerns the well known Lp-estimates for A-harmonic maps.

Theorem 2.5. Let A : RN×n ×RN×n → R be a symmetric bilinear form as in Theorem 2.4 and

let u ∈ W1,q(BR,RN ). If h ∈ W1,2

u (BR,RN ) minimizes the functional

v 7→

∫

BR

A[∇v,∇v] dx,

and if q ∈ [2,∞), there exists a constant Kq = Kq(L, n, q) > 0 such that, for every constant

vector z0 ∈ RN×n,

‖∇h− z0‖Lq(BR,RN×n) ≤ Kq‖∇u− z0‖Lq(BR,RN×n).

On the other hand, if u ∈ W1,∞(Ω,RN ), there exists a constant A∞ = A∞(L, n) > 0 such

that, for every constant vector z0 ∈ RN×n,

[∇h− z0]BMO(BR,RN×n) ≤ A∞‖∇u− z0‖L∞(BR,RN×n).

The proof for q = 2 follows from the strong quasiconvexity condition. The proof for q ∈(2,∞] can be found in [31, Theorem 2.14 and Theorem 10.15].

3. WEAK LOCAL MINIMIZERS

3.1. An improved local minimality property with BMO variations. In this section we will

establish that weak local minimizers at which the second variation is strictly positive are, in fact,

W1,BMO-local minimizers, in the terminology of [7]. This was first observed for homogeneous

integrands in [41, Theorem 6.1] (see also [7, Theorem 4.4]). The impact of small W1,BMO-

variations has been studied in [48], while interesting results concerning the regularity of ex-

tremals with BMO-small gradient recently appeared in [38].

Definition 3.1. Let φ ∈ L1(Ω,RN×n). We say that φ is of bounded mean oscillation if and only

if

supB(x,r)⊆Ω

−

∫

B(x,r)

|φ− (φ)x,r|dy <∞.

In this case, we define the semi-norm

[φ]BMO(Ω,RN×n) := supB(x,r)⊆Ω

−

∫

B(x,r)

|φ− (φ)x,r|dy <∞

and we set

BMO(Ω,RN×n) :=

φ ∈ L1(Ω,RN×n) : [φ]BMO(Ω,RN×n) <∞

.


In order to simplify the notation later on, for (x, v, w) ∈ Ω×Rn×R

N×n we define the bilinear

form L(x, u, z) by

L(x, u, z)[(v,w), (v , w)] :=Fuu(x, u, z)v · v + Fuz(x, u, z)v · w + Fuz(x, u, z)v · w

+ Fzz(x, u, z)w · w(3.1)

for every v, v ∈ RN and every w, w ∈ R

N×n.

We shall also denote F (x) := F (x, u(x),∇u(x)) and we adopt the corresponding notation

for all the partial derivatives of F .

For the last part of this section we will assume that u ∈ W1,∞(Ω,RN ) is a weak local mini-

mizer of F at which the second variation is uniformly strictly positive, meaning that there exists

a constant c0 > 0 such that, for all ϕ ∈ W1,∞0 (Ω,RN ),

∫

Ω(Fuu(x)[ϕ(x), ϕ(x)] + 2Fuz(x)[∇ϕ(x), ϕ(x)] + Fzz(x)[∇ϕ(x),∇ϕ(x)]) dx

=

∫

ΩL(x, u(x),∇u(x))[(ϕ,∇ϕ), (ϕ,∇ϕ)] dx(H5)

≥c0||∇ϕ||22.

Lemma 3.2. Let F : Ω×RN ×R

N×n → R be such that (H0w), (H1), (H1w) and (H2) hold for

some p ≥ 2. Let u ∈ W1,∞(Ω,RN ) and define the function G : Ω× RN × R

N×n → R by

G(x, y, z) :=F (x, u(x) + y,∇u(x) + z)− F (x)− Fy(x)[y]− Fz(x)[z]

=

∫ 1

0(1− t)L(x, u(x) + ty,∇u(x) + tz)[(y, z), (y, z)],

where, for each (x, y, z) ∈ Ω× RN × R

N×n, the bilinear form L(x, u, z) is defined as in (3.1).

Then, there exists a constant c = c(‖u‖1,∞) > 0 such that, for each x ∈ Ω, y, y ∈ RN and

z, z ∈ RN×d,

|G(x, y, z) −G(x, y, z)| ≤ c (Ap−1(y, z, y, z)|z − z|+Ap(y, z, y, z)|y − y|) ,

where

Ap(y, z, y, z) = |y|+ |y|+ |z|+ |z|+ |z|p + |z|p.

The proof of this result can be found in [8, Lemma 4.4] (see also [34]). It follows the truncation

strategy developed by Acerbi & Fusco in [1].

The regularity of the second partial derivatives of F can be phrased in terms of the existence

of a modulus of continuity for the function L, as we settle in the following lemma.

Lemma 3.3. Assume that F : Ω×RN ×R

N×n → R satisfies the condition (H0w) and let m > 1be fixed. There exists a modulus of continuity ωL : [0,∞) → [0, 1] such that ωL is increasing,

concave, ωL(t) ≥ 1 for every t ≥ 1, limt→0 ωL(t) = 0, and with the property that, for a

constant c = c(m) > 0 and for every (x, u, z), (x, v, w) ∈ Ω × RN × R

N×n satisfying that

|u|+ |v|+ |z|+ |w| ≤ m+ 1, it holds that

|L(x, u, z) − L(x, v, w)| ≤ c ωL(|u− v|+ |z − w|).

The proof follows a standard scheme and we only sketch it here for the convenience of the

reader:

10 J. CAMPOS CORDERO

Proof. By assumption the function L is continuous in its domain and it is bounded in any set of

the form Ω×B(0,m+ 1) ⊆ Ω× RN × R

N×n. Let

K := 1 + sup|u|+|z|≤m+1

|L(x, u, z)|.

We can then ensure that K is well defined. Define now

ω0(t) :=1

2Ksup

|L(x, u−v, z−w)| : |u−v|+|z−w| ≤ t and |u|+|v|+|z|+|w| ≤ m+1

.

ωL : [0,∞) → [0, 1] can then be given as the concave envelope of the function

ω(t) := maxω0(t),mint, 1.

The function ωL is a modulus of continuity for L in the set Ω×B(0,m+ 1) ⊆ Ω×RN ×R

N×n

and it satisfies all the desired properties.

We will require the following definition and the subsequent lemmata, which generalize the

Hardy-Littlewood-Fefferman-Stein maximal inequalities.

Definition 3.4. Let f : Rn → RN×n be an integrable map. We define the Hardy-Littlewood

maximal function by

f⋆(x) := supB(y,r)∋x

−

∫

B(y,r)

|f(y)|dy,

where the supremum is taken over all balls B(y, r) ⊆ Rn containing x. Similarly, the Fefferman-

Stein maximal function is given by

f#(x) := supB(y,r)∋x

−

∫

B(y,r)

|f(y)− (f)y,r|dy.

Lemma 3.5. Let Φ: [0,∞) → [0,∞) be a continuously increasing function with Φ(0) = 0. As-

sume, in addition, that Φ(t) = tpA(t) for some p > 1 and some increasing function A : [0,∞) →

[0,∞). Then, there exists a constant γ = γ(n, p) such that

(3.2)

∫

Rn

Φ(|f |)dx ≤

∫

Rn

Φ(f⋆)dx ≤ γ

∫

Rn

Φ(2|f |) dx

for all f ∈ L1(Rn,RN×n).

The proof of the first inequality in this lemma follows from the fact that Φ is increasing and

from Lebesgue’s Differentiation Theorem, which implies that |f(x)| ≤ f⋆(x) for almost every

x ∈ Rn. For a proof of the second inequality we refer the reader to [35, Lemma 5.1]. It is well

known that both notions of maximal functions are related in the following way.

Lemma 3.6. Let Φ: [0,∞) → [0,∞) be a continuously increasing function with Φ(0) = 0. Let

ε > 0 and f ∈ L1(Rn,RN×n). Then,

(3.3)

∫

Rn

Φ(f⋆) dx ≤5n

ε

∫

Rn

Φ

(

f#

ε

)

dx+ 2 · 53nε

∫

Rn

Φ(5n2n+1f⋆) dx.

If, in addition, we have that

supt>0

Φ(2t)

Φ(t)<∞,


we can further conclude that there is a constant γ1 = γ1(n) such that, for every f ∈ L1(Rn,RN×n)

satisfying that∫

Rn

Φ(f⋆)dx <∞, it holds that

(3.4)

∫

Rn

Φ(f⋆)dx ≤ γ1

∫

Rn

Φ(f#) dx.

The proof of (3.3) can be found, for example, in [41]. Inequality (3.4) follows easily from

(3.3) under the given extra assumptions.

An important remark to make at this point is the following:

Remark 3.7. If ϕ ∈ W1,1(Ω,RN ) is a Sobolev map, then for every x ∈ Ω we have, by Poincare

inequality, that for a constant c = c(n,N) > 0,

(3.5) ϕ#(x) := supQ∋x

−

∫

Q|ϕ− ϕQ|dy ≤ c sup

Q∋x−

∫

Q|∇ϕ|dy = c(∇ϕ)∗(x).

