Regularity for multi-phase variational problems

Available online at www.sciencedirect.com

ScienceDirect J. Differential Equations 267 (2019) 1631–1670 www.elsevier.com/locate/jde

Regularity for multi-phase variational problems Cristiana De Filippis a,∗ , Jehan Oh b a Mathematical Institute, University of Oxford, Andrew Wiles Building, Radcliffe Observatory Quarter, Woodstock

Road, Oxford, OX26GG, Oxford, United Kingdom b Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501 Bielefeld, Germany

Received 4 July 2018; revised 8 January 2019 Available online 5 March 2019

Abstract We prove C 1,ν -regularity for local minimizers of the multi-phase energy: ˆ |Dw|p + a(x)|Dw|q + b(x)|Dw|s dx,

w → 

under sharp assumptions relating the couples (p, q) and (p, s) to the Hölder exponents of the modulating coefficients a(·) and b(·), respectively. © 2019 Elsevier Inc. All rights reserved. Keywords: Regularity; Non-uniformly elliptic problems; Minimizer; Multi-phase

1. Introduction and results The aim of this paper is to analyze the regularity properties of non-autonomous variational integrals of the type ˆ w →

F (x, Dw) dx , 

* Corresponding author.

E-mail addresses: [email protected] (C. De Filippis), [email protected] (J. Oh). https://doi.org/10.1016/j.jde.2019.02.015 0022-0396/© 2019 Elsevier Inc. All rights reserved.

(1.1)

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where  ⊂ Rn is a bounded open domain with n ≥ 2, and emphasize a few new phenomena emerging when considering non-uniformly elliptic operators. Let us briefly review the situation. In the case of functionals satisfying standard polynomial growth and ellipticity of the type F (x, Dw) ≈ |Dw|p as for instance

∂zz F (x, Dw) ≈ |Dw|p−2 I d ,

and

(1.2)

ˆ w →

a(x)|Dw|p dx ,

0 < ν ≤ a(x) ≤ L ,

(1.3)



the regularity of minimizers is well-understood. In particular, assuming that the partial function x → F (x, ·) is Hölder continuous with some exponent (for example, a(·) is locally Hölder continuous in the case of (1.3)), then it turns out that the gradient of minima is locally Hölder continuous. This is a well established theory, both in the scalar and in the vectorial case, for which we refer for instance to [28,29,31,35]. The situation drastically changes when considering non-uniformly elliptic functionals. These are functionals so that the ellipticity ratio R(z, B) :=

supx∈B highest eigenvalue of ∂zz F (x, z) , infx∈B lowest eigenvalue of ∂zz F (x, z)

where B ⊂  is a ball, might become unbounded with |z|. This is the case, for instance, of the double phase functional given by ˆ W 1,H (·) () w →

|Dw|p + a(x)|Dw|q dx .

(1.4)



This functional has been introduced by Zhikov in the context of Homogenization and its integrand changes its growth – from p to q-rate – depending on the fact that x belongs to {a(·) = 0} or not (here is where the terminology double phase stems from). In the first case we have, following a terminology introduced in [11], the p-phase, in the other we have the (p, q)-phase. In the case of (1.4), the regularity of minimizers is dictated by a subtle interaction between the pointwise behaviour of the partial function x → F (x, ·) and the growth assumption satisfied by z → F (·, z). For instance, as established in the work of Baroni, Colombo and Mingione [2–4,11, 12], sufficient and necessary conditions for regularity of minimizers of the functional (1.4) are that a(·) ∈ C 0,α ()

and

q α ≤1+ . p n

(1.5)

Specifically, if (1.5) holds, then minimizers of the functional (1.4) are locally C 1,β0 -regular, for some β0 ∈ (0, 1), otherwise, they can be even discontinuous; see also [22,23]. After these contributions, functionals with double phase type have become a topic of intense study, see for instance [7–9,16,17,26,27,36,37]. The condition in (1.5) plays a role also when considering more general functionals of the type in (1.1), under so called (p, q)-growth conditions, i.e.: |Dw|p  F (x, Dw)  |Dw|q

and

|Dw|p−2 I d  ∂zz F (x, Dw)  |Dw|q−2 I d .

C. De Filippis, J. Oh / J. Differential Equations 267 (2019) 1631–1670

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For this we refer to [10,20,22]. Moreover, it intervenes in the validity of a corresponding Calderón-Zygmund theory [13,15]. We refer to the papers of Marcellini [32–34] for more on general functionals with (p, q)-growth. The aim of this paper is to study a significant generalization of the functional (1.4), considering a functional that exhibit three phases. We shall indeed consider the following Multi-Phase variational energy ˆ W

1,H (·)

() w → H(w, ) :=

H (x, Dw) dx,

1 < p
(1.6)



with H (x, z) := |z|p + a(x)|z|q + b(x)|z|s ,

(1.7)

and where the functions a(·) and b(·) satisfy the following assumptions a ∈ C 0,α (), a(·) ≥ 0, α ∈ (0, 1],

b ∈ C 0,β (), b(·) ≥ 0, β ∈ (0, 1].

(1.8)

As not to trivialize the problem, we specifically focus on the case in which the strict inequality (1.6)2 holds. The analysis of this functional then opens the way to that of functional exhibiting an arbitrary number of phases, and involves several subtle points. The main one can be described as follows. In the double phase case of the functional (1.4) the main game is to control the interaction between the potentially degenerate parte of the energy a(x)|Dw|q (here degenerate means that it can be a(x) ≡ 0) with the non-degenerate one |Dw|p , that always provides a solid rate of ellipticity. This is done in [4,11,12] via a careful comparison scheme built in order to distinguish between the two phases. Here the situation changes and the matter becomes more delicate. Indeed, the problem is to control the interaction between the two possibly degenerate parts of the energy, that is a(x)|Dw|q and b(x)|Dw|s . A new aspect in fact emerges here. We see that, in presence of a finer structure, conditions of the type in (1.5) can be in a sense relaxed. In fact, an immediate application of (1.5) would provide us with the conditions a(·), b(·) ∈ C 0,α () with q/p, s/p ≤ 1 + α/n, by considering the global regularity of x → F (x, ·). Instead, we see that the new condition coming into the play takes into account more precisely the way the presence of x affects the growth with respect to the gradient variable. Specifically, we shall assume that q α ≤1+ p n

and

s β ≤1+ . p n

(1.9)

In other words, less regularity is needed on the coefficient affecting the q-growth, intermediate part of the energy density. The main result of the paper is the following (see the next section for more definitions and notation): Theorem 1 (C 1,ν -local regularity). Let u be a local minimizer of the functional (1.6) under 1,ν (). assumptions (1.8) and (1.9). Then there exists ν = ν(data) ∈ (0, 1) such that u ∈ Cloc We remark that the sharpness of both conditions in (1.9) can be obtained by the same counterexamples in [22,23]. Moreover, as it is well-known from the regularity theory for the standard p-Laplacean case, the one in Theorem 1 is the maximal regularity obtainable for u. A worth singling-out intermediate result towards the proof of Theorem 1 is an intrinsic Morrey decay

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estimate, which reduces to a classical estimate in the case of the p-Laplacean and that extends to the multi-phase case the one proved in [4,11,12] for minima of functionals with a double phase. Theorem 2 (Intrinsic Morrey decay). Let u be a local minimizer of the functional (1.6) under assumptions (1.8) and (1.9). Then, for every ϑ ∈ (0, n), there exists a positive constant c = c(data(0 ), ϑ) such that the decay estimate ˆ H (x, Du) dx ≤ c

  n−ϑ ˆ

(1.10)

H (x, Du) dx

r

B

Br

holds whenever B ⊂ Br  0   are concentric balls with 0 <  ≤ r ≤ 1. Let us quickly describe the techniques we are employing to obtain the aforementioned theorems. The starting point is the recent proof of regularity of minimizers of double-phase variational problems appeared in [4], and based on a suitable use of harmonic type approximations lemmas (see also [12] for a first version). This is just a general blueprint we move from to treat the real new difficulty here. Indeed, as we are dealing here with the presence of several phase transitions, and we have to carefully handle the regularity of solutions on the zero sets {x ∈  : a(x) = 0} and {x ∈  : b(x) = 0}, that is, when the functional tends to lose part of its ellipticity properties and switch their kind of ellipticity. Therefore we have to handle the presence of two different transitions. We come up with a delicate scheme of alternatives and of nested exit time arguments, carefully controlling the interaction between the two phase transitions. It is then clear that the techniques introduced in this paper allow to prove regularity results for functionals with an arbitrary large numbers of phases, for instance, ˆ  w →

|Dw| + p



m 

 pi

ai (x)|Du|

dx ,

1 < p
i=1

with ai (·) ∈ C 0,αi (),

1<

pi αi ≤1+ , p n

1 < p < p 1 ≤ · · · ≤ pm .

(1.11)

Finally, let us compare our result with the one obtained in [21, Theorem 1.1]. Both essentially give optimal regularity for the gradient of minimizers, but there are some evident differences. Even if [21, Theorem 1.1] treats a larger class of functionals than the one covered by our Theorem 1, we do not require any differentiability for the coefficients a(·) and b(·), cf. assumption (1.8). Moreover, we allow various rates of ellipticity, as prescribed in (1.11). As anticipated would result in a trivialization of the problem. With above, asking that 1 < pq < ps ≤ 1 + min{α,β} n minor modification, see [4,16,30], our procedure also covers the regularity theory for minimizers of functionals with multi-phase structure, i.e. variational integrals of the type ˆ  W 1,H (·) () w → F(w, ) :=

fp (Dw) + 

1 < p < p 1 ≤ · · · ≤ pm ,

m  i=1

 ai (x)fpi (Dw)

dx,

C. De Filippis, J. Oh / J. Differential Equations 267 (2019) 1631–1670

with fp (z) ∼ |z|p , fpi (z) ∼ |z|pi for all z ∈ Rn , ai (·) ∈ C 0,αi () and 1 < i ∈ {1, · · · , m}.

pi p

1635

≤1+

αi n

for any

2. Notation and preliminaries In this section we establish some basic notation that we are going to use for the rest of the paper. As in the Introduction,  will denote an open, bounded subset of Rn with n ≥ 2. As usual, we shall denote by c a general constant larger than one, which can vary from line to line. Relevant dependencies from certain parameters will be emphasized using brackets, i.e.: c = c(n, p, q, s) means that c depends on n, p, q, s. We denote with Br (x0 ) = {x ∈ Rn : |x − x0 | < r} the n-dimensional open ball centered at x0 and with radius r > 0; when non relevant or clear from the context, we will omit to indicate the centre as follows: Br = Br (x0 ). When not differently specified, in the same context, balls with different radius will share the same center. If A ⊂ Rn is any measurable subset with finite and positive Lebesgue measure |A| > 0 and f : A → RN , N ≥ 1 is a measurable map, we shall denote its integral average over A as ˆ ˆ 1 (f )A := − f (x) dx = f (x) dx. |A| A

A

When A = Br , we shall write (f )r := (f )Br . The integrand H (·) has already been defined in (1.7). With abuse of notation we shall denote H (x, z) when z ∈ Rn and when z ∈ R, that is when z is a scalar, so that we shall intend both H :  × Rn → [0, ∞) and H :  × R → [0, ∞). The modulating coefficients a(·) and b(·) will always satisfy (1.8). Here we recall that, if f :  → R is any γ -Hölder continuous map with γ ∈ (0, 1) and A ⊂ , then its Hölder seminorm is defined as [f ]0,γ ;A :=

sup

x,y∈A, x=y

|f (x) − f (y)| , |x − y|γ

[f ]0,γ := [f ]0,γ ; .

With “∂” we denote the partial derivative with respect to the z-variable. We are going to use several tools from the Orlicz space setting, therefore we start with the following preliminaries. Definition 1. A function ϕ : [0, ∞) → [0, ∞) is said to be a Young function if it satisfies the following conditions: ϕ(0) = 0 and there exists the derivative ϕ  , which is right-continuous, non decreasing and satisfies ϕ  (0) = 0,

ϕ  (t) > 0

for t > 0,

and

lim ϕ  (t) = ∞.

t→∞

Remark 1. In order to extrapolate good regularity properties for minimizers of functionals with ϕ-growth, we need to assume something more. Precisely, from now on, in addition to the basic assumptions listed in Definition 1 we will also suppose that ϕ ∈ C 1 [0, ∞) ∩ C 2 (0, ∞) and that iϕ ≤

tϕ  (t) ≤ sϕ unifornly in t . ϕ  (t)

(2.1)

This is equivalent to the so-called 2 condition, since t → ϕ(t) is non decreasing, see [19], Section 2.

