Higher integrability for variational integrals with non-standard growth

Abstract

We consider autonomous integral functionals of the form

$$\begin{aligned} {\mathcal {F}}[u]:=\int _\varOmega f(D u)\,dx \quad \text{ where } u:\varOmega \rightarrow {\mathbb {R}}^N, N\ge 1, \end{aligned}$$

where the convex integrand f satisfies controlled (p, q)-growth conditions. We establish higher gradient integrability and partial regularity for minimizers of ${\mathcal {F}}$ assuming $\frac{q}{p}<1+\frac{2}{n-1}$, $n\ge 3$. This improves earlier results valid under the more restrictive assumption $\frac{q}{p}<1+\frac{2}{n}$.

1 Introduction

In this note, we study regularity properties of local minimizers of integral functionals

$$\begin{aligned} {\mathcal {F}}[u]:=\int _\varOmega f(D u)\,dx, \end{aligned}$$

(1)

where $\varOmega \subset {\mathbb {R}}^n$, $n\ge 3$, is a bounded domain, $u:\varOmega \rightarrow {\mathbb {R}}^N$, $N\ge 1$ and $f:{\mathbb {R}}^{N\times n}\rightarrow {\mathbb {R}}$ is a sufficiently smooth integrand satisfying (p, q)-growth of the form

Assumption 1

There exist $0<\nu \le L<\infty$ such that $f\in C^2({\mathbb {R}}^{N\times n})$ satisfies for all $z,\xi \in {\mathbb {R}}^{N\times n}$

$$\begin{aligned} {\left\{ \begin{array}{ll} \nu |z|^p\le f(z)\le L(1+|z|^q),&{}\\ \nu |z|^{p-2}|\xi |^2\le \langle \partial ^2 f(z)\xi ,\xi \rangle \le L(1+|z|^2)^\frac{q-2}{2}|\xi |^2. \end{array}\right. } \end{aligned}$$

(2)

Regularity properties of local minimizers of (1) in the case $p=q$ are classical, see, e.g., [24]. A systematic regularity theory in the case $p<q$ was initiated by Marcellini in [27, 28], see [31] for an overview (for a more up-to-date overview see the introduction in [30]). In particular, Marcellini [29] proves (among other things):

• (A) If $N=1$, $2\le p<q$ and $\frac{q}{p}<1+\frac{2}{n}$, then every local minimizer $u\in W_\mathrm{loc}^{1,p}(\varOmega )$ of (1) satisfies $u\in W_\mathrm{loc}^{1,\infty }(\varOmega )$.

Local boundedness of the gradient implies that the non-standard growth of f and $\partial ^2f$ in (1) becomes irrelevant and higher regularity (depending on the smoothness of f) follows by standard arguments, see e.g. [27, Chapter 7].

Only very recently, Bella and the author improved in [6] the result (A) in the sense that ’n’ in the assumption on the ratio $\frac{q}{p}$ can be replaced by ’$n-1$’ for $n\ge 3$ (to be precise, [6] considers the non-degenerate version (4) of (2)). The argument in [6] relies on scalar techniques, e.g., Moser-iteration type arguments, and thus cannot be extended to the vectorial case $N>1$.

For the vectorial case $N>1$, Esposito, Leonetti and Mingione showed in [18] that

• (B) If $2\le p<q$ and $\frac{q}{p}<1+\frac{2}{n}$, then every local minimizer $u\in W_\mathrm{loc}^{1,p}(\varOmega ,{\mathbb {R}}^N)$ of (1) satisfies $u\in W_\mathrm{loc}^{1,q}(\varOmega ,{\mathbb {R}}^N)$.

To the best of our knowledge, there is no improvement of (B) with respect to the relation between the exponents p, q and the dimension n available in the literature. Here we provide such an improvement for $n\ge 3$.

Before we state the results, we recall a standard notion of local minimizer in the context of functionals with (p, q)-growth

Definition 1

We call $u\in W_\mathrm{loc}^{1,1}(\varOmega )$ a local minimizer of ${\mathcal {F}}$ given in (1) iff

$$\begin{aligned} f(Du)\in L^1_\mathrm{loc}(\varOmega ) \end{aligned}$$

and

$$\begin{aligned} \int _{\mathrm{supp}\,\varphi }f(D u)\,dx\le \int _{\mathrm{supp}\,\varphi }f(D u+D \varphi )\,dx \end{aligned}$$

for any $\varphi \in W^{1,1}(\varOmega ,{\mathbb {R}}^N)$ satisfying $\mathrm{supp}\;\varphi \Subset \varOmega$.