The main result of this section can now established and we state it as follows.

Theorem 3.8. Let F : Ω × RN × R

N×n → R be a function satisfying (H0w), (H1), (H1w) and

(H2) for some 2 ≤ p < ∞. Let u ∈ W1,∞(Ω,RN ) be an extremal with strictly positive second

variation. In other words, assume that

(3.6)

∫

Ω

(Fy(x, u,∇u)[ϕ] + Fz(x, u,∇u)[∇ϕ]) dx = 0

and that (H5) is satisfied. Then, there exists a δ∗ > 0 such that

∫

Ω(F (x, u+ ϕ,∇u+∇ϕ)− F (x, u,∇u)) dx ≥ c

∫

Ω|∇ϕ|2 dx

for every ϕ ∈ W1,∞0 (Ω,RN ) satisfying [∇ϕ]BMO ≤ δ∗.

Proof. Let ϕ ∈ W1,∞0 (Ω,RN ). We define the sets:

A :=x ∈ Ω : |ϕ(x)| + |∇ϕ(x)| ≤ 1;

B :=x ∈ Ω : |ϕ(x)| + |∇ϕ(x)| > 1.


Then, we have that

∫

Ω(F (x, u+ ϕ,∇u+∇ϕ)− F (x, u,∇u)) dx

=

∫

Ω(F (x, u+ ϕ,∇u+∇ϕ)− F (x, u,∇u)− Fy(x, u,∇u)[ϕ] − Fz(x, u,∇u)[∇ϕ]) dx

=

[∫

Ω(F (x, u+ ϕ,∇u+∇ϕ)− F (x, u,∇u)− Fy(x, u,∇u)[ϕ] − Fz(x, u,∇u)[∇ϕ]) 1B dx

−1

2

∫

ΩL(x, u,∇u)[(ϕ,∇ϕ)(ϕ,∇ϕ)]1B dx

]

+

[∫

Ω

∫ 1

0(1− t)L(x, u+ tϕ,∇u+ t∇ϕ)[(ϕ,∇ϕ)(ϕ,∇ϕ)]1A dt dx

−

∫

Ω

∫ 1

0(1− t)L(x, u,∇u)[(ϕ,∇ϕ)(ϕ,∇ϕ)]1A dt dx

]

+1

2

∫

ΩL(x, u,∇u)[(ϕ,∇ϕ)(ϕ,∇ϕ)] dx

=: I + II + III.

Note first that, by assumption (H5),

III ≥ c0

∫

Ω|∇ϕ|2 dx.

Now let ωL : [0,∞) → [0,∞) be a modulus of continuity for L on the set

(x, y, z) ∈ Ω× RN × R

N×n : |y|+ |z| ≤ ‖∇u‖∞ + 1

and such that it satisfies the properties given by Lemma 3.3. Using in particular that ωL is

increasing, we can then estimate the term II as follows:

II ≥− c

∫

ΩωL (|ϕ| + |∇ϕ|) (|ϕ|2 + |∇ϕ|2)1A dx

≥− c

∫

ΩωL (|ϕ| + |∇ϕ|) (|ϕ|2 + |∇ϕ|2 + |ϕ|2p + |∇ϕ|2p)1A dx.

The need to consider the second inequality will become evident once we estimate the remaining

term. Indeed, using Lemma 3.2 and the fact that |ϕ(x)| + |∇ϕ(x)| > 1 for x ∈ B, we obtain,


for a constant c > 0 that depends on ‖u‖1,∞, that

I ≥− c

∫

Ω(1 + |ϕ|+ |∇ϕ|+ |∇ϕ|p−1)|∇ϕ|1B dx

− c

∫

Ω(1 + |ϕ|+ |∇ϕ|+ |∇ϕ|p)|ϕ|1B dx

≥− c

∫

Ω(|ϕ|+ |∇ϕ|+ |∇ϕ|p−1)|∇ϕ|1B dx

− c

∫

Ω(|ϕ|+ |∇ϕ|+ |∇ϕ|p)|ϕ|1B dx

≥− c

∫

Ω(|ϕ|2 + |∇ϕ|2 + |ϕ|2p + |∇ϕ|2p)1B dx

=− c

∫

ΩωL(|ϕ|+ |∇ϕ|)(|ϕ|2 + |∇ϕ|2 + |ϕ|2p + |∇ϕ|2p)1B dx.

For the last inequality before the equal sign we have used that, if a, b > 0, then ab ≤ 12 (a

2 + b2)and, in addition, that if 1 ≤ q ≤ s <∞, then for every a > 1, it holds that aq ≤ as. On the other

hand, the last identity follows from the fact that ωL(t) = 1 for t ≥ 1.

Compiling all the estimates above, and using that for s ≥ 1 and a, b > 0, (as+bs) ≤ cs(a+b)s,

we can further obtain that, for a new constant c > 0,

∫

Ω(F (x, u+ ϕ,∇u+∇ϕ)− F (x, u,∇u)) dx

≥c0

∫

Ω|∇ϕ|2 dx− c

∫

ΩωL(|ϕ| + |∇ϕ|)

[

(|ϕ|+ |∇ϕ|)2 + (|ϕ| + |∇ϕ|)2p]

dx.(3.7)

We now define the function Φs(t) := ωL(t)ts, with s ∈ 2, 2p. Note that, since Φs is increasing,

then for every a, b ≥ 0 we have that

(3.8) Φs(a+ b) ≤ Φs(2a) + Φs(2b).

This implies that

(3.9) Φs(|ϕ|+ |∇ϕ|) ≤ Φs(2|ϕ|) + Φs(2|∇ϕ|).

On the other hand, by applying to the function 2ϕ Lemmata 3.5, and 3.6, Remark 3.7 and then

Lemma 3.6 one more time, and assuming that ϕ takes the value of 0 outside of Ω, we whereby

obtain that, for constants c = c(n,N) > 0 and γ = γ(n, p) > 0,

(3.10)

∫

Rn

Φs(2|ϕ|) dx ≤ γ

∫

Rn

Φs(c(∇ϕ)#) dx.

From (3.8), (3.10) and after applying Lemmata 3.5 and 3.6 once again, but this time to the

function 2∇ϕ, we obtain that

(3.11)

∫

Rn

Φs(|ϕ| + |∇ϕ|) dx ≤ γ

∫

Rn

Φs(c(∇ϕ)#) dx.


From inequalities (3.7) and (3.11) we obtain that∫

Ω(F (x, u+ ϕ,∇u+∇ϕ)− F (x, u,∇u)) dx

≥ c0

∫

Ω|∇ϕ|2 dx− c

∫

Rn

ωL(c(∇ϕ)#)[

((∇ϕ)#)2 + ((∇ϕ)#)2p]

dx(3.12)

≥ c0

∫

Rn

((∇ϕ)#)2 dx− c

∫

Rn

ωL(c(∇ϕ)#)[

((∇ϕ)#)2 + ((∇ϕ)#)2p]

dx.

For the second inequality above we have used (3.2) and (3.4) once again.

Note that, if δ∗ ∈ (0, 1) is such that (∇ϕ)# < δ∗, the fact that 2p ≥ 2 will imply that that

((∇ϕ)#)2p < ((∇ϕ)#)2. Then, (3.12) becomes∫

Ω(F (x, u+ ϕ,∇u+∇ϕ)− F (x, u,∇u)) dx

≥ c0

∫

Rn

((∇ϕ)#)2 dx− c

∫

Rn

ωL(c(∇ϕ)#)((∇ϕ)#)2 dx.

Finally, since ωL is continuous at t = 0 and ωL(0) = 0, we can ensure that there exists δ∗ ∈ (0, 1)

such that, if 0 ≤ t < δ∗, then

cωL(ct) < c0.

Whereby, for every ϕ ∈ W1,∞0 (Ω,RN ) satisfying that (∇ϕ)# < δ∗, we will have that, for a

constant c = c(n,N, p) > 0,∫

Ω(F (x, u+ ϕ,∇u+∇ϕ)− F (x, u,∇u)) dx ≥ c

∫

Rn

((∇ϕ)#)2 dx.

Using again Lemmata 3.5 and 3.6, we can further conclude that, for such ϕ ∈ W1,∞0 (Ω,RN ),

∫

Ω(F (x, u+ ϕ,∇u+∇ϕ)− F (x, u,∇u)) dx ≥ c

∫

Ω|∇ϕ|2 dx.

This concludes the proof of the theorem.

4. THE REGULARITY RESULT

The proof of Theorem 1.2 will follow in a classical way from the following decay estimate for

the mean oscillations of the local minimizer u:

Proposition 4.1. Assume that F and u are as in Theorem 1.2. For every m > 1 there exist γ ∈(0, β), α ∈ (2γ, 1), Rδ ∈ (0, 1), τ0 ∈ (0, 14), and ε1 ∈ (0, 1), all depending on n,N,L, F ′′ and

m, such that for every ε ∈ (0, ε1), if ∈ (0, Rδ), |(u)x0,|+ |(∇u)x0,| < m and E(x0, ) < ε,then

(4.1) E(x0, τ0) ≤ c0R2γ + τα0 E(x0, ).