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Definition 2. Let ϕ be a Young function in the sense of Definition 1 and Remark 1. Given  ⊂ Rn , the Orlicz space Lϕ () is defined as ⎧ ⎫ ˆ ⎨ ⎬ Lϕ () := u :  → R such that ϕ(|u|) dx < ∞ ⎩ ⎭ 

and, consequently,

 W 1,ϕ () := u ∈ W 1,1 () ∩ Lϕ () such that Du ∈ Lϕ (, RN ) . 1,ϕ

1,ϕ

The definitions of the variants W0 () and Wloc () come in an obvious way from the one of W 1,ϕ (). In connection to H (·), we also consider the following Orlicz-Musielak-Sobolev space

 W 1,H (·) () := u ∈ W 1,1 () : H (·, Du) ∈ L1 () ,

(2.2) 1,p

with local variant defined in an obvious way and W01,H (·)() = W 1,H (·) () ∩ W0 (); we refer to [4,26,27] for more on such spaces. For later uses, we introduce also the auxiliary Young functions ⎧ H0 (z) := |z|p + a0 |z|q + b0 |z|s , ⎪ ⎪ ⎪ ⎨H s (z) := |z|p + b |z|s , 0 0 q p + a |z|q , ⎪ (z) := |z| H 0 ⎪ ⎪ ⎩ 0p H0 (z) := |z|p .

(2.3)

The values of the constants a0 , b0 ≥ 0 will vary according to the necessities, in particular they shall often assume the values a0 = ai (Br ) and b0 = bi (Br ), where ai (Br ) := inf a(x) and bi (Br ) := inf b(x). x∈Br

x∈Br

(2.4)

In accordance to this terminology, we also mention the auxiliary Young functions HB−r (z) := |z|p + ai (Br )|z|q + bi (Br )|z|s ,     H − (z) := |z|p + inf a(x) |z|q + inf b(x) |z|s x∈

x∈

In the following we will often use the vector field Vt (z) := |z|(t−2)/2 z,

t ∈ {p, q, s}.

(2.5)

C. De Filippis, J. Oh / J. Differential Equations 267 (2019) 1631–1670

We recall from [19] important features of (2.5): there exists c = c(n, t) > 0 such that   |Vt (z1 ) − Vt (z2 )|2 ≤ c |z1 |t−2 z1 − |z2 |t−2 z2 · (z1 − z2 ) , |Vt (z1 ) − Vt (z2 )| ∼ (|z1 | + |z2 |)

t−2 2

|z1 − z2 |,

1637

(2.6) (2.7)

where the constants implicit in (2.7) depend only on n, t and, for all z ∈ Rn |Vt (z)|2 = |z|t

(2.8)

.

We also introduce the following auxiliary functions ⎧ 2 2 2 2 ⎪ ⎪V0 (z1 , z2 ) := |Vp (z1 ) − Vp (z2 )| + a0 |Vq (z1 ) − Vq (z2 )| + b0 |Vs (z1 ) − Vs (z2 )| , ⎪ ⎨V s (z , z )2 := |V (z ) − V (z )|2 + b |V (z ) − V (z )|2 , p 1 p 2 0 s 1 s 2 0 1 2 ⎪V0q (z1 , z2 )2 := |Vp (z1 ) − Vp (z2 )|2 + a0 |Vq (z1 ) − Vq (z2 )|2 , ⎪ ⎪ ⎩ p V0 (z1 , z2 )2 := |Vp (z1 ) − Vp (z2 )|2 .

(2.9)

Let us also recall some important tools in regularity. The first one is an iteration lemma from [24]. Lemma 1. Let h : [, R0 ] → R be a non-negative bounded function and 0 < θ < 1, 0 ≤ A, 0 < β. Assume that h(r) ≤ A(d −r)−β +θ h(d) for  ≤ r < d ≤ R0 . Then h() ≤ cA/(R0 −)−β holds, where c = c(θ, β) > 0. Along the proof we shall make an intensive use of the regularity properties of ϕ-harmonic maps, so we recall definition and some reference estimates from [19]. 1,ϕ

Definition 3. Let U   be an open set and u0 ∈ Wloc (, RN ) be any function. With 1,ϕ ϕ-harmonic map, we mean a map h ∈ u0 + W0 (U, RN ) solving the Dirichlet problem 1,ϕ u0 + W0 (U, RN ) w

ˆ → min

ϕ(|Dw|) dx. U

The next proposition reports a Lipschitz type estimate for the gradient of ϕ-harmonic maps. Proposition 1. [19, Lemma 5.8] Let  ⊂ Rn be open and ϕ ∈ C 2 (0, ∞) ∩ C 1 [0, ∞) be a Young function satisfying (2.1). If h ∈ W 1,ϕ (, RN ) is ϕ-harmonic on  in the sense of Definition 3, then for any ball Br with B2r   there holds ˆ sup ϕ(|Dh|) ≤ c − ϕ(|Dh|) dx, Br

B2r

where c depends only on n, N, iϕ , sϕ . We conclude this section by giving the definition of a local minimizer of (1.6).

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C. De Filippis, J. Oh / J. Differential Equations 267 (2019) 1631–1670 1,H (·)

Definition 4. A map u ∈ Wloc () is a local minimizer of the variational integral (1.6) if and only if H (·, Du) ∈ L1loc () and the minimality condition H(u, supp(u −v)) ≤ H(v, supp(u −v)) 1,1 () and supp(u − v) ⊂ . is satisfied whenever v ∈ Wloc 3. First regularity results In this section we collect a few basic regularity results which can be proved with minor adjustments to the proofs contained in [4,11,12,26,36]. We start with the proof of Sobolev-Poincaré inequality. Lemma 2 (Sobolev-Poincaré inequality). Let 1 < p
⎞1 ⎛ d  ˆ  ⎟ ⎜ q−p s−p d ≤ c 1 + [a]0,α DwLp (Br ) + [b]0,β DwLp (Br ) ⎝ − H (x, Dw) dx ⎠ .

(3.1)

Br 1,H (·)

Furthermore, the same is still true with w − (w)r replaced by w if we consider w ∈ W0

(Br ).

Proof. We first look at the case sup a(x) ≤ 4[a]0,α r α x∈Br

and

sup b(x) ≤ 4[b]0,β r β ,

(3.2)

x∈Br

and notice that, by virtue of (1.9), q∗ < p and s∗ < p, where 

 nt t∗ := max , 1 , t ∈ {p, q, s}. n+t Then it follows from the classical Sobolev-Poincaré inequality, Holder inequality and (1.9) that ⎛ ⎞q q∗ ˆ ˆ ˆ |w − (w)r |q |w − (w)r |q ⎟ α α⎜ q∗ − a(x) dx ≤4[a]0,α r − dx ≤ c[a]0,α r ⎝ − |Dw| dx ⎠ rq rq Br

Br

Br

⎞ q−p ⎛ ⎞1 p da ˆ ˆ ⎟ ⎟ ⎜ ⎜ pd ≤c[a]0,α ⎝ − |Dw|p dx ⎠ ⎝ − |Dw| a dx ⎠ ⎛

Br

Br

C. De Filippis, J. Oh / J. Differential Equations 267 (2019) 1631–1670

⎞1 da ˆ ⎟ q−p ⎜ pda ≤c[a]0,α DwLp (Br ) ⎝ − |Dw| dx ⎠ ,

1639

⎛

(3.3)

Br

with c = c(n, q) and da :=

q∗ p

< 1. In a similar fashion,

⎞s ⎛ s∗ ˆ ˆ ˆ s s |w − (w) | |w − (w)r | r ⎟ ⎜ − b(x) dx ≤4[b]0,β r β − dx ≤ c[b]0,β r β ⎝ − |Dw|s∗ dx ⎠ rs rs Br

Br

Br

⎞ s−p ⎛

⎛

ˆ ⎟ ⎜ ≤c[b]0,β ⎝ − |Dw|p dx ⎠ Br

p

⎞1 db ˆ ⎟ ⎜ pd ⎝ − |Dw| b dx ⎠ Br

⎞1 db ˆ ⎟ ⎜ s−p pdb ≤c[b]0,β DwLp (Br ) ⎝ − |Dw| dx ⎠ , ⎛

(3.4)

Br

where c = c(n, s) and db :=

s∗ p

< 1. In addition, it is clear that

⎛ ⎞p ⎛ ⎞1 p∗ d0 ˆ ˆ ˆ p |w − (w)r | ⎜ ⎟ ⎜ ⎟ − dx ≤ c ⎝ − |Dw|p∗ dx ⎠ = c ⎝ − |Dw|pd0 dx ⎠ , rp Br

Br

for c = c(n, p) and d0 := (3.3)-(3.4), we get

p∗ p

Br

< 1. We remark from (1.6) that p∗
  ˆ w − (w)r − H x, dx r Br

⎞1 ⎛ db   ˆ ⎟ ⎜ q−p s−p pdb ≤ c 1 + [a]0,α DwLp (Br ) + [b]0,β DwLp (Br ) ⎝ − |Dw| dx ⎠ Br

⎞1 db ˆ   ⎟ ⎜ q−p s−p db ≤ c 1 + [a]0,α DwLp (Br ) + [b]0,β DwLp (Br ) ⎝ − H (x, Dw) dx ⎠ , ⎛

(3.5)

Br

where c = c(n, p, q, s). We now turn to the case sup a(x) > 4[a]0,α r α and sup b(x) > 4[b]0,β r β , x∈Br

x∈Br

(3.6)

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C. De Filippis, J. Oh / J. Differential Equations 267 (2019) 1631–1670

and notice that inf a(x) ≥

x∈Br

1 1 sup a(x) and inf b(x) ≥ sup b(x). x∈Br 2 x∈Br 2 x∈Br

(3.7)

So, if we set a0 := a(x0 ) and b0 := b(x0 ) for x0 ∈ Br , we obtain 

  a(x)t q ≤ 2 infx∈Br a(x) t q ≤ 2a0 t q ≤ 4a(x)t q   b(x)t s ≤ 2 infx∈Br b(x) t s ≤ 2b0 t s ≤ 4b(x)t s

for all x ∈ Br , t ≥ 0.

(3.8)

The content of the above display and the Sobolev Poincare inequality valid for the Young function H0 , see [18, Theorem 7], gives ⎞1 ⎛ d1     ˆ ˆ ˆ w − (w)r w − (w)r ⎟ ⎜ d1 − H x, dx ≤ 2 − H0 dx ≤ c ⎝ − H0 (Dw) dx ⎠ r r Br

Br

Br

⎛

⎞1 d1 ˆ ⎜ ⎟ ≤ c ⎝ − H (x, Dw)d1 dx ⎠ ,

(3.9)

Br

with c = c(n, p, q, s) and d1 = d1 (n, p, q, s) < 1. Let us examine the case sup a(x) > 4[a]0,α r α

sup b(x) ≤ 4[b]0,β r β

and

x∈Br

(3.10)

x∈Br

and notice that, by (3.10)1 , (3.7)1 and (3.8)1 hold true. This, (3.4) and the Sobolev-Poincaré q inequality valid for the Young function H0 render that ⎞1 ⎛ db    ˆ ˆ  ˆ w − (w)r w − (w) r ⎟ ⎜ q s−p − H x, dx ≤2 − H0 dx + c[b]0,β DwLp (Br ) ⎝ − |Dw|pdb dx ⎠ r r Br

Br

⎛

Br

⎞

⎞1 db ˆ ˆ ⎜ ⎟ ⎟ q s−p ⎜ dq pdb ≤c⎝ − H0 (Dw) dx ⎠ + c[b]0,β DwLp (Br ) ⎝ − |Dw| dx ⎠ ⎛

1 dq

Br

Br

⎛

⎞1

d2   ˆ ⎟ ⎜ s−p d2 ≤c 1 + [b]0,β DwLp (Br ) ⎝ − H (x, Dw) dx ⎠ ,

(3.11)

Br

with c = c(n, p, q, s), dq = dq (n, p, q) < 1 and d2 := max{dq , db } < 1. Finally, we look at the case

C. De Filippis, J. Oh / J. Differential Equations 267 (2019) 1631–1670

sup a(x) ≤ 4[a]0,α r α

and

x∈Br

sup b(x) > 4[b]0,β r β .