The main result of the present paper is

Theorem 2

Let $\varOmega \subset {\mathbb {R}}^n$, $n\ge 3$, and suppose Assumption 1 is satisfied with $2\le p<q<\infty$ such that

$$\begin{aligned} \frac{q}{p}<1+\frac{2}{n-1}. \end{aligned}$$

(3)

Let $u\in W_\mathrm{loc}^{1,1}(\varOmega ,{\mathbb {R}}^N)$ be a local minimizer of the functional ${\mathcal {F}}$ given in (1). Then, $u\in W_\mathrm{loc}^{1,q}(\varOmega ,{\mathbb {R}}^N)$.

Higher gradient integrability is a first step in the regularity theory for integral functionals with (p, q)-growth, see [7, 11, 19, 20] for further higher integrability results under (p, q)-conditions. Clearly, we cannot expect to improve from $W_\mathrm{loc}^{1,q}$ to $W_\mathrm{loc}^{1,\infty }$ for $N>1$, since this even fails in the classic setting $p=q$, see [34]. Direct consequences of Theorem 2 are higher differentiability and a further improvement in gradient integrability in the form:

• (i) (Higher differentiability). In the situation of Theorem 2 it holds $|\nabla u|^\frac{p-2}{2} \nabla u\in W^{1,2}_\mathrm{loc}(\varOmega )$, see Theorem 5.

• (ii) (Higher integrability). Sobolev inequality and (i) imply $\nabla u\in L^{\kappa p}_\mathrm{loc}(\varOmega ,{\mathbb {R}}^{N\times n})$ with $\kappa =\frac{n}{n-2}$. Note that $\kappa p>q$ provided $\frac{q}{p}<1+\frac{2}{n-2}$.

A further, on first glance less direct, consequence of Theorem 2 is partial regularity of minimizers of (1), see, e.g.,  [1, 7, 10, 32], for partial regularity results under (p, q)-conditons. For this, we slightly strengthen the assumptions on the integrand and suppose

Assumption 3

There exist $0<\nu \le L<\infty$ such that $f\in C^2({\mathbb {R}}^{N\times n})$ satisfies for all $z,\xi \in {\mathbb {R}}^{N\times n}$

$$\begin{aligned} {\left\{ \begin{array}{ll} \nu |z|^p\le f(z)\le L(1+|z|^q),&{}\\ \nu (1+|z|^2)^\frac{p-2}{2}|\xi |^2\le \langle \partial ^2 f(z)\xi ,\xi \rangle \le L(1+|z|^2)^\frac{q-2}{2}|\xi |^2. \end{array}\right. } \end{aligned}$$

(4)

In [7], Bildhauer and Fuchs prove partial regularity under Assumption 3 with $\frac{q}{p}<1+\frac{2}{n}$ ( [7] contains also more general conditions including, e.g., the subquadratic case). Here we show

Theorem 4

Let $\varOmega \subset {\mathbb {R}}^n$, $n\ge 3$, and suppose Assumption 3 is satisfied with $2\le p<q<\infty$ such that (3). Let $u\in W_\mathrm{loc}^{1,1}(\varOmega ,{\mathbb {R}}^N)$ be a local minimizer of the functional ${\mathcal {F}}$ given in (1). Then, there exists an open set $\varOmega _0\subset \varOmega$ with $|\varOmega \setminus \varOmega _0|=0$ such that $\nabla u\in C^{0,\alpha }(\varOmega _0,{\mathbb {R}}^{N\times n})$ for each $0<\alpha <1$.

We do not know if (3) in Theorems 2 and 4 is optimal. Classic counterexamples in the scalar case $N=1$, see, e.g., [23, 28], show that local boundedness of minimizers can fail if $\frac{q}{p}$ is to large depending on the dimension n. In fact, [28, Theorem 6.1] and the recent boundedness result [26] show that $\frac{1}{p}-\frac{1}{q}\le \frac{1}{n-1}$ is the sharp condition ensuring local boundedness in the scalar case $N=1$ (for sharp results under additional structure assumptions, see, e.g., [14, 22]).

For non-autonomous functionals, i.e., $\int _\varOmega f(x,Du)\,dx$, rather precise sufficiently & necessary conditions are established in [20], where the conditions on p, q and n has to be balanced with the (Hölder)-regularity in space of the integrand. However, if the integrand is sufficiently smooth in space, the regularity theory in the non-autonomous case essentially coincides with the autonomous case, see [10]. Currently, regularity theory for non-autonomous integrands with non-standard growth, e.g. p(x)-Laplacian or double phase functionals are a very active field of research, see, e.g., [2, 12, 13, 15,16,17, 25, 33].