There are several steps to follow in order to achieve this decay estimate. Furthermore, we

recall that the minimality assumption is essential for a regularity result of this nature, since maps

that are merely solutions to the weak Euler-Lagrange equations associated to the problem, need

not satisfy the stated partial regularity property [41, 44].

In order to make use of the local minimality condition, we will need to build suitable test

functions that are sufficiently small in W1,q(Ω,RN ). With this in mind, we make the following

observation.


Lemma 4.2. Take q ∈ [1,∞] and assume that u ∈ W1,qloc(Ω,R

N ). Let BR = B(x0, R), with

R ≤ 1, and suppose that Br = B(y0, r) ⊆ BR. Furthermore, let ζ ∈ (0, 1) and assume that ρ is

a cut-off function such that

1Bζr≤ ρ ≤ 1Br and |∇ρ| ≤

1

r(1− ζ).

In addition, for a given m > 0, let z0 := (∇u)BR∈ R

N×n and assume that |z0| ≤ m. Finally,

let a : Rn → RN be an affine function of the form a(x) := z0 · (x− y0) + (u)y0,r and define

ϕ := ρ(u− a).

If q ∈ [1,∞), then for any given δ > 0 there exist a constant c > 0, and a radius Rδ =Rδ(c, ζ,m, q) ∈ (0, 1) such that, if 0 < R < Rδ and Br ⊆ BR, then

(4.2) ‖∇ϕ‖Lq(Ω,RN×n) < δ.

On the other hand, if q = ∞ and δ > 0 is given, there exists δ0 = δ0(δ, c, ζ,m, q) > 0 such that,

if δ1 ∈ (0, δ0) and (1.3) holds, then we can find a radius Rδ = Rδ(c, ζ,m, q) ∈ (0, 1) with the

property that, if 0 < R < Rδ and Br ⊆ BR, then

(4.3) ‖∇ϕ‖L∞(Ω,RN×n) < δ.

Proof. If 1 ≤ q <∞, then Poincare inequality implies that there is a constant c = c(n,N, q,m)such that

(4.4) ‖∇ϕ‖Lq(Br ,RN×n) ≤ c

(

1 +1

1− ζ

)

‖∇u− z0‖Lq(Br ,RN×n).

If Br ⊆ BR, it clearly follows that

(4.5) ‖∇ϕ‖Lq(Br ,RN×n) ≤ c

(

1 +1

1− ζ

)

‖∇u− z0‖Lq(BR,RN×n).

Note that, under the assumptions that u ∈ W1,qloc(Ω,R

N ) and |z0| ≤ m, for a given δ > 0 we will

be able to make ‖∇ϕ‖Lq(Ω,RN×n) < δ by taking B(y0, r) ⊆ B(x0, R), with R > 0 sufficiently

small. This concludes the proof of (4.2) for the case q ∈ [1,∞).If q = ∞, our aim is to prove that, if Br ⊆ BR, then for a constant c = c(N) > 0,

(4.6) ‖∇ϕ‖L∞(Br ,RN×n) ≤ c

(

1 +1

1− ζ

)

‖∇u− z0‖L∞(BR,RN×n).

In order to do this, we will first show that

(4.7) ‖∇ϕ‖L∞(Br ,RN×n) ≤ c

(

1 +1

1− ζ

)

‖∇u− z0‖L∞(Br ,RN×n).

Inequality (4.6) will then follow after taking now the L∞ norm from the right hand side on the

larger ball BR.

In order to show (4.7), we follow the ideas for the calculations made in [41]. We deal with

each coordinate function of u − a and, for a fixed 1 ≤ k ≤ N , we take xk ∈ B(y0, r) and

yk ∈ B(y0, r) such that:

• (u− a)(k)(xk) = (u− a)(k)y0,r = 0;

• |(u− a)(k)(yk)| = supB(y0,r) |(u− a)(k)|.


Then, by the Fundamental Theorem of Calculus applied over the segment [xk, yk] to each coor-

dinate function, we obtain that

supB(y0,r)

|u− a| ≤N max1≤k≤N

∣

∣

∣

∣

∫ yk

xk

∇(u− a)(k)∣

∣

∣

∣

≤ 2Nr ‖∇(u− a)‖L∞(Br ,RN×n)

=2Nr ‖∇u− z0‖L∞(Br ,RN×n).

This readily implies (4.7).

On the other hand, if δ > 0 is given and we take B(y0, r) ⊆ B(x0, R) and z0 := (∇u)BR,

inequality (4.6) implies that, if 0 < δ1 < δc(1+1/(1−ζ)) and (1.3) holds, there exists Rδ =

Rδ(c, ζ,m, q) > 0 such that, if 0 < R < Rδ, then

‖∇ϕ‖L∞(Ω,RN×n) = ‖∇ϕ‖L∞(Br ,RN×n) ≤ c

(

1 +1

1− ζ

)

‖∇u− z0‖L∞(BR,RN×n) < δ,

and (4.3) holds.

We remark that, in what follows, ζ ∈ (0, 1) will be fixed to be a suitable constant depending

on other fixed parameters, none of which will depend on δ.

4.1. Higher integrability. The next step is to establish a preliminary higher integrability result.

For the rest of this section we assume that F satisfies conditions (H0)-(H4) for some p ≥ 2.

Lemma 4.3 (Preliminary higher integrability). Let q ∈ [1,∞] and let u ∈ W1,p(Ω,RN ) ∩

W1,qloc(Ω,R

N ) be a W1,q-local minimizer of F . Furthermore, if q = ∞, assume that (1.3) holds

for δ1 ∈ (0, δ0), with δ0 as the one given by Lemma 4.2. Then, for every m > 0 there exists some

Rδ = Rδ(m,n,N, q, L) > 0 such that, if 2R0 ∈ (0, Rδ), B2R0 = B(x0, 2R0) ⊆ Ω is a ball,

and |(∇u)x0,2R0 | ≤ m, then there exist ε0 > 0 and c = c(m,n,N,L) > 0 such that, for every

q0 ∈ [p, p + ε0),

−

∫

B 12R0

|∇u|q0 dx

1q0

≤ c

(

−

∫

BR0

|∇u|p dx

) 1p

+ c.

Before proceeding to proving the lemma, we remark that the need to make the assumption

that it is for the radius 2R0 that it holds |(∇u)x0,2R0 | ≤ m will become clear while establishing

(4.28) in the proof of Theorem 4.7. Furthermore, fixing this constant vector at this stage will be

necessary in order to use the local minimality condition for the case q = ∞. We shall elaborate

further on this at the end of the following proof.

Proof. Let BR0 = B(x0, R0) ⊆ Ω be a ball of radius R0 and let Br = B(y, r) ⊆ BR0 .

We define z0 := (∇u)x0,2R0 and assume that it satisfies |z0| ≤ m. Furthermore, we let

a(x) := z0 · (x− y) + (u)y,r.

On the other hand, for a ζ ∈ (0, 1) yet to be determined, and that will depend exclusively on a

constant θ = θ(n,L) > 0, we let ρ be a cut-off function such that


1

r(1− ζ).

Define

ϕ := ρ(u− a), ψ := (1− ρ)(u− a).


Then, by the strong quasiconvexity of G and assumption (H3), which at this step turns out to

be crucial, we obtain that

∫

Br

(|∇ϕ|p +G(z0)) dx ≤

∫

Br

G(z0 +∇ϕ) dx

≤

∫

Br

F (x, u, z0 +∇ϕ)dx(4.8)

=

∫

Br

F (x, u,∇u−∇ψ) dx

=

∫

Br

F (x, u,∇u) dx

+

∫

Br

(F (x, u,∇u−∇ψ)− F (x, u,∇u)) dx.

To estimate the first term on the right hand side we use the local minimality of u. Indeed, if

‖∇ϕ‖Lq(Ω,RN×n) < δ, then

∫

Br

F (x, u,∇u) dx

≤

∫

Br

F (x, u− ϕ,∇u−∇ϕ) dx

=

∫

Br

F (x, ψ + a,∇ψ +∇a) dx

=

∫

Br\Bζr

F (x, ψ + a,∇ψ + z0) dx+

∫

Bζr

F (x, a, z0) dx(4.9)

(H1)≤ L

∫

Br\Bζr

|∇ψ + z0|p dx+ crn

≤ c

∫

Br\Bζr

(

|∇u− z0|p +

|u− a|p

rp(1− ζ)p

)

dx+ crn.

Here, c = c(L, p,m). On the other hand, by (2.1), and taking into account that ψ = 0 in Bζr and

that |z0| ≤ m, we obtain that

∫

Br

(F (x, u,∇u−∇ψ)− F (x, u,∇u)) dx

≤ c

∫

Br\Bζr

(

1 + |∇u|p−1 + |∇ψ|p−1)

|∇ψ|dx(4.10)

≤ c

∫

Br\Bζr

(

|∇u|p +|u− a|p

rp(1− ζ)p

)

dx+ crn.