1641

(3.12)

x∈Br

Inequality (3.12)2 validates (3.7)2 and (3.8)2 , therefore, using this time (3.3), the Sobolev Poincaré inequality holding for the Young function H0s we get ⎞1 ⎛ da   ˆ ˆ w − (w)r ⎟ ⎜ q−p pda − H x, dx ⎠ dx ≤ c[a]0,α DwLp (Br ()) ⎝ − |Dw| r Br

Br

  ˆ w − (w)r + 2 − H0s dx r Br

⎞1 da ˆ ⎟ ⎜ q−p da ≤ c[a]0,α DwLp (Br ()) ⎝ − H (x, Dw) dx ⎠ ⎛

Br

⎛

⎞1 ds ˆ ⎜ ⎟ s ds + c ⎝ − H0 (Dw) dx ⎠ Br

⎞1 ⎛ d3  ˆ ⎟ ⎜ q−p d3 ≤ c 1 + [a]0,α DwLp (Br ()) ⎝ − H (x, Dw) dx ⎠ , (3.13) 

Br

with c = c(n, p, q, s), ds = ds (n, p, s) < 1 and d3 := max{da , ds } < 1. Setting d := max{db , d1 , d2 , d3 } < 1, from estimates (3.5), (3.9), (3.11) and (3.13) we obtain (3.1). 2 Remark 2. An inequality of the type of (3.1) holds for general Sobolev maps w ∈ W 1,H (·) such that w ≡ 0 on a set A such that |A| ≥ γ |Br |. Precisely, we have that ⎞1 ⎛ d ˆ ˆ  w ⎟ ⎜ d − H x, dx ≤ c ⎝ − H (x, Dw) dx ⎠ , r Br

(3.14)

Br

where d < 1 is the same as the one appearing in (3.1) and c = c(γ , n, p, q, s, [a]0,α , [b]0,β , DwLp (Br ) ). Next, we have a Caccioppoli-type inequality, which proof can be retrieved from those in [11, 26]. 1,H (·) Lemma 3 (Caccioppoli inequalities). Let u ∈ Wloc () be a local minimizer of (1.6), with a(·), b(·) and p, q, s satisfy (1.8) and (1.9) respectively. Then there exists a constant c = c(n, p, q, s) > 0 such that

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C. De Filippis, J. Oh / J. Differential Equations 267 (2019) 1631–1670

  ˆ ˆ u − (u)r − H (x, Du) dx ≤ c − H x, dx, r −

B

(3.15)

Br

and for κ ∈ R, ˆ

ˆ H (x, D(u − κ)± ) dx ≤ c

B

Br



(u − κ)± H x, r −

 dx.

(3.16)

A direct consequence of (3.15) and (3.1) is the following inner local higher integrability result. Lemma 4 (Gehring’s lemma). There are c = c(n, p, q, s, [a]0,α , [b]0,β , DuLp (Br ) ) > 0 and a positive integrability exponent δg = δg (n, p, q, s, [a]0,α , [b]0,β , DuLp (Br ) ) such that if u ∈ 1,p Wloc () is a local minimizer of the integral functional (1.6) under assumptions (1.8)-(1.9), then ⎛ 1+δ

H (·, Du) ∈ Lloc g ()

and

⎞ 1 1+δg ˆ ˆ ⎜ ⎟ 1+δg dx ⎠ ≤ c − H (x, Du) dx, ⎝ − H (x, Du) Br/2

Br

for all Br  .

(3.17) 1,p(1+δ )

g From (3.17)1 it follows that u ∈ Wloc (), so u ∈ W 1,p(1+δg ) (0 ) for 0  . Moreover, by Hölder inequality, (3.17) is true if δg is replaced by any σ ∈ (0, δg ). The next one is an up to the boundary higher integrability result for a solution of Dirichlet problems related to the multi-phase energy H (·). Clearly, when a(·) ≡ a0 = const and b(·) ≡ b0 = const, it extends p q to the auxiliary Young functions H0 , H0 , H0s and H0 . In this case, [a]0,α = [a0 ]0,α = 0 and [b]0,β = [b0 ]0,β = 0, so constants and exponents do not depend either on [a]0,α , [b]0,β nor on DvLp (Br ) .

Lemma 5 (Higher integrability up to the boundary). Let assumptions (1.8)-(1.9) be in force, 1,H (·) (Br ) be a solution to the Dirichlet problem Br  0  , and v ∈ Wu u + W01,H (·) (Br ) v

ˆ → min

H (x, Dw) dx,

(3.18)

Br

and δ0 > 0 be such that u ∈ W 1,H (·) 1+σg (Br ) and W 1,H (·)

1+δ0

(Br ). Then there exists 0 < σg < δ0 , so that v ∈

ˆ ˆ − H (x, Dv)1+σ dx ≤ c − H (x, Du)1+σ dx for all σ ∈ (0, σg ], Br

(3.19)

Br

where c = c(n, p, q, s, [a]0,α , [b]0,β , H (·, Du)L1 (Br ) ) and σg = σg (n, p, q, s, [a]0,α , [b]0,β , H (·, Du)L1 (Br ) ).

C. De Filippis, J. Oh / J. Differential Equations 267 (2019) 1631–1670

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Proof. With x0 ∈ Br , let us fix a ball B (x0 ) ⊂ Rn . We start with the case in which it is |B (x0 ) \ |B (x )| Br | > 10 0 . Let us fix /2 < t < s <  and take a cut-off function η ∈ Cc1 (Bs (x0 )) such that 1,H (·) (Br ) and η|∂Bs (x0 ) = 0, the χBt (x0 ) ≤ η ≤ χBs (x0 ) and |Dη| ≤ 2/(s − t). Since v − u ∈ W0 function v −η(v −u) coincides with v on ∂Br and on ∂Bs (x0 ) in the sense of traces and therefore, by the minimality of v and the features of η we obtain ˆ H (x, Dv) dx Bs (x0 )∩Br

≤c

⎧ ⎪ ⎨

ˆ

ˆ H (x, Dv) dx +

⎪ ⎩

(Bs (x0 )\Bt (x0 ))∩Br

Bs (x0 )∩Br

⎫ ⎪   ⎬ v−u H (x, Du) + H x, dx , ⎪ r ⎭

with c = c(n, p, q, s). By the classical hole-filling technique and Lemma 1, we can conclude that ˆ

ˆ H (x, Dv) dx ≤ c

B/2 (x0 )∩Br

  v−u H (x, Du) + H x, dx, r

(3.20)

B (x0 )∩Br 1,p

for c = c(n, p, q, s). Notice that we can extend v − u as zero outside Br since u − v ∈ W0 (Br ), so there are no discontinuities on ∂(Br ∩ B (x0 )) and recall that |B (x0 )| ≥ |B (x0 ) \ Br | > |B (x0 )| 10 . Poincaré’s inequality (3.14) applies, thus getting 

ˆ −

H x,

B (x0 )∩Br

v−u r



⎫ ⎧⎛ ⎞1 d ⎪ ⎪ ⎪ ⎪ ˆ ˆ ⎬ ⎨ ⎟ ⎜ d H (x, Dv) dx ⎠ + − H (x, Du) dx , dx ≤ c ⎝ − ⎪ ⎪ ⎪ ⎪ ⎭ ⎩ B (x0 )∩Br B (x0 )∩Br (3.21)

with c = c(n, p, q, s, [a]0,α , [b]0,β , H (·, Du)L1 (Br ) ). Here we dispensed c from the depen-

dence of DvLp (Br ) by using the minimality of v, the fact that v − u ∈ W01,H (·) (Br ) and observing that the constant appearing in (3.1) depends in an increasing fashion from the Lp -norm of the gradient. Merging (3.20) and (3.21) we obtain ˆ − B/2 (x0 )∩Br

⎧⎛ ⎫ ⎞1 d ⎪ ⎪ ⎪ ⎪ ˆ ˆ ⎨ ⎬ ⎟ ⎜ d H (x, Dv) dx ≤ c ⎝ − H (x, Dv) dx ⎠ + − H (x, Du) dx . ⎪ ⎪ ⎪ ⎪ ⎩ B (x0 )∩Br ⎭ B (x0 )∩Br

We next consider the situation when it is B (x0 )  Br , in which case the proof is analogous to the one for the interior case. As mentioned in Remark 2, we can assume that the exponent d < 1 from (3.1) and (3.14) is the same. The two cases can be combined via a standard covering argument. In fact, let us define

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 V (x) =

x ∈ Br x ∈ Rn \ Br

H (x, Dv(x))d 0

 and U (x) =

H (x, Du(x)) 0

x ∈ Br , x ∈ Rn \ Br

we get ˆ − B/2 (x0 )

⎧⎛ ⎫ ⎞1 d ⎪ ⎪ ⎪ ⎪ ˆ ⎨ ˆ ⎬ 1 ⎜ ⎟ V (x) d dx ≤ c ⎝ − V (x) dx ⎠ + − U (x) dx , ⎪ ⎪ ⎪ ⎪ ⎩ B (x0 ) ⎭ B (x0 )

with c = c(n, p, q, s, [a]0,α , [b]0,β , H (·, Du)L1 (Br ) ) and 0 < d < 1. At this point the conclusion follows by a standard variant of Gehring’s lemma, see [25, Theorem 3 and Proposition 1, Chapter 2]. 2 Furthermore, u is locally bounded. 1,H (·)

Lemma 6. Let u ∈ Wloc () be a local minimizer of (1.6) under assumptions (1.8) and (1.9). Then u is locally bounded in  and for any 0   there holds that uL∞ (0 ) ≤ c(n, p, q, s, [a]0,α , [b]0,β , H (·, Du)L1 (0 ) ).

(3.22)

Proof. If n < p(1 + δg ), where δg is the higher integrability exponent coming from Lemma 4, 0,

p(1+δg )−n

then there is nothing to prove. In fact, by Morrey’s embedding theorem u ∈ C p(1+δg ) (), therefore we can assume n ≥ p(1 + δg ). The boundedness result can be obtained as in [11, Section 10], as a consequence of (3.16) or by noticing that the generalized Young function in (1.7) under the assumptions (1.8) and (1.9) satisfies hypotheses (A0), (A1), (AInc) and (ADec) of [26, Theorem 1.3]. In fact, with the notation used in [26] (keep in mind also the terminology described in Section 2), it is easy to see that H + (δ) ≤ 1 ≤ H − (1) for δ :=

− p1

−

1

−

1

, (1 + aL∞ () ) q−p , (1 + bL∞ () ) s−p ∈ (0, 1). (A1) is true by choosing δ := 1 1   − q−p  − s−p n(s−p) − p1 − p1 − p1 α− n(q−p) β− p p [a]0,α ωn , , 2 diam() [b]0,β ωn min 3 , 2 diam() ∈ (0,1), min 3 

where ωn is the volume of the unit ball B1 ⊂ Rn . (AInc) clearly holds with γ − = p > 1 and (ADec) is verified by γ + = s ≥ q > p > 1. 2 4. Different alternatives For later uses, recall the quantities introduced in (2.4), which will play an important role along the proof. In fact, when dealing with those so called non uniformly elliptic problems, the question of the degeneracy of the coefficients is crucial. Precisely we will look at four different scenarios: ⎧ deg(Br ) : ai (Br ) ≤ 4[a]0,α r α−γa ⎪ ⎪ ⎪ ⎨ degα (Br ) : ai (Br ) ≤ 4[a]0,α r α−γa ⎪ degβ (Br ) : ai (Br ) > 4[a]0,α r α−γa ⎪ ⎪ ⎩ ndeg(Br ) : ai (Br ) > 4[a]0,α r α−γa

and and and and

bi (Br ) ≤ 4[b]0,β r β−γβ bi (Br ) > 4[b]0,β r β−γb bi (Br ) ≤ 4[b]0,β r β−γb bi (Br ) > 4[b]0,β r β−γb ,

C. De Filippis, J. Oh / J. Differential Equations 267 (2019) 1631–1670

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where  γa :=

0 n(q−p) p

α−

+

nδg (q−p) 2p(1+δg )

if

n ≥ p(1 + δg )

if

n < p(1 + δg )

(4.1)

and  γb :=

if n ≥ p(1 + δg )

0 β−

n(s−p) p

+

nδg (s−p) 2p(1+δg )

if n < p(1 + δg )

,

(4.2)

where δg is the higher integrability exponent given by Lemma 4. The above four cases, suitably combined, will render the desired regularity. To shorten the notation, we shall summarize the dependencies from the characteristics of the integrand we are dealing with, as data(0 )  ⎧ ⎪ n, p, q, s, [a]0,α , [b]0,β , H (·, Du)L1 (0 ) , H (·, Du)L1+δg (0 ) ⎪ ⎪ ⎪ ⎨ if n ≥ p(1 + δ ) g  , :=  ⎪ n, p, q, s, [a] ⎪ 0,α , [b]0,β , [u]C 0,λg (0 ) , H (·, Du)L1 (0 ) , H (·, Du)L1+δg (0 ) ⎪ ⎪ ⎩ if n < p(1 + δg )  ! n, p, q, s, [a]0,α , [b]0,β , H (·, Du)L1 (0 ) if n ≥ p(1 + δg ) ! , data0 (0 ) := if n < p(1 + δg ) n, p, q, s, [a]0,α , [b]0,β , [u]0,λg ;0 and data := (n, p, q, s, α, β) . n Here, λg := 1 − p(1+δ is the Hölder continuity exponent coming from Sobolev-Morrey’s emg) bedding theorem when n < p(1 + δg ) and 0   is any open set compactly contained in . This will be helpful, since all the existing results we are going to use are of local nature. Exploiting the different phases (deg)-(ndeg) we obtain the various forms of Caccioppoli’s inequality contained in Lemma 3. We collect them in the next Corollary. Moreover, the constants a0 and b0 p q appearing in the definition of the auxiliary Young functions H0 , H0 , H0s and H0 will take the values a0 = ai (Br ) and b0 = bi (Br ). 1,H (·)

Corollary 3. Let u ∈ Wloc () be a local minimizer of (1.6) under assumptions (1.8)-(1.9), and Br , r ∈ (0, 1) be any ball such that B2r  0  . Then the following is verified:   ˆ ˆ p u − (u)2r deg(Br ) ⇒ − H (x, Du) dx ≤ c − H0 dx, r Br

B2r

  ˆ ˆ s u − (u)2r degα (Br ) ⇒ − H (x, Du) dx ≤ c − H0 dx, r Br

(4.3)

B2r

(4.4)

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  ˆ ˆ q u − (u)2r degβ (Br ) ⇒ − H (x, Du) dx ≤ c − H0 dx, r Br

(4.5)

B2r

  ˆ ˆ u − (u)2r ndeg(Br ) ⇒ − H (x, Du) dx ≤ c¯ − H0 dx, r Br

(4.6)

B2r

for c = c(data0 (0 )) and c¯ = c(n, ¯ p, q, s, [a]0,α , [b]0,β ). Proof. First, notice that, by (1.9), γa ≥ 0 and γb ≥ 0. Moreover, if n ≥ p(1 + δg ) we see that α − γa + p − q ≥

n(q − p) − (q − p) ≥ δg (q − p) > 0, p

(4.7)

β − γb + p − s ≥

n(s − p) − (s − p) ≥ δg (s − p) > 0, p

(4.8)

while, if n < p(1 + δg ), α − γa + (λg − 1)(q − p) =

nδg (q − p) > 0, 2p(1 + δg )

(4.9)

β − γb + (λg − 1)(s − p) =

nδg (s − p) > 0. 2p(1 + δg )

(4.10)

Assume deg(Br ). We observe that for any x ∈B2r , a(x) ≤ 8[a]0,α r α−γa and b(x) ≤ 8[b]0,β r β−γb , since γa , γb ≥ 0 and r ∈ (0, 1). If n ≥ p(1 + δg ), from (3.15), Lemma 6, (4.7) and (4.8) we get,   ˆ ˆ u − (u)2r − H (x, Du) dx ≤ c − H x, dx r Br

B2r

ˆ  q−p ≤ c − 1 + 8[a]0,α r α−γa +p−q uL∞ (0 ) B2r

+ 8[b]0,β r

β−γb +p−s

s−p uL∞ (0 )

  ˆ p u − (u)2r dx, ≤ c − H0 r

 "" u − (u) ""p 2r " " " dx " r

B2r

where c = c(data0 (0 )). To determine the dependencies of c we also used (3.22). On the other hand, if n < p(1 + δg ) proceding as before but using Sobolev-Morrey’s theorem and (4.9), (4.10) instead of (4.7), (4.8), we obtain   ˆ ˆ u − (u)2r − H (x, Du) dx ≤c − H x, dx r Br

B2r

C. De Filippis, J. Oh / J. Differential Equations 267 (2019) 1631–1670

ˆ  q−p ≤c − 1 + 8[a]0,α r α−γa +(λg −1)(q−p) [u] 0,λg C

(0 )

B2r

+ 8[b]0,β r

β−γb +(λg −1)(s−p)

  ˆ p u − (u)2r ≤c − H0 dx, r

1647

s−p [u] 0,λg (0 ) C

 "" u − (u) ""p 2r " " " dx " r

B2r

where c = c(data0 (0 )). Now suppose degα (Br ). If n ≥ p(1 + δg ), we see from (3.15), (4.7), (4.8) and Lemma 6 that   ˆ ˆ u − (u)2r − H (x, Du) dx ≤c − H x, dx r Br

B2r

ˆ   "" u − (u) ""p 2r " q−p α−γa +p−q uL∞ (0 ) "" ≤c − 1 + 8[a]0,α r " dx r B2r

" " " " ˆ " u − (u)2r "s " u − (u)2r "s " " " " dx + c − |b(x) − bi (Br )| " " + bi (Br ) " " r r B2r

ˆ ≤c − B2r

ˆ ≤c −

" " "s "s " " " " " " u − (u)2r "p " " + [b]0,β r β " u − (u)2r " + bi (Br ) " u − (u)2r " dx " " " " " " " r r r " " " "s   ˆ " u − (u)2r "p " " " " + bi (Br ) " u − (u)2r " dx ≤ c − H s u − (u)2r dx, 0 " " " " r r r

B2r

B2r

since, being r ∈ (0, 1), r β ≤ r β−γb . Here, c = c(data0 (0 )). If n < p(1 + δg ) we have, by exploiting (4.9) and (4.10),   ˆ ˆ u − (u)2r − H (x, Du) dx ≤c − H x, dx r Br

B2r

ˆ  q−p ≤c − 1 + 8[a]0,α r α−γa +(λg −1)(q−p) [u] 0,λg C

 "" u − (u) ""p 2r " " " dx (0 ) " r

B2r

" " " " ˆ " u − (u)2r "s " u − (u)2r "s " " " " dx + c − |b(x) − bi (Br )| " " + bi (Br ) " " r r B2r

ˆ ≤c −

" " " " "s "s " u − (u)2r "p " " " " " " + [b]0,β r β " u − (u)2r " + bi (Br ) " u − (u)2r " dx " " " " " " r r r

B2r

" " "s   ˆ " ˆ " " " u − (u)2r "p " + bi (Br ) " u − (u)2r " dx ≤ c − H s u − (u)2r dx, ≤c − "" 0 " " " r r r B2r

B2r

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with c = c(data0 (0 )). If degβ (Br ) is in force, then, as before, for n ≥ p(1 + δg ), we have   ˆ ˆ u − (u)2r − H (x, Du) dx ≤c − H x, dx r Br

B2r

ˆ   "" u − (u) ""p 2r " s−p β−γb +p−s uL∞ (0 ) "" ≤c − 1 + 8[b]0,β r " dx r B2r

" " " "q ˆ " u − (u)2r "q " " " + ai (Br ) " u − (u)2r " dx + c − |a(x) − ai (Br )| "" " " " r r B2r

" " " "q   ˆ " u − (u)2r "p " " " " + ai (Br ) " u − (u)2r " dx ≤ c − H q u − (u)2r dx, 0 " " " " r r r

ˆ ≤c − B2r

B2r

where c = c(data0 (0 )). Moreover, if n < p(1 + δg ) we obtain   ˆ ˆ u − (u)2r − H (x, Du) dx ≤c − H x, dx r Br

B2r

ˆ  s−p ≤c − 1 + 8[b]0,β r β−γb +(λg −1)(s−p) [u] 0,λg C

 "" u − (u) ""p 2r " " " dx (0 ) " r

B2r

" " " " ˆ " u − (u)2r "q " u − (u)2r "q " " " " dx + c − |a(x) − ai (Br )| " " + ai (Br ) " " r r B2r

ˆ ≤c −

" " " "q   ˆ " " u − (u)2r "p " " " + ai (Br ) " u − (u)2r " dx ≤ c − H q u − (u)r dx, 0 " " " " r r r

B2r

B2r

with c = c(data0 (0 )). Finally, if ndeg(Br ) holds, then by (3.15), (1.8), the fact that either if n ≥ p(1 + δg ) or if n < p(1 + δg ), α ≥ α − γa and β ≥ β − γb , and the very definition of ndeg(Br ) we have   ˆ ˆ u − (u)2r − H (x, Du) dx ≤ c − H dx r Br

B2r

ˆ ≤c − B2r

" " " "(q−p)+p " u − (u)2r "p " " " " + |a(x) − ai (Br )| " u − (u)2r " " " " " 2r r

" " " " " u − (u)2r "(s−p)+p " u − (u)2r "q " " " + |b(x) − bi (Br )| "" + a (B ) i r " " " r r " "s " u − (u)2r " " dx + bi (Br ) "" " r

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" " " "q "s ˆ " " u − (u)2r "p " " " " " + [a]0,α r α " u − (u)2r " + [b]0,β r β " u − (u)2r " dx ≤ c − "" " " " " " r r r B2r

    ˆ ˆ u − (u)2r u − (u)2r dx ≤ c¯ − H0 dx, + c − H0 r r B2r

B2r

with c¯ = c(n, ¯ p, q, s, [a]0,α , [b]0,β ).

2

We conclude this section by recalling a quantitative Harmonic-approximation type result from [4]. We shall report it in the form that better fits our necessities. Lemma 7. Let Br ⊂ Rn be a ball, ε ∈ (0, 1), H˜ be one of the Young functions defined in (2.3) ˜ and v ∈ W 1,H (B2r ) be a map satisfying the following estimates: ˆ − H˜ (Dv) dx ≤ c˜1 ,

(4.11)

B2r

and ˆ − H˜ (Dv)1+σ0 dx ≤ c˜2 ,

(4.12)

Br

where c˜1 , c˜2 ≥ 1 and σ0 > 0 are fixed constants. Moreover, assume that " " " "ˆ " " " " " − ∂ H˜ (Dv) · Dϕ dx " ≤ εDϕL∞ (Br ) for all ϕ ∈ Cc∞ (Br ), " " " " Br

(4.13)

˜ for some ε ∈ (0, 1). Then there exists a function h˜ ∈ v + W01,H (Br ) such that the following conditions are satisfied:

ˆ ˜ · Dϕ dx = 0 for all ϕ ∈ Cc∞ (Br ), − ∂ H˜ (D h)

(4.14)

Br

ˆ ˜ 1+σ1 dx ≤ c(n, p, q, s, σ0 )c˜2 , − H˜ (D h)

(4.15)

Br

ˆ # ˜ 2 dx ≤ cε m , D h) − V(Dv,

(4.16)

Br

where V˜ is the corresponding auxiliary function defined in (2.9), c = c(n, p, q, s, c˜1 , c˜2 ), σ1 = σ1 (n, p, q, s, σ0 ) ∈ (0, σ0 ), m = m(n, p, q, s, σ0 ) > 0.

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p q Proof. The cases of H˜ = H0 , H0 , H0s are treated in [4, Lemma 1], so we focus on H˜ = H0 . The proof we provide is in some sense a simplified version of the original one since we do not need a powerful result such as [13, Theorem 5.1]. In fact we can recover some extra boundary integrability from Lemma 5. Define h0 ∈ W 1,H0 (Br ) to be the solution to the Dirichlet problem ˆ v + W01,H0 (Br ) w → min H0 (Dw) dx. (4.17) Br

By minimality (4.14) is verified, since it is the Euler-Lagrange equation associated to the above variational problem. Moreover, it follows from (4.11) that ˆ ˆ − H0 (Dh0 ) dx ≤ − H0 (Dv) dx ≤ 2n c˜1 . (4.18) Br

Br

Now, by the previous inequality, Lemma 5 with a(·) ≡ const and b(·) ≡ const, and (4.12), we obtain ˆ ˆ 1+σg 1+σ 1+σg − H0 (Dh0 ) dx ≤c − H0 (Dv)1+σg dx ≤ cc˜2 0 := c˜3 , Br

(4.19)

Br

for some 0 < σg < σ0 , which is (4.15) with σ1 = σg . Here c˜3 = c˜3 (n, p, q, s, σg , c˜1 , c˜2 ). Set w = 1,H h0 − v ∈ W0 0 (Br ) and let λ ≥ 1 to be fixed later and consider wλ ∈ W01,∞ (Br ), the Lipschitz truncation of w given by the main result in [1] and satisfying Dwλ L∞ (Br ) ≤ c(n)λ

and

{wλ = w} ⊂ {M(|Dw|) > λ} ∪ negligible set.