From (4.8)-(4.10) and using that G is locally bounded, we deduce that

∫

Bζr

|∇u|p dx ≤ c

∫

Br\Bζr

(

|∇u|p +|u− a|p

rp(1− ζ)p

)

dx+ crn.


We now use Wydman’s hole-filling strategy to obtain that, for some θ ∈ (0, 1), with θ depending

exclusively on m,L and p, the following holds:∫

Bζr

|∇u|p dx ≤ θ

∫

Br

(

|∇u|p +|u− a|p

rp(1− ζ)p

)

dx+ θrn.

Therefore, if we fix θ < ζn, for θ := θζn ∈ (0, 1) we have that

−

∫

Bζr

|∇u|p dx ≤ θ−

∫

Br

(

|∇u|p +|u− a|p

rp(1− ζ)p

)

dx+ θ.

We now apply Poincare-Sobolev inequality to the second term on the right hand side and use that

|z0| ≤ m to obtain that

−

∫

Bζr

|∇u|p dx ≤ θ−

∫

Br

|∇u|p dx+c(θ, n)

(1− ζ)p

(

−

∫

Br

|∇u|pnn+p dx

)n+pn

+ c(m, θ).(4.11)

We note that this will all hold as long as ‖∇ϕ‖Lq(Ω,RN×n) < δ. In order to achieve this, we use

Lemma 4.2 (with R = 2R0) and the fact that u ∈ W1,qloc(Ω,R

N ) to conclude that, for our fixed

choice of ζ = ζ(m,n,L), we can find Rδ = Rδ(m,n,N, q, L) ∈ (0, 1) such that, if q ∈ [1,∞],then for all 0 < R0 < Rδ, if Br ⊆ B(x0, R0), then ‖∇ϕ‖Lq(Ω,RN×n) < δ.

We remark that, for the case q = ∞, we are using here that z0 = (∇u)x0,2R0 and that (1.3)

holds in order to use the estimate given by Lemma 4.2.

Whereby, considering that inequality (4.11) holds for every Br ⊆ B(x0, R0), we can fi-

nally apply Gehring’s Lemma in the version of Theorem 2.2 and obtain that there exist ε0 =ε0(m,n, θ, p) > 0 and c = c(m,n, θ, p) > 0 such that, for q0 ∈ [p, p + ε0),

−

∫

BR02

|∇u|q0 dx

1q0

≤ c

(

−

∫

BR0

|∇u|p dx

) 1p

+ c.

This concludes the proof of the Lemma.

4.2. A Caccioppoli inequality of the first kind. We are ready to establish a Caccioppoli in-

equality that, combined with the higher integrability from Lemma 4.3, will pave the way to

finally obtain the decay in the mean oscillations that we pursue. For the results in this section we

also assume that F satisfies the conditions (H0)-(H4) for some p ≥ 2.

The Caccioppoli inequality will fundamentally rely on the local minimality property satisfied

by u. For this reason, and considering the comments made in Lemma 4.2, inequality (4.12) will

be valid for a fixed ζ ∈ (0, 1), that will depend on other fixed parameters. This implies that,

just as in [41], and opposite to what happens in the case of global minimizers, as in [20, 28], we

cannot iterate (4.12) in order to obtain a Caccioppoli inequality of the first kind.

As discussed in the introduction, Kristensen and Taheri overcame this problem in the case of

homogeneous integrands by means of a measure theoretical compactness argument for the blown-

up sequence. However, given the need of a direct argument that is brought by the dependence

on u of F , we will instead make use of Theorem 2.2 to derive from (4.12) a reverse Holder

inequality in Theorem 4.5.

Theorem 4.4. Let q ∈ [1,∞] and let u ∈ W1,p(Ω,RN ) ∩W1,qloc(Ω,R

N ) be a W1,q-local mini-

mizer of F . If q = ∞, assume further that (1.3) holds for some δ1 ∈ (0, δ0), with δ0 > 0 given

by Lemma 4.2.


Then, for every m > 0 there exist Rδ = Rδ(n,N, q, L,m) ∈ (0, 1), as well as ζ, θ ∈ (0, 1),

with ζ = ζ(m,n,L, F ′′) and θ = θ(m,n,L, F ′′), such that, if R0 ∈ (0, Rδ), B(x0, 2R0) ⊆ Ω,

z0 := (∇u)x0,2R0 and u0 ∈ RN satisfy |u0| + |z0| ≤ m, then for every B(y0, r) ⊆ B(x0, R0)

and for a : Rn → RN the affine map given by a(x) := z0 · (x− y0) + (u)y0,r, we have that

−

∫

Bζr

|V (∇u− z0)|2 dx

≤ θ−

∫

Br

|V (∇u− z0)|2 dx+ θ−

∫

Br

∣

∣

∣

∣

V

(

u− a

r(1− ζ)

)∣

∣

∣

∣

2

dx(4.12)

+ θ−

∫

Br

ϑ(|u0|, 2|x− y0|2 + 2|u− u0|

2) (1 + |∇u|p + |z0|p) dx

+ θ−

∫

Br

ϑ(|u0|, 2|u− a|2) (1 + |z0|p) dx.

Proof of Theorem 4.4. Let m > 0 be arbitrary and fix (x0, u0, z0) ∈ Ω × RN × R

N×n, with

|u0|+ |z0| ≤ m. Take Br := B(y0, r) ⊆ B(x0, R0) = BR0 , with R0 to be determined.

Furthermore, let ζ ∈ (0, 1) and assume that ρ is a cut-off function satisfying that


1

r(1− ζ).

Define a(x) := z0 · (x− y0) + (u)y0,r,

ϕ := ρ(u− a), and ψ := (1− ρ)(u− a).

In addition, let F (z) := F (y0, u0, z0 + z)− F (y0, u0, z0)− Fz(y0, u0, z0)[z].Then, by first using (H2) and then Lemma 2.1, we obtain the following:

∫

Br

|V (∇ϕ)|2 dx ≤

∫

Br

F (∇ϕ) dx =

∫

Br

F (∇u− z0 −∇ψ) dx

=

∫

Br

F (∇u− z0) dx+

∫

Br

(

F (∇u− z0 −∇ψ)− F (∇u− z0))

dx

≤

∫

Br

F (∇u− z0) dx

+ c

∫

Br

(

|∇ψ|+ |∇u− z0|+ |∇ψ|p−1 + |∇u− z0|p−1)

|∇ψ|dx.

For the first term on the right hand side above, we note that

∫

Br

F (∇u− z0) dx

=

∫

Br

(

F (y0, u0,∇u)− F (y0, u0, z0)− Fz(y0, u0, z0)[∇u− z0])

dx

=

∫

Br

F (x, u,∇u) dx+

∫

Br

(

F (y0, u0,∇u)− F (x, u,∇u))

dx

−

∫

Br

(

F (y0, u0, z0) + Fz(y0, u0, z0)[∇u− z0])

dx.


On the other hand, if ‖∇ϕ‖Lq < δ, then the local minimality property of u implies that∫

Br

F (x, u,∇u) dx ≤

∫

Br

F (x, u− ϕ,∇u−∇ϕ) dx

=

∫

Br

F (y0, u0,∇ψ + z0) dx

+

∫

Br

(F (x, ψ + a,∇ψ + z0)− F (y0, u0,∇ψ + z0)) dx

=

∫

Br

F (∇ψ) dx+

∫

Br

F (y0, u0, z0) dx+

∫

Br

Fz(y0, u0, z0)[∇ψ] dx

+

∫

Br

(F (x, ψ + a,∇ψ + z0)− F (y0, u0,∇ψ + z0)) dx.

Compiling all the estimates above we obtain that∫

Br

|V (∇ϕ)|2 dx

≤

∫

Br

F (∇ψ) dx+ c

∫

Br

(

|∇ψ|+ |∇u− z0|+ |∇ψ|p−1 + |∇u− z0|p−1)

|∇ψ|dx

+

∫

Br

(F (y0, u0,∇u)− F (x, u,∇u)) dx

+

∫

Br

Fz(y0, u0, z0)[∇ψ] dx−

∫

Br

Fz(y0, u0, z0)[∇ϕ +∇ψ] dx

+

∫

Br

(

F (x, ψ + a,∇ψ + z0)− F (y0, u0,∇ψ + z0))

dx.

Using Lemma 2.1 for the first term and Young’s inequality for the second term on the right hand

side, together with ϕ ∈ W1,p0 (Br,R

N ) and the definition of V , this implies that, for ζ ∈ (0, 1),∫

Bζr

|V (∇u− z0)|2 dx ≤ c

∫

Br\Bζr

|V (∇ψ)|2 dx

+ c

∫

Br\Bζr

|V (∇u− z0)|2 dx

+

∫

Br

(F (y0, u0,∇u)− F (x, u,∇u)) dx(4.13)

+

∫

Br

(

F (x, ψ + a,∇ψ + z0)− F (y0, u0,∇ψ + z0))

dx.