(4.20)

Using such properties, the fact that t → H0 (t) is increasing, Markov’s inequality, (4.12), (4.19) and the maximal theorem we deduce that ˆ |{wλ = w}| |Br ∩ {M(|Dw|) ≥ λ}| 1 − H0 (M(|Dw|))1+σg dx ≤ ≤ |Br | |Br | H0 (λ)1+σg Br

ˆ ˆ c c 1+σg − H (Dw) dx ≤ − H0 (Dh0 )1+σg + H0 (Dv)1+σg dx ≤ 0 H0 (λ)1+σg H0 (λ)1+σg Br

Br

⎡

≤

c H0 (λ)1+σg

⎤ ⎛ ⎞ 1+σg 1+σ0 ˆ ⎢ˆ ⎥ ⎜ ⎟ ⎢ ⎥ ⎢ − H0 (Dh0 )1+σg dx + ⎝ − H0 (Dv)1+σ0 dx ⎠ ⎥ ⎣ ⎦

1+σg 1+σ0

≤

c(c˜3 + c˜2 H0

(λ)1+σg

Br

)

≤

Br

c(c˜3 + c˜2 ) , H0 (λ)1+σg

(4.21)

where c = c(n, p, q, s, σg , σ0 ). Now we test (4.14) against wλ , which is admissible by density, to get

C. De Filippis, J. Oh / J. Differential Equations 267 (2019) 1631–1670

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ˆ (I) := − (∂H0 (Dh0 ) − ∂H0 (Dv)) · Dwλ χ{wλ =w} dx Br

ˆ ˆ = − − ∂H0 (Dv) · Dwλ dx − − (∂H0 (Dh0 ) − ∂H0 (Dv)) · Dwλ χ{w=wλ } dx Br

Br

=: (II) + (III). The properties of H0 and (2.6) give ˆ (I) ≥ c − V0 (Dv, Dh0 )χ{wλ =w} dx,

(4.22)

Br

where c = c(n, p, q, s) > 0. Moreover, by (4.13) and (4.20)1 we see that |(II)| ≤ cελ,

(4.23)

with c = c(n). Before estimating term (III), we recall a standard Young type inequality holding for H0 , see [4]: for all σ ∈ (0, 1), xy ≤ σ 1−s H0 (x) + σ H0∗ (y),

(4.24)

∗ (y) = sup where x>0 {yx − H0 (x)} is the convex conjugate of H0 . Furthermore, there holds:  H0  H (t) ∗ 0 H0 ≤ H0 (t), see [5] for more details. Now, using (4.20)1 , (4.11), (4.18), (4.24) and (4.21) t we estimate, for a certain fixed σ ∈ (0, 1),

 ˆ  H0 (Dh0 ) H0 (Dv) |(III)| ≤sDwλ L∞ (Br ) − + χ{wλ =w} dx |Dh0 | |Dv| Br

   ˆ   cH0 (Dwλ L∞ (Br ) ) |{wλ = w}| ∗ H0 (Dh0 ) ∗ H0 (Dv) + H0 dx + ≤σ − H0 |Dh0 | |Dv| σ s−1 |Br | Br

ˆ ≤σ − H0 (Dh0 ) + H0 (Dv) dx + Br

c σ s−1 H0 (λ)σg

≤ 2n+1 σ c˜1 +

c σ s−1 λpσg

,

(4.25)

where we also used the fact that H0 (λ) ≥ λp since λ ≥ 1. Here c = c(n, p, q, s, σg ). Collecting (4.22), (4.23) and (4.25) we obtain ˆ   − V0 (Dv, Dh0 )2 χ{wλ =w} dx ≤c ελ + σ + σ 1−s λ−pσg ,

(4.26)

Br

for c = c(c˜1 , c˜2 , c˜3 , n, p, q, s, σg ) and σ ∈ (0, 1) to be fixed. For θ ∈ (0, 1), by Hölder’s inequality and (4.26) we estimate

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C. De Filippis, J. Oh / J. Differential Equations 267 (2019) 1631–1670

⎞1 θ ˆ   ⎟ ⎜ 2θ 1−s −pσ ⎝ − V0 (Dv, Dh0 ) χ{wλ =w} dx ⎠ ≤ c ελ + σ + σ λ g . ⎛

Br

Again, by Hölder’s inequality, (4.21), (4.11) and (4.18) we have ⎞1 θ   1−θ ˆ ˆ |{wλ = w}| θ ⎟ ⎜ 2θ − V (Dv, Dh ) χ dx ≤c − V0 (Dv, D0 )2 dx ⎠ ⎝ 0 0 {wλ =w} |Br | ⎛

Br

Br

≤cH0 (λ)−

(1+σg )(1−θ) θ

ˆ p(1−θ) − H0 (Dh0 ) + H0 (Dv) dx ≤ cλ− θ , Br

(4.27) pσg

where c = c(c˜1 , c˜2 , c˜3 , n, p, q, s, σg ). Setting in (4.26) and (4.27) λ = ε −1/2 and σ = ε 4(s−1) we obtain that ⎞1 θ ˆ ⎟ ⎜ 2θ 2m ⎝ − V0 (Dv, Dh0 ) dx ⎠ ≤ cε , ⎛

(4.28)

Br

 pσ pσg with c = c(c˜1 , c˜2 , c˜3 , n, p, q, s, σg ) and m := 12 min 12 , 4 g , 4(s−1) , p(1−θ) . Notice that in the 2θ above estimates we still have a degree of freedom in θ . Applying Hölder’s inequality with expo2(1+σ ) nents 1+2σgg and 2(1 + σg ) we obtain ⎛ ⎞ 1+2σg ⎛ ⎞ 1 1+σg 2(1+σg ) ˆ ˆ ˆ 2(1+σg ) ⎜ ⎟ ⎟ ⎜ 2 2(1+σ ) g dx − V0 (Dv, Dh0 ) dx ≤ ⎝ − V0 (Dv, Dh0 ) 1+2σg dx ⎠ ⎠ ⎝ − V0 (Dv, Dh0 ) Br

Br

Br

⎞ 1 1+σg ˆ ⎟ m⎜ 1+σg ≤cε ⎝ − (H0 (Dh0 ) + H0 (Dv)) dx ⎠ ≤ cε m , ⎛

Br

with c = c(c˜1 , c˜2 , c˜3 , n, p, q, s, σg ). Here we used (4.12), (4.19), and (4.28) with θ = Recalling (2.8), we can conclude from the previous estimate that ˆ − V0 (Dv, Dh0 )2 dx ≤ cε m , Br

which is what we wanted. 2

1+σg 1+2σg

< 1.

C. De Filippis, J. Oh / J. Differential Equations 267 (2019) 1631–1670

1653

5. Morrey decay and Theorem 2 The proof of Theorem 2 goes in two moments: first, we prove that a suitable manipulation of a local minimizer u of (1.6) satisfies the assumptions of Lemma 7, then we exploit this to start an iteration which will eventually render the announced decay. Step 1: Quantitative harmonic approximation. Define the quantities ⎛

⎞1 p ˆ u ⎜ ⎟ E := E(u, B2r ) = ⎝ − H (x, Du) dx ⎠ and v := , E B2r 1,H (·)

where u ∈ Wloc () is a local minimizer of (1.6) and B2r  0   is any ball of radius r ≤ 12 . From now on, we will consider the following auxiliary Young functions ⎧ H0 (z) := |z|p + ai (Br )|z|q + bi (Br )|z|s , ⎪ ⎪ ⎪ ⎪ ⎪ H˜ 0 (z) := |z|p + ai (Br )E q−p |z|q + bi (Br )E s−p |z|s , ⎪ ⎪ ⎪ s p s ⎪ ⎪ ⎨H0 (z) := |z| + bi (Br )|z| , s p s−p |z|s , H˜ 0 (z) := |z| + bi (Br )E ⎪ ⎪ q ⎪ H0 (z) := |z|p + ai (Br )|z|q , ⎪ ⎪ ⎪ q ⎪ ⎪ H˜ (z) := |z|p + ai (Br )E q−p |z|q , ⎪ ⎪ ⎩ 0p H0 (z) := |z|p ,

(5.1)

and ⎧ V0 (z1 , z2 )2 := |Vp (z1 ) − Vp (z2 )|2 + ai (Br )|Vq (z1 ) − Vq (z2 )|2 + bi (Br )|Vs (z1 ) − Vs (z2 )|2 , ⎪ ⎪ ⎪ ⎨V s (z , z )2 := |V (z ) − V (z )|2 + b (B )|V (z ) − V (z )|2 , p 1 p 2 i r s 1 s 2 0 1 2 ⎪V0q (z1 , z2 )2 := |Vp (z1 ) − Vp (z2 )|2 + ai (Br )|Vq (z1 ) − Vq (z2 )|2 , ⎪ ⎪ ⎩ p V0 (z1 , z2 )2 := |Vp (z1 ) − Vp (z2 )|2 , where ai (·) and bi (·) are defined as in (2.4). Since u is a local minimizer of (1.6), a straightforward computation shows that v is a local minimizer of the functional ˜ H(w, ) :=

ˆ |Dw|p + a(x)E q−p |Dw|q + b(x)E s−p |Dw|s dx. 

Then, by scaling, it is easy to see that Lemma 4 holds true also for v with the same extra integrability exponent δg = δg (n, p, q, s, [a]0,α , [b]0,β , DuLp (Br ) ) as u. For any open set U   it satisfies the Euler-Lagrange equation 0=

ˆ 

 p|Dv|p−2 + qa(x)E q−p |Dv|q−2 + sb(x)E s−p |Dv|s−2 Dv · Dϕ dx

U

for all ϕ ∈ Cc∞ (U ).

(5.2)

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C. De Filippis, J. Oh / J. Differential Equations 267 (2019) 1631–1670

Moreover, if H˜ denotes H0 , H˜ 0 , H˜ 0s or H˜ 0 , we see from the definition of v that p

q

ˆ ˆ − H˜ (Dv) dx ≤ E −p − H (x, Du) dx ≤ 1, B2r

(5.3)

B2r

which is (4.11) and, by (5.3) and Lemma 4 we obtain, for some σ˜ g ∈ (0, δg ), ˆ ˆ − H˜ (Dv)1+σ˜ g dx ≤ − H (x, Dv)1+σ˜ g dx Br

Br

⎞−(1+σ˜ g ) ⎛ ⎞ ˆ ˆ ⎟ ⎟ ⎜ ⎜ 1+σ˜ = ⎝ − H (x, Du) dx ⎠ ⎝ − H (x, Du) g dx ⎠ ≤ c, ⎛

B2r

(5.4)

Br

where c = c(n, p, q, s, [a]0,α , [b]0,β , DuLp (0 ) ) is the constant appearing in Lemma 4 and this verifies (4.12). So we see that conditions (4.11)-(4.12) of Lemma 7 are matched with σ0 = σ˜ g no matter what degeneracy (or non-degeneracy) condition holds on B2r . Clearly we have no problems of integrability, since σ˜ g < δg , which is the exponent coming from Lemma 4. We now define σ a = α − γa −

n(q − p) p(1 + δg )

and σb = β − γb −

n(s − p) . p(1 + δg )

(5.5)

A simple computation shows that σa and σb are both positive numbers. We first assume deg(Br ). From (5.2) we deduce that " " " "ˆ ˆ ˆ " " " " p q−p q−1 s−p − a(x)|Dv| |Dϕ| dx + sE − b(x)|Dv|s−1 |Dϕ| dx " − ∂H0 (Dv) · Dϕ dx " ≤ qE " " " " Br Br Br =: (I)deg + (II)deg . From the very definition of condition deg, Lemma 4, (5.3), Hölder’s inequality and (5.5) we get (I)deg ≤cE

q−p q

r

α−γa q

ˆ q−1 DϕL∞ (Br ) − (E q−p a(x)) q |Dv|q−1 dx Br

⎞ q−p ⎞ q−1 ⎛ pq q ˆ ˆ α−γa ⎟ ⎟ ⎜ ⎜ q−p−q q ≤cDϕL∞ (Br ) ⎝ − H (x, Du) dx ⎠ r q ⎝− E a(x)|Du| dx ⎠ ⎛

B2r

≤cDϕL∞ (Br ) H (·, Du)

Br q−p pq L1+δg (0 )

r

α−γa q

n(q−p) − pq(1+δ g)

≤ cDϕL∞ (Br ) r

σa q

with c1 = c1 (n, p, q, [a]0,α , H (·, Du)L1+δg (0 ) ). In a totally similar way we obtain

(5.6)

C. De Filippis, J. Oh / J. Differential Equations 267 (2019) 1631–1670

(II)deg ≤cE

s−p s

r

1655

ˆ s−1 DϕL∞ (Br ) − (E s−p b(x)) s |Dv|s−1 dx

β−γb s

Br

⎞ s−p ⎞ s−1 ⎛ ps s ˆ ˆ β−γb ⎜ ⎟ ⎟ ⎜ s−p−s s ≤cDϕL∞ (Br ) ⎝ − H (x, Du) dx ⎠ r s ⎝− E b(x)|Du| dx ⎠ ⎛

B2r

≤cDϕL∞ (Br ) H (·, Du)

Br s−p ps L1+δg (0 )

r

β−γb s

n(s−p) − ps(1+δ g)

≤ c2 DϕL∞ (Br ) r

σb s

(5.7)

where c2 = c2 (n, p, s, [b]0,β , H (·, Du)L1+δg (0 ) ). Now we define σ˜ p := 12 min{q −1 σa , s −1 σb } > 0 and fix a threshold radius R˜ ∗1 such that max{c1 , c2 }(R˜ ∗1 )σ˜p ≤ 12 and assume that 0 < r ≤ min{R˜ ∗1 , 1}. In correspondence of such a choice, by (5.6) and (5.7) we can conclude that " " " "ˆ " " " " p " − ∂H0 (Dv) · Dϕ dx " ≤r σ˜ p DϕL∞ (Br ) , " " " " Br

(5.8)

so the assumptions of Lemma 7 are matched and there exists a H0 -harmonic map h˜ p satisfying in p

p

1,H p particular (4.16). It is clear that, if hp := E h˜ p , then hp is still H0 -harmonic, hp −u ∈ W0 0 (Br ) and, by (4.16),

ˆ ˆ p − V0 (Du, Dhp )2 dx ≤ cr mp − H (x, Du) dx, Br

(5.9)

B2r

for c = c(n, p, q, s, [a]0,α , [b]0,β , H (·, Du)L1+δg (0 ) ) and mp = mp (n, p, q, s, [a]0,α , [b]0,β , H (·, Du)L1 (0 ) ). Suppose now that degα (Br ) holds. Then, by (5.2) we obtain " " " "ˆ " " " " s " − ∂ H˜ 0 (Dv) · Dϕ dx " " " " " Br ˆ ˆ q−p q−1 s−p − a(x)|Dv| |Dϕ| dx + sE − (b(x) − bi (Br ))|Dv|s−1 |Dϕ| dx ≤ qE Br

Br

ˆ ˆ q−p q−1 β s−p ∞ ∞ ≤ qE DϕL (Br ) − a(x)|Dv| dx + 2s[b]0,β r E DϕL (Br ) − |Dv|s−1 dx Br

=: (I)degα + (II)degα .

Br

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As in (5.6), we estimate ⎛

(I)degα

⎛ ⎞ q−p ⎞ q−1 pq q ˆ ˆ α−γa ⎜ ⎜ ⎟ ⎟ q−p−q q ≤cDϕL∞ (Br ) ⎝ − H (x, Du) dx ⎠ r q ⎝− E a(x)|Du| dx ⎠ B2r

Br

≤cDϕL∞ (Br ) H (·, Du)

q−p pq L1+δg (0 )

r

α−γa q

n(q−p) − pq(1+δ g)

≤ cDϕL∞ (Br ) r

σa q

(5.10)

with c1 = c1 (n, p, q, [a]0,α , H (·, Du)L1+δg (0 ) ), and ⎞ ˆ ⎟ ⎜ s−p β−γb s−1 r ) s |Dv|s−1 dx ⎠ ⎝ − (E ⎛ (II)degα ≤cDϕL∞ (Br ) E

s−p s

β

rs+

γb (s−1) s

Br

⎛

≤cDϕL∞ (Br ) r

≤cr

γb (s−1) 1 +s s

γb (s−1) s

r

  n(s−p) β− p(1+δ g)

1 s

  ˆ n(s−p) β− p(1+δ ⎜ g)

⎞ s−1 s

⎟ −p s ⎝ − E bi (Br )|Du| dx ⎠ B2r

DϕL∞ (Br ) ,

(5.11)

 where c2 = c2 (n, p, s, [b]0,β , H (·, Du)L1+δg (0 ) ). Define σ˜ s := 12 min σqa , γb (s−1) + 1s β − s  n(s−p) > 0 and fix a threshold radius R˜ ∗2 such that max{c1 , c2 }(R˜ ∗2 )σ˜s ≤ 12 and assume that p(1+δg ) 0 < r ≤ min{R˜ ∗1 , R˜ ∗2 , 1}. In correspondence of such a choice, by (5.10) and (5.11) we can conclude that " " " "ˆ " " " " s (5.12) " − ∂ H˜ 0 (Dv) · Dϕ dx " ≤r σ˜ s DϕL∞ (Br ) , " " " " Br so the assumptions of Lemma 7 are matched and there exists a H˜ 0s -harmonic map h˜ s satisfying 1,H s in particular (4.16). Clearly, if hs := E h˜ s , then hs is H0s -harmonic, hs − u ∈ W0 0 (Br ) and, by (4.16), ˆ ˆ s 2 ms − V0 (Du, Dhs ) dx ≤ cr − H (x, Du) dx, (5.13) Br

B2r

where c = c(n, p, q, s, [a]0,α , [b]0,β , H (·, Du)L1+δg (0 ) ) and ms = ms (n, p, q, s, [a]0,α , [b]0,β , H (·, Du)L1 (0 ) ). This time assume degβ (Br ) holds. Then, by (5.2) we obtain " " " "ˆ ˆ " " " " q s−p ˜ − b(x)|Dv|s−1 |Dϕ| dx " − ∂ H0 (Dv) · Dϕ dx " ≤ sE " " " " Br Br

C. De Filippis, J. Oh / J. Differential Equations 267 (2019) 1631–1670

+ qE

q−p

1657

ˆ − (a(x) − ai (Br )) |Dv|q−1 |Dϕ| dx Br

=: (I)degβ + (II)degβ . As in (5.7), we estimate ⎛ (I)degβ ≤ cDϕL∞ (Br ) E

s−p s

r

β−γb s

⎞ s−1 s ˆ ⎜ ⎟ −p s ⎝ − E b(x)|Du| ⎠ Br

≤ cDϕL∞ (Br ) H (·, Du)

s−p sp L1+δg (0 )

1

rs

  n(s−p) β−γb − p(1+δ g)

≤ cr

σb s

DϕL∞ (Br ) ,

(5.14)

with c1 = c1 (n, p, s, [b]0,β , H (·, Du)L1+δg (0 ) ), and ⎛ (II)degβ ≤cDϕL∞ (Br ) E

q−p q

α

rq

+ γa (q−1) q

⎞ ˆ q−1 ⎜ ⎟ q−p α−γa q r ) |Dv|q−1 dx ⎠ ⎝ − (E Br

≤cDϕL∞ (Br ) r

≤cr

γa (q−1) 1 +q q



γa (q−1) q

n(q−p) α− p(1+δ g)

1

rq



n(q−p) α− p(1+δ g)



⎞ q−1 q ˆ ⎟ ⎜ −p q ⎝ − E ai (Br )|Du| dx ⎠ ⎛

B2r



DϕL∞ (Br ) ,

(5.15)

 where c2 = c2 (n, p, q, [a]0,α , H (·, Du)L1+δg (0 ) ). Let σ˜ q := 12 min σsb , γa (q−1) + q1 α − q  n(q−p) > 0 and fix a threshold radius R˜ ∗3 such that max{c1 , c2 }(R˜ ∗3 )σ˜ q ≤ 12 and assume that p(1+δg ) 0 < r ≤ min{R˜ ∗1 , R˜ ∗2 , R˜ ∗3 , 1}. In correspondence of such a choice, by (5.10) and (5.11) we can conclude that " " " "ˆ " " " " q (5.16) " − ∂ H˜ 0 (Dv) · Dϕ dx " ≤r σ˜ q DϕL∞ (Br ) , " " " " Br so the assumptions of Lemma 7 are satisfied and there exists a H˜ 0 -harmonic map h˜ q satisfying q

q

1,H q in particular (4.16). Clearly, if hq := E h˜ q , then hq is H0 -harmonic, hq − u ∈ W0 0 (Br ) and, by (4.16), ˆ ˆ q 2 mq − V0 (Du, Dhq ) dx ≤ cr − H (x, Du) dx, (5.17) Br

B2r

where c = c(n, p, q, s, [a]0,α , [b]0,β ) and mq = mq (n, p, q, s, [a]0,α , [b]0,β ). Finally, suppose ndeg(Br ) holds. Then, by (5.2) we obtain

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" " " "ˆ ˆ " " " " " − ∂ H˜ 0 (Dv) · Dϕ dx " ≤ sE s−p − (b(x) − bi (Br )) |Dv|s−1 |Dϕ| dx " " " " Br Br ˆ q−p + qE − (a(x) − ai (Br )) |Dv|q−1 |Dϕ| dx Br

=: (I)ndeg + (II)ndeg . As in (5.11) we estimate ⎞ ˆ ⎟ ⎜ s−p β−γb s−1 r ) s |Dv|s−1 dx ⎠ ⎝ − (E ⎛ (I)ndeg ≤cDϕL∞ (Br ) E

s−p s

r

β γb (s−1) s+ s

Br

≤cDϕL∞ (Br ) r

≤cr

γb (s−1) 1 +s s



γb (s−1) s

n(s−p) β− p(1+δ g)

1

rs



n(s−p) β− p(1+δ g)



⎞ s−1 s ˆ ⎟ ⎜ −p s ⎝ − E bi (Br )|Du| dx ⎠ ⎛

B2r



DϕL∞ (Br ) ,

(5.18)

with c1 = c1 (n, p, s, [b]0,β , β, H (·, Du)L1+δg (0 ) ), and, keeping (5.15) in mind, ⎞ ˆ q−1 ⎟ ⎜ q−p α−γa q r ) |Dv|q−1 dx ⎠ ⎝ − (E ⎛ (II)ndeg ≤cDϕL∞ (Br ) E

q−p q

α

rq

+ γa (q−1) q

Br

⎞ q−1

⎛

≤cDϕL∞ (Br ) r

≤cr

γa (q−1) 1 +q q

γa (q−1) q

r

  n(q−p) α− p(1+δ g)

  ˆ n(q−p) 1 q α− p(1+δg ) ⎜

⎝− E

q

−p

⎟ ai (Br )|Du| dx ⎠ q

B2r

DϕL∞ (Br ) ,

(5.19)

where c2 = c2 (n, p, q, [a]0,α , α, H (·, Du)L1+δg (0 ) ). Let σ˜ 0 :=

     1 γa (q − 1) 1 n(q − p) γb (s − 1) 1 n(s − p) min + α− , + β− > 0, 2 q q p(1 + δg ) s s p(1 + δg )

fix another threshold radius R˜ ∗4 such that max{c1 , c2 }(R˜ ∗4 )σ˜ 0 ≤ 12 and assume that 0 < r ≤ min{R˜ ∗1 , R˜ ∗2 , R˜ ∗3 , R˜ ∗4 , 1}. In correspondence of such a choice, by (5.18) and (5.19) we can conclude that " " " "ˆ " " " " (5.20) " − ∂ H˜ 0 (Dv) · Dϕ dx " ≤r σ˜ 0 DϕL∞ (Br ) , " " " " Br

C. De Filippis, J. Oh / J. Differential Equations 267 (2019) 1631–1670

1659

so the assumptions of Lemma 7 are satisfied and there exists a H˜ 0 -harmonic map h˜ 0 satisfying 1,H in particular (4.16). Clearly, if h0 := E h˜ 0 , then h0 is H0 -harmonic, h0 − u ∈ W0 0 (Br ) and, by (4.16), ˆ ˆ 2 m0 − V0 (Du, Dh0 ) dx ≤ cr − H (x, Du) dx, Br

(5.21)

B2r

where c = c(n, p, q, s, [a]0,α , [b]0,β , H (·, Du)L1+δg (0 ) ) and m0 = m0 (n, p, q, s, [a]0,α , [b]0,β , H (·, Du)L1 (0 ) ). Summarizing we got ˆ ˆ p 2 mp deg(Br ) ⇒ − V0 (Du, Dhp ) dx ≤ cr − H (x, Du) dx Br

B2r

ˆ ˆ degα (Br ) ⇒ − V0s (Du, Dhs )2 dx ≤ cr ms − H (x, Du) dx Br

B2r

ˆ ˆ q degβ (Br ) ⇒ − V0 (Du, Dhq )2 dx ≤ cr mq − H (x, Du) dx Br

B2r

Br

B2r

ˆ ˆ 2 m0 ndeg(Br ) ⇒ − V0 (Du, Dh0 ) dx ≤ cr − H (x, Du) dx

where the above holds for 0 < r ≤ R˜ ∗ := min{R˜ ∗1 , R˜ ∗2 , R˜ ∗3 , R˜ ∗4 , 1}, and all the quantities involved are as described before. Finally, for the sake of clarity, we let m := min{mp , mq , ms , m0 }. Now ! take a ball Br with 0 < r ≤ 12 R˜ ∗ such that B2r  0  . Fix τp ∈ 0, 18 and assume deg(Br ) and deg(Bτp r ). Notice that, by virtue of deg(Bτp r ), there holds ai (B2τp r ) ≤ 8[a]0,α (τp r)α−γa and bi (B2τp r ) ≤ 8[b]0,β (τp r)β−γb .