Now we use assumption (H4) and the fact that ϕ ∈ W1,p0 (Br,R

N ) to obtain from above that∫

Bζr

|V (∇u− z0)|2 dx

≤ c

∫

Br\Bζr

|V (∇ψ)|2 dx+ c

∫

Br\Bζr

|V (∇u− z0)|2 dx

+ c

∫

Br

ϑ(|u0|, |x− y0|2 + |u− u0|

2) (1 + |∇u|p) dx

+ c

∫

Br

ϑ(|u0|, |x− y0|2 + |u− u0|

2 + |u− a|2) (1 + |∇ψ|p + |z0|p) dx.


Since ϑ ≤ 2L, ψ = 0 in Bζr, and | · |p ≤ |V |2, this implies that

∫

Bζr

|V (∇u− z0)|2 dx

≤ c

∫

Br\Bζr

|V (∇ψ)|2 dx+ c

∫

Br\Bζr

|V (∇u− z0)|2 dx

+ c

∫

Br

ϑ(|u0|, |x− y0|2 + |u− u0|

2) (1 + |∇u|p) dx

+ c

∫

Br

ϑ(|u0|, |x− y0|2 + |u− u0|

2 + |u− a|2) (1 + |z0|p) dx.

Note that we have made a hole for all the terms where |∇ψ| appears as a factor. We emphasize

that performing this step is crucial to ensure that we will later be able to suitably apply the higher

integrability from Lemma 4.3. Observe here that, since ϑ is increasing and non-negative, then

for all s > 0 and for all a, b > 0,

(4.14) ϑ(s, a+ b) ≤ ϑ(s, 2a) + ϑ(s, 2b).

We now use the definition of ψ in the inequality above, (4.14) and that ϑ is increasing, to obtain

that

∫

Bζr

|V (∇u− z0)|2 dx

≤ c

∫

Br\Bζr

|V (∇u− z0)|2 dx+ c

∫

Br

∣

∣

∣

∣

V

(

u− a

r(1− ζ)

)∣

∣

∣

∣

2

dx

+ c

∫

Br

ϑ(|u0|, 2|x− y0|2 + 2|u− u0|

2) (1 + |∇u|p + |z0|p) dx

+ c

∫

Br

ϑ(|u0|, 2|u− a|2) (1 + |z0|p) dx.

We fill in the hole and conclude that, for some θ ∈ (0, 1),

∫

Bζr

|V (∇u− z0)|2 dx

≤ θ

∫

Br

|V (∇u− z0)|2 dx+ θ

∫

Br

∣

∣

∣

∣

V

(

u− a

r(1− ζ)

)∣

∣

∣

∣

2

dx

+ θ

∫

Br

ϑ(|u0|, 2|x− y0|2 + 2|u− u0|

2) (1 + |∇u|p + |z0|p) dx

+ θ

∫

Br

ϑ(|u0|, 2|u− a|2) (1 + |z0|p) dx.


Henceforth, if we assume that θ < ζn, this implies that, for θ = θ/ζn ∈ (0, 1),

−

∫

Bζr

|V (∇u− z0)|2 dx

≤ θ−

∫

Br

|V (∇u− z0)|2 dx

+ θ−

∫

Br

∣

∣

∣

∣

V

(

u− a

r(1− ζ)

)∣

∣

∣

∣

2

dx

+ θ−

∫

Br

ϑ(|u0|, 2|x− y0|2 + 2|u− u0|

2) (1 + |∇u|p + |z0|p) dx

+ θ−

∫

Br

ϑ(|u0|, 2|u− a|2) (1 + |z0|p) dx.

We recall that this will hold provided we can use the W1,q-local minimality of u to obtain (4.13),

meaning that we need ‖∇ϕ‖Lq(Ω,RN×n) < δ. By Lemma 4.2 and the fact that u ∈ W1,qloc(Ω,R

N ),

as well as the assumptions that |z0| ≤ m and (1.3), we can argue as at the end of Lemma

4.3 and, whereby, find Rδ = Rδ(n,N, q, L,m) ∈ (0, 1) such that, for every r ∈ (0, Rδ),‖∇ϕ‖Lq(Ω,RN×n) < δ and, hence, (4.12) holds.

4.3. A reverse Holder inequality. We will now use the previous Caccioppoli inequality to ob-

tain a reverse Holder inequality. It is important to note at this stage that, in order to do so, we

need to obtain an expression where the term

θ−

∫

Br

|V (∇u− z0)|2 dx

is improved into an estimate with a higher exponent. It is for this purpose that we wish to apply

Theorem 2.2. Once again, for this section we assume that F satisfies the conditions (H0)-(H4)

for some p ≥ 2.

On the other hand, we note that in (4.12), we also have the lower order term u− a on the right

hand side. This is already a higher integrable term, by Poincare-Sobolev compactness theorem.

However, in order to take advantage of this, we have made a dependent on r > 0 in the statement

of Theorem 4.4. It is therefore crucial that both appearances of u − a have been made so that

they are only being multiplied by constants and, whereby, we can readily apply Poincare-Sobolev

inequality. This allows us to obtain an estimate where the integrands do not depend anymore on

r > 0 and, finally, apply Gehring’s Lemma in its generalized version of Theorem 2.2. We

proceed to implement these ideas in order to obtain the following result.

Theorem 4.5 (A reverse Holder inequality). Let q ∈ [1,∞] and let u ∈ W1,p(Ω,RN ) ∩

W1,qloc(Ω,R

N ) be a W1,q-local minimizer of F . Furthermore, if q = ∞, assume that (1.3)

holds for some δ1 > 0 taken as in Lemma 4.2. Then, for every m > 0 there exist Rδ =

Rδ(n,N, q, L,m) ∈ (0, 1), a constant c = c(m,n,L, F ′′) > 0 and q1 > 2, such that, if

2R0 < Rδ, B2R0 = B(x0, 2R0) ⊆ Ω is a ball, and |(u)2R0 | + |(∇u)2R0 | ≤ m, then we have


that, for z0 = (∇u)2R0 ,

−

∫

BR02

|V (∇u− z0)|q1 dx

2q1

≤ c−

∫

B2R0

|V (∇u− z0)|2 dx(4.15)

+ c ϑ0

(

|(u)x0,2R0 |, R20 +R2

0 −

∫

B2R0

|∇u− z0|2 dx

)

+ cR2β0 .

Here, ϑ0(u, t) := ϑκ(u, ct) for a constant c = c(m,n) > 0 and an exponent κ = κ(n, p, θ,m) ∈(0, 1), with θ as in Theorem 4.4.

Proof. Let m > 0, and let Rδ, ζ, θ ∈ (0, 1) be as in Theorem 4.4. Take R0 ∈ (0, 12Rδ) and a ball

B2R0 = B(x0, 2R0) ⊆ Ω.

Define u0 := (u)2R0 , z0 := (∇u)2R0 , and assume that (u0, z0) ∈ RN ×R

N×n are such that

|u0|+ |z0| ≤ m.

Furthermore, let Br = B(y0, r) ⊆ BR0 .

Now, use that ϑ is concave for the last term on the right hand side of (4.12), and then Poincare-

Sobolev inequality for all the appearances of u − a, as well as the increasing nature of ϑ. We

henceforth obtain that

−

∫

Bζr

|V (∇u− z0)|2 dx ≤ θ−

∫

Br

|V (∇u− z0)|2 dx

+ c

(

−

∫

Br

|V (∇u− z0)|2nn+2 dx

)n+2n

(4.16)

+ θ−

∫

Br

ϑ(|u0|, 2|x− y0|2 + 2|u− u0|

2) (1 + |∇u|p + |z0|p) dx

+ θ (1 + |z0|p) ϑ

(

|u0|, 2cr2

(

−

∫

Br

|∇u− z0|2nn+2 dx

)n+2n

)

.

To estimate the last term on the right hand side, note that by the definition on ϑ and the fact that,

for t > 0, (1 + t)β ≤ (1 + t), the assumption that |u0|+ |z0| ≤ m implies that

θ (1 + |z0|p) ϑ

(

|u0|, 2cr2

(

−

∫

Br

|∇u− z0|2nn+2 dx

)n+2n

)

≤ c(m, θ)r2β(

−

∫

Br

|∇u− z0|2nn+2 dx

)n+2n

β

(4.17)

≤ c(m, θ)r2β

(

1 +

(

−

∫

Br

|∇u− z0|2nn+2 dx

)n+2n

)β

≤ c(m, θ)R2β0

(

1 +

(

−

∫

Br

|V (∇u− z0)|2nn+2 dx

)n+2n

)

.


From (4.16), (4.17) and Br ⊆ BR0 , R0 ∈ (0, 1), we finally obtain the following estimate:

−

∫

Bζr

|V (∇u− z0)|2 dx

≤ θ−

∫

Br

|V (∇u− z0)|2 dx+ c

(

−

∫

Br

|V (∇u− z0)|2nn+2 dx

)n+2n

+ θ−

∫

Br

ϑ(|u0|, 2R20 + 2|u− u0|

2) (1 + |∇u|p + |z0|p) dx+ cR2β

0 .