(5.22) p

We fix ϑ ∈ (0, n) and we estimate, by (4.3), (5.22), (3.1), Proposition 1 with ϕ = H0 , (2.8) and (5.9), ˆ B2τp r



ˆ H (x, Du) dx ≤c

p

H0

u − (u)τp r 4τp r



ˆ dx ≤ c

B4τp r

≤c

⎧ ⎪ ⎨ ˆ ⎪ ⎩

B4τp r

≤c

⎧ ⎪ ⎨ ˆ ⎪ ⎩

B4τp r

p

H0 (Du) dx

B4τp r

p

ˆ

V0 (Du, Dhp )2 dx + B4τp r

|Vp (Dhp )|2 dx

⎫ ⎪ ⎬ ⎪ ⎭

⎫ ⎪ ⎬ p p V0 (Du, Dhp )2 dx + |B4τp r | sup H0 (Dhp ) ⎪ B4τp r ⎭

1660

C. De Filippis, J. Oh / J. Differential Equations 267 (2019) 1631–1670

≤c

⎧ ⎪ ⎨ˆ ⎪ ⎩

ˆ

p

p

V0 (Du, Dhp )2 dx + τpn

Br

⎫ ⎪ ⎬

H0 (Dhp ) dx Br

⎪ ⎭

 ˆ H (x, Du) dx, ≤τpn−ϑ cr m τ ϑ−n + cτpϑ

(5.23)

B2r

where c = c(data(0 ), ϑ). For the ease of exposition we set  := 2r and adjusting the constants in (5.23) we get ˆ

 ˆ H (x, Du) dx ≤ τpn−ϑ cm τpϑ−n + cτpϑ H (x, Du) dx.

Bτp 

B

Selecting τp in such a way that cτpϑ ≤ 12 and a threshold radius R∗1 ∈ (0, R˜ ∗ ] such that cR m τpϑ−n ≤ 12 , we can conclude that, for all  ∈ (0, R∗1 ) and all ϑ ∈ (0, n), ˆ

ˆ H (x, Du) dx

Bτp 

≤ τpn−ϑ

(5.24)

H (x, Du) dx. B

! Now fix τs ∈ 0, 18 , assume degα (Br ) and that ai (Bτs r ) ≤ 4[a]0,α (τs r)α−γa , where r < 12 R∗1 . In this situation we have ai (B2τs r ) ≤ 8[a]0,α (τs r)α−γa and bi (Br ) > 4[b]0,β r β−γb .

(5.25)

For ϑ ∈ (0, n), by (4.4), (5.25), (3.1), Proposition 1 with ϕ = H0s , (2.8) and (5.13) we obtain ˆ H (x, Du) dx ≤c B2τs r

" " "s ˆ " " " " u − (u)4τs r "p " + bi (B2τ r ) " u − (u)4τs r " dx " s " " " " τs r τs r

B4τs r

ˆ

±c ˆ

" " " u − (u)4τs r "s " " dx bi (Br ) " " τs r

B4τs r

≤c



H0s

u − (u)4τs r τs r



ˆ dx ≤ c

B4τs r

≤c

⎧ ⎪ ⎨ ˆ ⎪ ⎩

B4 τs r

≤c

⎧ ⎪ ⎨ ˆ ⎪ ⎩

B4 τs r

H0s (Du) dx

B4τs r

ˆ V0s (Du, Dhs )2 dx + B4τs r

H0s (Dhs ) dx

⎫ ⎪ ⎬ ⎪ ⎭

⎫ ⎪ ⎬ s 2 s V0 (Du, Dhs ) dx + |B4τs r | sup H0 (Dhs ) ⎪ B4τs r ⎭

C. De Filippis, J. Oh / J. Differential Equations 267 (2019) 1631–1670

≤c

⎧ ⎪ ⎨ˆ

ˆ V0s (Du, Dhs )2 dx + τsn

⎪ ⎩

Br

≤τsn−ϑ cr m τsϑ−n + cτsϑ

!

H0s (Dhs ) dx Br

ˆ

1661

⎫ ⎪ ⎬ ⎪ ⎭ (5.26)

H (x, Du) dx,

B2r

where c = c(data(0 ), ϑ). Again, we name  := 2r thus getting ˆ H (x, Du) dx ≤ τsn−ϑ cm τsϑ−n + cτsϑ

!

Bτs 

ˆ H (x, Du) dx,

B

where, as before, ϑ ∈ (0, n) is arbitrary. Choose τs small enough so that cτsϑ < 12 and a threshold R∗2 , 0 < R∗2 ≤ R∗1 such that c(R∗1 )m τsϑ−n ≤ 12 . Hence, for all  ∈ (0, R∗2 ] and all ϑ ∈ (0, n) we get ˆ

ˆ H (x, Du) dx

≤ τsn−ϑ

Bτs 

(5.27)

H (x, Du) dx. B

! Consider τq ∈ 0, 18 , assume degβ (Br ) and that bi (Bτq r ) ≤ 4[b]0,β (τq r)β−γb , where r < 12 R∗2 . Now, ai (Br ) > 4[a]0,α (2r)α−γa and bi (B2τq r ) ≤ 8[b]0,β (τq r)β−γb

(5.28) q

holds true. For ϑ ∈ (0, n), by (3.15), (5.28), (4.5), (3.1), Proposition 1 with ϕ = H0 and (5.17) we obtain ˆ B2τq r

" " " ˆ " " u − (u)4τq r "q " u − (u)4τq r "p " " " dx " H (x, Du) dx ≤c " + ai (B2τq r ) " " " τq r τq r B4τq r

ˆ

±c

" " " u − (u)4τq r "q " dx ai (Br ) "" " τq r

B4τq r

ˆ ≤c



q H0

u − (u)4τq r τq r



ˆ

B4τq r

≤c

⎧ ⎪ ⎨ˆ ⎪ ⎩

B4τq r

≤c

⎧ ⎪ ⎨ˆ ⎪ ⎩

B4τq r

q

dx ≤ c

H0 (Du) dx

B4τs t

q

ˆ

V0 (Du, Dhq )2 dx + B4τq r

q

H0 (Dhq ) dx

⎫ ⎪ ⎬ ⎪ ⎭

⎫ ⎪ ⎬ q q V0 (Du, Dhq )2 dx + |B4τq r | sup H0 (Dhq ) ⎪ B4τq r ⎭

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C. De Filippis, J. Oh / J. Differential Equations 267 (2019) 1631–1670

≤c

⎧ ⎪ ⎨ˆ ⎪ ⎩

Br

≤τqn−ϑ

ˆ

q

V0 (Du, Dh0 )2 dx + τqn 

cr m τqϑ−n

+ cτqϑ

⎫ ⎪ ⎬

q

H0 (Dhq ) dx Br

ˆ

⎪ ⎭ (5.29)

H (x, Du) dx, B2r

where c = c(data(0 ), ϑ). Again, we set  = 2r thus obtaining ˆ H (x, Du) dx

≤ τqn−ϑ



cm τqϑ−n

+ cτqϑ

Bτq 

ˆ H (x, Du) dx, B

where, as before, ϑ ∈ (0, n) is arbitrary. Take τq sufficiently small so that cτqϑ < 12 and a threshold R∗3 , 0 < R∗3 ≤ R∗2 such that c(R∗3 )m τqϑ−n ≤ 12 . Hence, for all  ∈ (0, R∗3 ] and all ϑ ∈ (0, n) we get ˆ

ˆ H (x, Du) dx ≤ τqn−ϑ

Bτq 

(5.30)

H (x, Du) dx. B

! Finally, select τ0 ∈ 0, 18 , assume ndeg(Br ), where r ≤ 12 R∗3 . In this scenario we observe that ai (Br ) > 4[a]0,α (r)α−γa and bi (Br ) > 4[b]0,β (r)β−γb .

(5.31)

For ϑ ∈ (0, n), by (4.6), (5.31), (3.1), Proposition 1 with ϕ = H0 , (2.8) and (5.21) we obtain ˆ B2τ0 r

" " " ˆ " " u − (u)4τ0 r "q " u − (u)4τ0 r "p " " " " H (x, Du) dx ≤c " + ai (B2τ0 r ) " " " τ0 r τ0 r B4τ0 r

" " " " ˆ " u − (u)4τ0 r "s " u − (u)4τ0 r "q " " " " + bi (B2τ0 r ) " ai (Br ) " " dx ± c " τ0 r τ0 r B4τ0 r

" " " u − (u)4τ0 r "s " dx + bi (Br ) "" " τ0 r   ˆ ˆ u − (u)4τ0 r H0 H0 (Du) dx dx ≤ c ≤c 4τ0 r B4τ0 r

≤c

⎧ ⎪ ⎨ˆ ⎪ ⎩

B4τ0 r

≤c

⎧ ⎪ ⎨ˆ ⎪ ⎩

B4τ0 r

B4τ0 s

ˆ V0 (Du, Dh0 )2 dx + B4τ0 r

H0s (Dh0 ) dx

⎫ ⎪ ⎬ ⎪ ⎭

⎫ ⎪ ⎬ V0 (Du, Dh0 )2 dx + |B4τ0 r | sup H0 (Dh0 ) ⎪ B4τ0 r ⎭

C. De Filippis, J. Oh / J. Differential Equations 267 (2019) 1631–1670

≤c

⎧ ⎪ ⎨ˆ ⎪ ⎩

ˆ V0 (Du, Dh0 )2 dx + τ0n

Br

H0s (Dh0 ) dx Br

1663

⎫ ⎪ ⎬ ⎪ ⎭

 ˆ H (x, Du) dx, ≤τ0n−ϑ cr m τ0ϑ−n + cτ0ϑ

(5.32)

B2r

where c = c(data(0 ), ϑ). Again, we set  = 2r thus obtaining ˆ

 ˆ H (x, Du) dx ≤ τ0n−ϑ cm τ0ϑ−n + τ0ϑ H (x, Du) dx,

Bτ0 

B

where, as before, ϑ ∈ (0, n) is arbitrary. Take τ0 sufficiently small so that cτ0ϑ < 12 and a threshold R∗4 , 0 < R∗4 ≤ R∗3 such that c(R∗4 )m τ0ϑ−n ≤ 12 . Hence, for all  ∈ (0, R∗4 ] and all ϑ ∈ (0, n) we get ˆ

ˆ H (x, Du) dx ≤ τ0n−ϑ

Bτ0 

(5.33)

H (x, Du) dx. B

Step 2: doble nested exit time and iteration. Now we are in position to develop the announced double nested exit time argument, which will connect estimates (5.24), (5.27), (5.30) and (5.33). Take B   with  ∈ (0, R∗ ], where R∗ = mini∈{1,2,3,4} {R∗i }. For κ ∈ N ∪ {0}, we consider condition deg(Bτpκ+1  ) and define the exit time index

 tp = min κ ∈ N : deg(Bτpκ+1  ) fails . For any κ ∈ {1, · · · , tp } we apply repeatedly (5.24) to obtain ˆ

ˆ H (x, Du) dx ≤ τpκ(n−ϑ)

Bτpκ 

(5.34)

H (x, Du) dx. B

The failure of deg(Bτpκ+1  ) at κ = tp , opens three different scenarios: either degα (B

tp +1

τp

or degβ (B

tp +1

τp



) or directly ndeg(B

tp +1

τp





)

) is in force. Since the last condition is stable for

shrinking balls, and the first two are described by similar procedures, we shall focus on the occurrence of degα (B tp +1 ). Let us introduce a second exit time index τp







ts = min κ ∈ N : degα (B

tp +1

τsκ+1 τp



) fails .

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C. De Filippis, J. Oh / J. Differential Equations 267 (2019) 1631–1670

Iterating (5.27) with B replaced by B

tp +1

τp



, we obtain

ˆ

ˆ H (x, Du) dx ≤ τsκ(n−ϑ)

B

B

tp +1 τsκ τp 

If degα (B

tp +1

τsκ+1 τp



(5.35)

H (x, Du) dx. tp +1 τp 

) fails at κ = ts , the only chance we have is to look at ndeg(B

Condition ndeg is stable, so we can iterate (5.33) for κ ∈ N, thus getting ˆ



).

ˆ

κ(n−ϑ)

H (x, Du) dx ≤ τ0 B

tp +1

τsts +1 τp

H (x, Du) dx.

(5.36)

B

t +1 tp +1 τ0κ τss τp 

t +1 tp +1 τs s τp 

Now we only need to fillet estimates (5.33)-(5.36). For 0 < ς <  ≤ R∗ we consider the following five cases. t +1 ¯  ≤ ς < τ κ¯ . We Case (i):  > ς ≥ τpp . Then there is κ¯ ∈ {0, · · · , tp } such that τpκ+1 p obtain from (5.34) that, ˆ ˆ H (x, Du) dx ≤ H (x, Du) dx Bς

Bτ κ¯  p

¯ ≤ τpκ(n−ϑ)

ˆ H (x, Du) dx B

¯ ≤ τp(κ+1)(n−ϑ) τpϑ−n

ˆ H (x, Du) dx ≤ c

 n−ϑ ˆ ς H (x, Du) dx, (5.37) 

B

B

where c = c(data(0 ), ϑ). t +1 t +1 Case (ii): τpp  > ς ≥ τs τpp . We see that, by (5.37), ˆ

ˆ H (x, Du) dx ≤

Bς

H (x, Du) dx

B

tp +1 τp 

(t +1)(n−ϑ) ≤ cτp p

ˆ H (x, Du) dx B

t +1 n−ϑ ϑ−n ) τs

ˆ

= c(τs τpp

H (x, Du) dx ≤ c B

 n−ϑˆ ς H (x, Du) dx, (5.38)  B

with c = c(data(0 ), ϑ). t +1 t +1 t +1 Case (iii): τs τpp  > ς ≥ τsts +1 τpp . So there is κ¯ ∈ {1, · · · , ts } so that τsκ¯ τpp  > ς ≥ t +1

¯ τ p τsκ+1 p

. We have, by (5.35) and (5.37),

C. De Filippis, J. Oh / J. Differential Equations 267 (2019) 1631–1670

ˆ

1665

ˆ H (x, Du) dx ≤

Bς

H (x, Du) dx

B

tp +1 τsκ¯ τp 

ˆ

¯ ≤τsκ(n−ϑ)

H (x, Du) dx

B

tp +1 τp 

(t +1)(n−ϑ)

¯ τsϑ−n τp p ≤τs(κ+1)(n−ϑ)

ˆ H (x, Du) dx B

 n−ϑ ˆ ς H (x, Du) dx, ≤c 

(5.39)

B

where c = c(data(0 ), ϑ). t +1 t +1 Case (iv): τsts +1 τpp  > ς ≥ τsts +1 τpp τ0 . By (5.39) we obtain ˆ

ˆ H (x, Du) dx ≤

Bς

H (x, Du) dx

B

t +1 tp +1 τs s τp 

t +1 n−ϑ

≤c(τsts +1 τpp

ˆ

)

H (x, Du) dx B

t +1 ≤cτ0ϑ−n (τ0 τsts +1 τpp )n−ϑ

ˆ

 n−ϑ ˆ ς H (x, Du) dx ≤ c H (x, Du) dx, 

B

B

(5.40) with c = c(data(0 ), ϑ). t +1 t +1 ¯ τsts +1 τpp ≤ Case (v): τsts +1 τpp τ0  > ς > 0. This condition renders a κ¯ ∈ N such that τ0κ+1 t +1

ς < τ0κ¯ τsts +1 τpp

. We then estimate, using (5.36) and (5.40),

ˆ

ˆ H (x, Du) dx ≤

Bς

H (x, Du) dx

B

t +1 tp +1 τ0κ¯ τss τp 

κ(n−ϑ) ¯

ˆ

≤ τ0

H (x, Du) dx

B

t +1 tp +1 τs s τp 

t +1 κ(n−ϑ) ¯ (τsts +1 τpp )(n−ϑ) ≤ cτ0

ˆ H (x, Du) dx B

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C. De Filippis, J. Oh / J. Differential Equations 267 (2019) 1631–1670

≤ τ0ϑ−n

 n−ϑ ˆ  n−ϑ ˆ ς ς H (x, Du) dx = c H (x, Du) dx,   B

(5.41)

B

where c = c(data(0 ), ϑ). As mentioned before, the procedure is the same if, after deg occurs degβ instead of degα and it is actually easier if, from deg we jump directly to ndeg. All in all we can conclude that, for all 0 < ς <  ≤ R∗ and all ϑ ∈ (0, n) there holds ˆ H (x, Du) dx ≤ c

 n−ϑ ˆ ς H (x, Du) dx, 

Bς

(5.42)

B

with c = c(data(0 ), ϑ). Now, if 0 < R∗ ≤ ς <  ≤ 1, we get  n−ϑ  n−ϑ ˆ ς  H (x, Du) dx ≤ H (x, Du) dx  ς

ˆ Bς

B

 n−ϑ  n−ϑ ˆ  n−ϑ ˆ ς ς  ≤ H (x, Du) dx ≤ c H (x, Du) dx,  R∗  B

B

(5.43) where c = c(data(0 ), ϑ), by recalling the dependencies of R∗ . Finally, if 0 < ς < R∗ ≤  ≤ 1, by (5.42) and (5.43) we have 

ˆ H (x, Du) dx ≤c Bς

 ≤c

ς R∗ ς R∗

n−ϑ ˆ H (x, Du) dx BR∗

n−ϑ 

R∗ 

n−ϑ ˆ H (x, Du) dx = c

 n−ϑ ˆ ς H (x, Du) dx, 

B

B

(5.44) for c = c(data(0 ), ϑ). Collecting estimates (5.42)-(5.44) we conclude that, for all 0 < ς <  ≤ 1 there holds ˆ H (x, Du) dx ≤ c Bς

with c = c(data(0 ), ϑ).

 n−ϑ ˆ ς H (x, Du) dx,  B

(5.45)

C. De Filippis, J. Oh / J. Differential Equations 267 (2019) 1631–1670

1667

6. Gradient continuity From (5.45) and a standard covering argument, we can conclude that for every open subset 0   and κ > 0 there exists a constant c = c(data(0 ), κ) such that ˆ − H (x, Du) dx ≤ cr −κ (6.1) Br

holds for every ball Br  0  , r ≤ 1. Now, if h is any of the harmonic maps given by Lemma 7 and H˜ is one of the Young functions listed in (2.3) with a0 = ai (Br ) or b0 = bi (Br ), then, the theory in [30] applies rendering ˆ   pν˜ ˆ   pν˜ ˆ − H˜ (Dh − (Dh)B ) dx ≤ c − H˜ (Dh) dx ≤ c − H (x, Du) dx, r r B

Br

(6.2)

Br

where c and ν˜ depend at the most from n, p, q, s. Moreover, for Br  0 with 0 < r ≤ R∗ , where R∗ is the threshold radius introduced in the previous section, we obtain from Lemma 7 and (6.1) that ˆ ˆ 2 m # − V(Du, Dh) dx ≤ cr − H (x, Du) dx ≤ cr m−κ = cr κ0 , Br

B2r

# is the corresponding auxiliary function defined in (2.9) and the conby fixing κ := m2 , where V stant c depends on data(0 ). Arguing exactly as in [14, Proposition 3.3], we get ˆ − H˜ (Du − Dh) dx ≤ cr κ1 (6.3) Br

for some positive exponent κ1 = κ1 (n, p, q, s, [a]0,α , [b]0,β , H (·, Du)L1 (0 ) ). In this case, c = c(data(0 )). Now, for 0 <  < r ≤ R∗ , by (6.3), the minimality of h, (6.1) and (6.2) we see that ⎧ ⎫ ⎪ ⎪ ˆ ˆ ⎨ˆ ⎬ − |Du − (Du) |p dx ≤c − |Dh − (Dh) |p dx + − |Du − Dh|p dx ⎪ ⎪ ⎩ ⎭ B

B

B

⎧ ⎫ ⎪ ⎪  n ˆ ⎨ˆ ⎬ r ≤c − H˜ (Dh − (Dh) ) dx + − H˜ (Du − Dh) dx ⎪ ⎪  ⎩ ⎭ B

Br

⎫ ⎧ ⎪ ⎪  n ˆ ⎬ ⎨  pν˜ ˆ r − H (x, Du) dx + r m − H (x, Du) dx ≤c ⎪ ⎪  ⎭ ⎩ r Br

  ≤c pν˜ r −pν˜ −κ + −n r n+κ1 ,

B2r

(6.4)

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C. De Filippis, J. Oh / J. Differential Equations 267 (2019) 1631–1670

with c = c(data(0 ), κ). Now, first notice that there is no loss of generality in supposing p ν˜ ≤ κ1 1. Setting  = r 1+ 4n and κ := κ18npν˜ in (6.4), we easily obtain ˆ κ1 p ν˜ − |Du − (Du) |p dx ≤ c 16n ,

(6.5)

B

for all  ∈ (0, R∗ ), with c = c(data(0 )). Now, by the integral characterization of Hölder con0,ν κ1 ν˜ (, Rn ) for ν = 16n . The tinuity due to Campanato and Meyers we can conclude that Du ∈ Cloc full proof of Theorem 1 is still not complete, since ν depends on data(0 ), while we announced that the Hölder continuity exponent of Du depends only on data. So we will retain that, after n a covering argument, Du ∈ L∞ loc (, R ), therefore the non-uniform ellipticity of (1.6) becomes immaterial. Now, for Br  0  , no matter what degeneracy condition holds, we compare u to h ∈ W 1,H 0 (Br ) solution to the Dirichlet problem ˆ 1,H (·) u + W0 (Br ) w → min h0 (Dw) dx, (6.6) Br

where H0 (z) := |z|p + ai (Br )|z|q + bi (Br )|z|s . Notice that, for a functional like the one in (6.6), the Bounded Slope Condition holds, see [6], so there exists c = c(n, p, q, s, DuL∞ (Br ) ) such that DhL∞ (Br ) ≤ c.

(6.7)

By strict convexity we obtain ˆ ˆ − V0 (Du, Dh)2 dx ≤ c − H0 (Du) − H0 (Dh) dx Br

Br

⎧ ⎪ ˆ ⎨ˆ =c − H0 (Du) − H (x, Du) dx + − H (x, Du) − H (x, Dh) dx ⎪ ⎩ Br Br ⎫ ⎪ ˆ ⎬ + − H (x, Dh) − H0 (Dh) dx ≤ cr γ , ⎪ ⎭

(6.8)

Br

with γ := min{α, β} and c = c(p, q, s, [a]0,α , [b]0,β , aL∞ () , bL∞ () , DuL∞ (0 ) ). We got this last estimate by using (1.8), the boundedness of DuL∞ and (6.7). Now we jump loc () back to (6.4), thus getting ⎧ ⎫ ⎪ ⎪ ˆ ⎨ˆ ⎬   pν˜ ˆ − |Du − (Du) |p dx ≤c − H (x, Du − Dh) dx + − H (x, Du) dx ⎪ ⎪ r ⎩ ⎭ B

B

  ≤c −n r n+γ + pν˜ r −pν˜ ,

Br

(6.9)

C. De Filippis, J. Oh / J. Differential Equations 267 (2019) 1631–1670

1669

with c = c(data(0 ), DuL∞ (0 ) ). Equalizing in (6.9) as we did to get (6.4), we have ˆ − |Du − (Du) |p dx ≤ cνp , B

with ν =

γ ν˜ n+pν˜ .

This means, by the integral characterization of Hölder continuity due to Cam-

0,ν panato and Mayers, that Du ∈ Cloc (), and, recalling that ν˜ = ν(n, ˜ p, q, s), we see that now ν = ν(data). This concludes the proof.

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