We can whereby apply Gehring’s Lemma from Theorem 2.2 to obtain that there exists ε1 =

ε1(n, p, θ,m) > 0 such that, for q1 ∈ [2, 2 + ε1) and a constant c = c(n, θ,m),

−

∫

BR02

|V (∇u− z0)|q1 dx

2q1

≤ c−

∫

BR0

|V (∇u− z0)|2 dx

+ c

(

−

∫

BR0

ϑq12 (|u0|, 2R

20 + 2|u− u0|

2)(

1 + |∇u|pq12

)

dx

)2q1

+ cR2β0(4.18)

≤ c−

∫

BR0

|V (∇u− z0)|2 dx

+ c

(

−

∫

BR0

ϑ(|u0|, 2R20 + 2|u− u0|

2)(

1 + |∇u|pq12

)

dx

)2q1

+ cR2β0 .

For the last inequality we have used that, since ϑ ≤ 2L, then for η > 1, ϑη ≤ c(L, η)ϑ.

We now take ε0 > 0 as the one given by Lemma 4.3 and fix q1 ∈ (2, 2 + ε1) such that pq12 ∈

(p, p + ε0). Then, we further choose q2 ∈(pq1

2 , p + ε0)

.

Using the exponent q3 = 2q2pq1

> 1, we now apply Holder’s inequality to the second term of

(4.18). This, and ϑq3

q3−1 ≤ c(L, q3)ϑ, lead to

−

∫

BR02

|V (∇u− z0)|q1 dx

2q1

≤ c−

∫

BR0

|V (∇u− z0)|2 dx

+ c

(

−

∫

BR0

ϑ(|u0|, 2R20 + 2|u− u0|

2) dx

)

2(q3−1)q1q3

(

−

∫

BR0

(1 + |∇u|q2) dx

)pq2

+ cR2β0 .

From this point onwards we shall simplify the notation by removing the dependence of ϑ on |u0|.Now, we use that ϑ is concave and apply Lemma 4.3 to the right hand side above. Whereby, we


conclude that

−

∫

BR02

|V (∇u− z0)|q1 dx

2q1

≤ c−

∫

BR0

|V (∇u− z0)|2 dx

+ c ϑ

(

2R20 + 2−

∫

BR0

|u− u0|2 dx

)

2(q3−1)

q1q3

(

−

∫

B2R0

(1 + |∇u|p) dx

)

+ cR2β0(4.19)

≤ c−

∫

BR0

|V (∇u− z0)|2 dx

+ c ϑ

(

2R20 + 2n+1−

∫

B2R0

|u− u0|2 dx

)

2(q3−1)q1q3

(

−

∫

B2R0

(1 + |∇u|p) dx

)

+ cR2β0 .

Finally, we apply Poincare inequality to the argument of ϑ. It is at this point that we use that

u0 = (u)2R0 . This, together with the fact that |z0| ≤ m, lead to the following estimate, for a

constant c > 0:

−

∫

BR02

|V (∇u− z0)|q1 dx

2q1

≤ c−

∫

BR0

|V (∇u− z0)|2 dx

+ c ϑ

(

2R20 + 2n+1R2

0 −

∫

B2R0

|∇u|2 dx

)

2(q3−1)q1q3

(

−

∫

B2R0

(1 + |∇u|p) dx

)

+ cR2β0

≤ c−

∫

BR0

|V (∇u− z0)|2 dx

+ c ϑ

(

cR20 + cR2

0 −

∫

B2R0

|∇u− z0|2 dx

)

2(q3−1)q1q3

(

−

∫

B2R0

(1 + |∇u− z0|p) dx

)

+ cR2β0 .

The desired conclusion follows after rescaling the constant c > 0, as well as using that ϑ ≤ 2Land | · |p ≤ |V (·)|2. Indeed, we obtain that

−

∫

BR02

|V (∇u− z0)|q1 dx

2q1

≤ c−

∫

B2R0

|V (∇u− z0)|2 dx+ c ϑ0

(

R20 +R2

0 −

∫

B2R0

|∇u− z0|2 dx

)

+ cR2β0 .

For the last inequality we have defined ϑ0(t) := ϑ(ct)κ, with κ = 2(q3−1)q1q3

∈ (0, 1). Note that

κ = κ(n, p,m,L), since ε0 and ε1 depend on those parameters. This completes the proof of the

theorem.


4.4. Linearization of the problem. Let (x0, u0) ∈ Ω × RN be fixed. We define the frozen

integrand F 0 : RN×n → R by

F 0(z) := F (x0, u0, z).

Furthermore, we consider the second order Taylor polynomial of F 0 centred at z0 ∈ RN×n,

which is given by

(4.20) P (z) := F 0(z0) + F 0z (z0)[z − z0] +

1

2F 0zz(z0)[z − z0, z − z0].

We record here that, for every z, w ∈ RN×n,

(4.21) P (z) := P (w) + P ′(w)[z − w] +1

2F 0zz(z0)[z − w, z − w].

We then have the following well known approximation estimate:

Lemma 4.6. Assume that F satisfies the conditions (H0)-(H4) for some p ≥ 2. Let m > 1 be

fixed and assume that |u0|+ |z0| ≤ m. There exists a modulus of continuity ω : [0,∞) → [0, 1]such that ω is increasing, concave, ω(t) ≥ 1 for every t ≥ 1, limt→0 ω(t) = 0, and with the

property that, for a constant c = c(m) > 0 and for every z ∈ RN×n,

(4.22) |F 0(z)− P (z)| ≤ c ω(|z − z0|2)|V (z − z0)|

2.

This result is well known in regularity theory and we shall only make a brief comment regard-

ing its proof.

Proof. The construction of the function ω was already sketched in the proof of Lemma 3.3.

In order to obtain (4.22), it is enough to use Taylor’s approximation theorem with the integral

form of the remainder for the case in which |z − z0| ≤ m+ 1. On the other hand, if |z − z0| >m+ 1 > 1, the proof follows by the truncation technique of Acerbi & Fusco in [1].

The following result is the key linearization strategy that will enable us to obtain partial reg-

ularity of u. We note that, unlike in the previous results, we assume here that q ∈ [2,∞]. We

need this assumption in order to use the Lp-estimates stated in Theorem 2.5. We remark that,

just as in [41], the case q ∈ [1, 2) should be obtained by establishing first that u satisfies a global

minimality property in small subsets of Ω.

Theorem 4.7 (Application of the linearization strategy). Assume that F satisfies the conditions

(H0)-(H4) for some p ≥ 2. Let u ∈ W1,p(Ω,RN ) ∩W1,qloc(Ω,R

N ) be a W1,q-local minimizer of

F , where q ∈ [2,∞].

If q ∈ [2,∞), then there exists R1 ∈ (0, Rδ

4 ) such that, for every ε ∈ (0, 14n ) and every

m > 1, there exists γ ∈ (m,n,L, p) ∈ (0, 1) such that, if ∈ (0, R1), B(x0, ) ⊆ Ω, |ux0,| +|(∇u)x0,| ≤ m, and E(x0, ) < ε, then it holds that for every τ ∈ (0, 14),

(4.23) E(x0, τ) ≤ cτ−n2γ + c(τ−nω1(ε) + τ2)E(x0, ),

where ω1(t) := ως(ct) for some ς ∈ (0, 1), c > 0 is a constant, and both c and ς are depending

on m,n,L, F ′′ and p.

On the other hand, if q = ∞, we assume in addition that (H0w) and (H1w) hold for F , and

that u satisfies (H5). Then, there exists δ1 > 0 such that, if (1.3) is satisfied, the decay stated in

(4.23) will remain to be true.


Proof. We let R > 0 and assume that 0 < 4R < Rδ. Let B(x0, 4R) ⊆ Ω. Furthermore, we

denote

u0 := (u)x0,4R and z0 := (∇u)x0,4R

in the polynomial P defined in (4.20) and suppose that |u0| + |z0| ≤ m for a given m > 1.

Assume also that E(x0, 4R) < ε, with ε ∈ (0, 14n ).

Let h ∈ W1,pu (BR,R

N ) be a minimizer of the functional given by

P (v) :=

∫

BR

P (∇v) dx.

Then, by the strong quasiconvexity of F 0,

−

∫

BR

|V (∇u−∇h)|2 dx ≤−

∫

BR

(

F 0(∇u−∇h+ z0)− F 0(z0))

dx

=−

∫

BR

(

F 0(∇u−∇h+ z0)− P (∇u−∇h+ z0))

dx(4.24)

+1

2−

∫

BR

F 0zz(z0)[∇u−∇h,∇u−∇h] dx.

We first note that, since u− h ∈ W1,p0 (BR,R

N ), and h is a P -extremal, (4.21) implies that

1

2−

∫

BR

F 0zz(z0)[∇u−∇h,∇u−∇h] dx

=−

∫

BR

(P (∇u)− P (∇h)) dx

=−

∫

BR

(

P (∇u)− F 0(∇u))

dx+−

∫

BR

(

F 0(∇u)− F (x, u,∇u))

dx(4.25)

+−

∫

BR

(F (x, u,∇u) − F (x, h,∇h)) dx+−

∫

BR

(

F (x, h,∇h) − F 0(∇h))

dx

+−

∫

BR

(

F 0(∇h)− P (∇h))

dx.

From (4.24) and (4.25), it follows that

−

∫

BR

|V (∇u−∇h)|2 dx

≤−

∫

BR

(

F 0(∇u−∇h+ z0)− P (∇u−∇h+ z0))

dx

+ −

∫

BR

(

P (∇u)− F 0(∇u))

dx(4.26)

+−

∫

BR

(

F 0(∇u)− F (x, u,∇u))

dx+−

∫

BR

(F (x, u,∇u) − F (x, h,∇h)) dx

+−

∫

BR

(

F (x, h,∇h) − F 0(∇h))

dx+−

∫

BR

(

F 0(∇h)− P (∇h))

dx

=I + II + III + IV + V+VI.

Observe here that if q ∈ [2,∞), inequality (2.5) and Theorem 2.5 imply that there exists a

constant K = K(m,n, q) > 0 such that

‖∇u−∇h‖Lq(BR,RN×n) ≤ ‖∇u‖Lq(BR,RN×n) + ‖∇h‖Lq(BR,RN×n) ≤ K‖∇u‖Lq(BR,RN×n).


Since u ∈ W1,qloc(Ω,R

N ), this implies that, for R1 ∈ (0, Rδ

4 ) small enough we will have that for

every R ∈ (0, R1),

‖∇u−∇h‖Lq(BR,RN×n) < δ

and hence, by the local minimality condition,

(4.27) IV < 0.

On the other hand, if q = ∞ Theorem 2.5 implies that

[∇h− z0]BMO(BR,RN×n) ≤ A∞‖∇u− z0‖L∞(BR,RN×n).

Furthermore, since u is a weak local minimizer, our assumptions enable us to apply Theorem 3.8

to conclude that u is a W1,BMO-local minimizer, meaning that for some δ∗ ∈ (0, 1) we will have

that, if h− u ∈ W1,2p0 (Ω,RN ) and [∇h−∇u]BMO(Ω,RN×n) < δ∗, then (4.27) will also hold.

Whereby, by choosing δ1 > 0 such that δ1 < min δ∗A∞+1 , δ0, where δ0 > 0 is taken as in

Lemma 4.2, assumption (1.3) will imply that, if R1 > 0 is small enough and R ∈ (0, R1), then

[∇h−∇u]BMO(BR,RN×n) ≤[∇h− z0]BMO(BR,RN×n) + [∇u− z0]BMO(BR,RN×n)

≤A∞‖∇u− z0‖L∞(BR,RN×n) + [∇u]BMO(BR,RN×n)

<A∞δ1 + δ1 < δ∗.

This means that, for our choice of δ1, the W1,BMO-local minimality can be applied and, whereby,

we have (4.27) for every q ∈ [2,∞].

To estimate III and V we will make use of (H4). We take q0 ∈ (p, p + ε0) as in Lemma 4.3

and apply Holder’s inequality to obtain the following:

III ≤ c−

∫

BR

ϑ(

|x− x0|2 + |u− u0|

2)

(1 + |∇u|p) dx

≤ c

(

−

∫

BR

ϑq0

q0−p(

R2 + |u− u0|2)

dx

)

q0−p

q0

(

−

∫

BR

(1 + |∇u|p)q0p dx

)pq0

≤ c ϑ

(

R2 +−

∫

B4R

|u− u0|2 dx

)

q0−p

q0

(

−

∫

BR

(1 + |∇u|q0) dx

)pq0

.

The last inequality follows after increasing the size of the ball of integration and, then, using that

ϑ ≤ 2L is concave.

We now apply Poincare inequality and the higher integrability from Lemma 4.3, which for q = ∞we can do because δ1 < δ0. This allows us to obtain that, for some constant c = c(m,n) > 0

and for ϑ1(t) := ϑ(ct)q0−p

q0 ,

III ≤ c ϑ

(

R2 +R2−

∫

B4R

|∇u|2 dx

)

q0−p

q0

−

∫

B2R

(1 + |∇u|p) dx

≤ c ϑ1

(

R2 +R2−

∫

B4R

|∇u− z0|2 dx

)

−

∫

B4R

(1 + |∇u− z0|p) dx.(4.28)

We note that we have made R = R02 while applying Lemma 4.3, which is consistent with our

assumption that |u0|+ |z0| ≤ m.


We move on to estimate V. By applying assumption (H4) and then Holder’s inequality, we have

that

V ≤ c−

∫

BR

ϑ(

|x− x0|2 + |h− u0|

2)

(1 + |∇h|p) dx

+ c−

∫

BR

ϑ(

R2 + 2|h− u|2 + 2|u− u0|2)

(1 + |∇h|p) dx

≤ c

(

−

∫

BR

ϑq0

q0−p(

R2 + 2|h − u|2 + 2|u− u0|2)

dx

)

q0−p

q0

(

−

∫

BR

(1 + |∇h|p)q0p dx

)pq0

≤ c ϑ

(

R2 +−

∫

BR

(

2|h− u|2 + 2|u− u0|2)

dx

)

q0−p

q0

(

−

∫

BR

(1 + |∇h|q0) dx

)pq0

≤ c ϑ1

(

R2 +−

∫

BR

|h− u|2 dx+−

∫

B4R

|u− u0|2 dx

)(

−

∫

BR

(1 + |∇h|q0) dx

)pq0

,

where the constant c > 0 that appears in the definition of ϑ1, may have changed.

As before, we now apply Poincare inequality to the functions h − u ∈ W1,p0 (BR,R

N ) and

u− u0, to estimate the first term on the right hand side. This, together with the Lp-estimates for

h from (2.5), leads to

V ≤ cϑ1

(

R2 +R2−

∫

BR

|∇h−∇u|2 dx+R2−

∫

B4R

|∇u|2 dx

)(

−

∫

BR

(1 + |∇u|q0) dx

)pq0

≤ cϑ1

(

R2 +R2−

∫

B4R

(

|∇h− z0|2 + |∇u− z0|

2 + |∇u|2)

dx

)(

−

∫

BR

(1 + |∇u|q0) dx

)pq0

.

For the last inequality above, the argument of ϑ1 has been rescaled again without changing the

notation.

By Theorem 2.5, Lemma 4.3, and the fact that |z0| ≤ m, we can further obtain that

V ≤ c ϑ1

(

R2 + 2R2−

∫

B4R

|∇u− z0|2 dx

)

−

∫

B2R

(1 + |∇u− z0|p) dx

≤ c ϑ1

(

R2 +R2−

∫

B4R

|∇u− z0|2 dx

)

−

∫

B4R

(1 + |∇u− z0|p) dx.(4.29)


We now estimate II + VI. Indeed, applying Lemma 4.6 and Holder’s inequality we obtain that,

for q1 ∈ (2, 2 + ε1) as in Theorem 4.5, the following holds:

II + VI ≤ c−

∫

BR

ω(|∇u− z0|2)|V (∇u− z0)|

2 dx+ c−

∫

BR

ω(∇h− z0)|V (∇h− z0)|2 dx

≤ c

(

−

∫

BR

ωq1

q1−2 (|∇u− z0|2)

)

q1−2q1

(

−

∫

BR

|V (∇u− z0)|q1

)2q1

+ c

(

−

∫

BR

ωq1

q1−2 (|∇h− z0|2)

)

q1−2

q1

(

−

∫

BR

|V (∇h− z0)|q1

)2q1

≤ c ω

(

−

∫

BR

|∇u− z0|2

)

q1−2q1

(

−

∫

BR

|V (∇u− z0)|q1

)2q1

+ c ω

(

−

∫

BR

|∇h− z0|2

)

q1−2q1

(

−

∫

BR

|V (∇h− z0)|q1

)2q1

.

For the last inequality we have used, once again, that ω ≤ 1 is concave. Using this one more

time, as well as the Lp-estimates from Theorem 2.5, we derive from above that

II + VI ≤ c ω

(

c−

∫

BR

|∇u− z0|2

)

q1−2q1

(

−

∫

BR

|V (∇u− z0)|q1

) 2q1

.

Therefore, by (4.15), we obtain that for ϑ1 as it was previously defined,

II + VI ≤ c ω

(

c−

∫

B4R

|∇u− z0|2

)

q1−2q1

−

∫

B4R

|V (∇u− z0)|2 dx

+ c ω

(

c−

∫

B4R

|∇u− z0|2

)

q1−2q1

ϑ1

(

R2 +R2−

∫

B4R

|∇u− z0|2 dx

)

(4.30)

+ c ω

(

c−

∫

B4R

|∇u− z0|2

)

q1−2q1

R2β.

Finally, we will now estimate term I. In what follows, we first use Lemma 4.6 and then we take

q1 ∈ (2, 2+ ε) as in the estimate of II + IV. We use Holder’s inequality and, once again, exploit

the fact that ω ≤ 1 is concave to obtain that

I ≤ c−

∫

BR

ω(|∇u−∇h|2)|V (∇u−∇h)|2 dx

≤ c ω

(

−

∫

BR

|∇u−∇h|2 dx

)

q1−2q1

(

−

∫

BR

|V (∇u−∇h)|q1 dx

) 2q1

.


Now, using Theorem 2.5, the previous inequality leads to

I ≤ c ω

(

c−

∫

BR

(

|∇u− z0|2 + |∇h− z0|

2)

dx

)

q1−2q1

(

−

∫

BR

|V (∇u− z0)|q1 dx

) 2q1

+ c ω

(

c−

∫

BR

(

|∇u− z0|2 + |∇h− z0|

2)

dx

)

q1−2q1

(

−

∫

BR

|V (∇h− z0)|q1 dx

)2q1

≤ c ω

(

c−

∫

BR

|∇u− z0|2 dx

)

q1−2q1

(

−

∫

BR

|V (∇u− z0)|q1 dx

)2q1

.

Finally, we let ω1(t) := ω(c t)q1−2q1 and note that, by the higher integrability from Theorem 4.5,

this leads to the following estimate:

I ≤ c ω1

(

c−

∫

B4R

|∇u− z0|2 dx

)

−

∫

B4R

|V (∇u− z0)|2 dx

+ c ω1

(

c−

∫

B4R

|∇u− z0|2 dx

)

ϑ1

(

R2 +R2−

∫

B4R

|∇u− z0|2 dx

)

(4.31)

+ c ω1

(

c−

∫

B4R

|∇u− z0|2

)

R2β .

Since ω1 ≤ 1 and ϑ1 ≤ c(m,n, p, L), it follows from the estimates (4.27)-(4.31) that, if R ∈(0, R1), u0 = (u)4R and z0 = (∇u)4R satisfy |u0|+ |z0| ≤ m and E(x0, 4R) < ε < 1, then for

a constant c = c(m) > 0,

I + II + III + IV + V+VI ≤ c ϑ1(2R2) + cR2β + ω1(ε)E(x0, 4R)

≤ cR2βκ + ω1(ε)E(x0, 4R).(4.32)

The last inequality is making use of the definition of ϑ1 and the fact that κ < 1.

For the last step of the proof we note that, by Lemma 2.3, inequality (2.5), and Theorem 2.4, it

follows that for every r ∈ (0, R4 ),

−

∫

Br

|V (∇u− (∇u)r)|2 dx ≤ c−

∫

Br

|V (∇u− (∇h)r)|2 dx

≤ c−

∫

Br

|V (∇u−∇h)|2 dx+ c−

∫

Br

|V (∇h− (∇h)r)|2 dx

≤ c

(

R

r

)n

−

∫

BR

|V (∇u−∇h)|2 dx(4.33)

+ c( r

R

)2−

∫

BR

|V (∇u− (∇u)R)|2 dx.

Combining the estimates (4.26), (4.32) and (4.33), and increasing again the size of the ball BR

in (4.33), we can conclude that

E(x0, r) ≤ c

(

R

r

)n

R2βκ + c

((

R

r

)n

ω1(ε) +( r

R

)2)

E(x0, 4R).

By writing = 4R, we can reinterpret this as the fact that, for every ε ∈ (0, 14n ) and m > 1,

if ∈ (0, 4R1), B(x0, ) ⊆ Ω, |(u)x0,| + |(∇u)x0,| < m, and E(x0, ) < ε, then for every

τ ∈ (0, 14),

E(x0, τ) ≤ cτ−n2βκ + c(τ−nω1(ε) + τ2)E(x0, ).


Proof of Proposition 4.1. The estimate follows readily from (4.23). Indeed, by taking the con-

stant c > 0 to be larger if necessary, we take γ := βκ, with κ ∈ (0, 1) as in Theorem 4.7,

α ∈ (2γ, 1), and τ0 ∈ (0, 14) such that

cτ20 =1

2τα0 ,

or, in other words, τ0 := (2c)−1

2−α .

Furthermore, for such a τ0 = τ0(m,n,N,L), we impose a further smallness condition on

ε > 0, so that

(4.34) cω1(ε)τ−n0 ≤

1

2τα0 .

This is possible because ω1 is increasing, continuous at 0 and ω1(0) = 0. Whereby, there exists

ε1 ∈ (0, 14n ) such that, for every ε ∈ (0, ε1), (4.34) holds.

We now let c0 := cτ−n0 and conclude that

E(x0, τ0) ≤ c02γ + τα0 E(x0, ).

This completes the proof of the proposition.

We conclude by establishing the proof of the regularity result. The proof follows a classical

iteration process and it fully relies on Proposition 4.1. However, since the exponent γ depends

on m > 1 in the previous statement, we include below the main steps of the iteration, for the

convenience of the reader.

Proof of Theorem 1.2. We begin by fixing m > 0 and noting that, if ε1 > 0 taken as in Propo-

sition 4.1 for the fixed number m + 1, and if 0 < ε < ε1 < 1, then for any x0 ∈ Ω0 and any

∈(

0, 12dist(x0, ∂Ω))

satisfying that

|(u)x0,|+ |(∇u)x0,| < m and E(x0, ) < ε,

there exists 0 ∈ (0, ) such that, for every x ∈ B(x0, 0), it also holds that

|(u)x,|+ |(∇u)x,| < m and E(x, ) < ε.

This means that, for every x ∈ B(x0, ) and for 0 < τ0 <14 as in Proposition 4.1, it holds that

E(x0, τ0) ≤ c02γ + τα0 E(x0, ).

A standard iteration process (see, for example, [31, Proposition 9.4]) leads to finding a constant

M =M(n,m,L, p) > 0 such that, for every x ∈ B(x0, ), and for every r ∈(

0, 14)

,

(4.35) E(x, r) ≤M

(

r

)γ

,

where γ is also as in Proposition 4.1. Therefore, by Campanato-Meyers’ classical characteriza-

tion of Holder continuity, it follows that u ∈ C1, γ2 (B(x0, 0),R

N ).In order to establish that, in fact, u ∈ C1,β(Ω0,R

N ), with β as in (H4), it is sufficient to go

again through the proof of Theorem 4.7 while exploiting the fact that ∇u is already known to be

continuous in Ω0, in the spirit of using Schauder estimates.


Indeed, following the notation of Theorem 4.7 we can take R > 0 so that B(x0, R) ⊆ Ω0, and

R > 0 small enough so that there exists a constant M1 =M1(m) > 0 such that

(4.36) ‖u‖C1,

γ2 (B(x0,R),RN )

+ ‖h‖C1,

γ2 (B(x0,R),RN )

≤M1.

Whereby, revisiting the estimates for each term on the right hand side of (4.26), we note that

there is no longer need to apply the higher integrability of ∇u and Holder inequality to estimate

III and V. Instead, we can use (4.36) to conclude that, since ϑ is concave, then for a constant

c = c(m,M1, n) > 0,

(4.37) III + V ≤ c ϑ(

cR2)

≤ cR2β.

In a similar fashion we can obtain that

(4.38) I + II + VI ≤ ω1(E(x0, 4R)E(x0, 4R) + cR2β,

where c > 0 now also depends on the modulus of continuity ω.

Compiling and applying these estimates to the corresponding inequality (4.33), we can con-

clude that, for every ε ∈ (0, 14n ) and m > 1, if ∈ (0, 4R1), B(x0, ) ⊆ Ω, |(u)x0,| +

|(∇u)x0,| < m, and E(x0, ) < ε, then for every τ ∈ (0, 14),

E(x0, τ) ≤cτ−n2β + c(τ−nω1(ε) + τ2)E(x0, ).

Whereby, we can argue again exactly as we did to obtain that u ∈ C1, γ2 (B(x0, R),R

N ) to

conclude now that, for some constant C > 0, and for any r ∈ (0, 14),

E(x0, r) ≤ Cr2β,

with the constant C depending in particular on , on β and, once again, onm > 1. This completes

the proof of Theorem 1.2.

FUNDING

This work was supported by Universidad Nacional Autonoma de Mexico - Direccion Gen-

eral de Asuntos del Personal Academico - Programa de Apoyo a Proyectos de Investigacion e

Innovacion Tecnologica [IA105221].

ACKNOWLEDGMENTS

The author wishes to thank Jan Kristensen for inspiring discussions. The author is also grateful

to the anonymous referee for providing very valuable corrections and for such a careful review

of the manuscript.

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Departamento de Matematicas, Facultad de Ciencias, Universidad Nacional Autonoma de Mexico,

Circuito exterior s/n, Ciudad Universitaria, C.P. 04510, Ciudad de Mexico, Mexico

e-mail: [email protected]

arXiv:2108.08869v1 [math.AP] 19 Aug 2021

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