Coming back to autonomous integral functionals: In [11] higher gradient integrability is proven assuming so-called ’natural’ growth conditions, i.e., no upper bound assumption on $\partial ^2f$, under the relation $\frac{q}{p}<1+\frac{1}{n-1}$. Moreover, in two dimensions we cannot improve the previous results on higher differentiability and partial regularity of, e.g., [7, 18], see [8] for a full regularity result under Assumption 3 with $n=2$ and $\frac{q}{p}<2$. Finally, we mention the recent paper [3] where optimal Lipschitz-estimates with respect to a right-hand side are proven for functionals with (p, q)-growth.

Let us briefly describe the main idea in the proof of Theorem 2 and from where our improvement compared to earlier results comes from. The main point is to obtain suitable a priori estimates for minimizers that may already be in $W_\mathrm{loc}^{1,q}(\varOmega ,{\mathbb {R}}^N)$. The claim then follows by a known regularization and approximation procedure, see, e.g., [18]. For minimizers $v\in W_\mathrm{loc}^{1,q}(\varOmega ,{\mathbb {R}}^N)$ a Caccioppoli-type inequality

$$\begin{aligned} \int \eta ^2 |D(|Dv|^\frac{p-2}{2}Dv)|^2\lesssim \int |\nabla \eta |^2(1+|Dv|^q) \end{aligned}$$

(5)

is valid for all sufficiently smooth cut-off functions $\eta$, see Lemma 1. Very formally, the Caccioppoli inequality (5) can be combined with Sobolev inequality and a simple interpolation inequality to obtain

$$\begin{aligned} \Vert Dv\Vert _{L^{\kappa p}}^p\lesssim \Vert D(|Dv|^\frac{p-2}{2}Dv)\Vert _{L^2}^2\lesssim \Vert Dv\Vert _{L^q}^q\lesssim \Vert Dv\Vert _{L^{\kappa p}}^{q\theta }\Vert Dv\Vert _{L^{p}}^{(1-\theta )q}, \end{aligned}$$

where $\theta =\frac{\frac{1}{p}-\frac{1}{q}}{\frac{1}{p}-\frac{1}{\kappa p}}\in (0,1)$ and $\kappa =\frac{n}{n-2}$. The $\Vert Dv\Vert _{L^{\kappa p}}$-factor on the right-hand side can be absorbed provided we have $\frac{q\theta }{p}<1$, but this is precisely the ’old’ (p, q)-condition $\frac{q}{p}<1+\frac{2}{n}$, this type of argument was previously rigorously implemented in, e.g., [7, 19]. Our improvement comes from choosing a cut-of function $\eta$ in (5) that is optimized with respect to v, which enables us to use Sobolev inequality on $n-1$-dimensional spheres wich gives the desired improvement, see Sect. 3. This idea has its origin in joint works with Bella [4, 5] on linear non-uniformly elliptic equations.

With Theorem 2 at hand, we can follows the arguments of [7] almost verbatim to prove Theorem 4. In Sect. 4, we sketch (following [7]) a corresponding $\varepsilon$-regularity result from which Theorem 4 follows by standard methods.

2 Preliminary results

In this section, we gather some known facts. We begin with a well-known higher differentiability result for minimizers of (1) under the assumption that $u\in W^{1,q}_\mathrm{loc}(\varOmega ,{\mathbb {R}}^N)$:

Lemma 1

Let $\varOmega \subset {\mathbb {R}}^n$, $n\ge 2$, and suppose Assumption 1 is satisfied with $2\le p<q<\infty$. Let $v\in W_\mathrm{loc}^{1,q}(\varOmega ,{\mathbb {R}}^N)$ be a local minimizer of the functional ${\mathcal {F}}$ given in (1). Then, $|Dv|^\frac{p-2}{2}Dv\in W_\mathrm{loc}^{1,2}(\varOmega ,{\mathbb {R}}^{N\times n})$ and there exists $c=c(\frac{L}{\nu },n,N,p,q)\in [1,\infty )$ such that for every $Q\in {\mathbb {R}}^{N\times n}$ and every $\eta \in C_c^1(\varOmega )$

$$\begin{aligned} \int _\varOmega \eta ^2|D(|Dv|^\frac{p-2}{2}Dv)|^2\,dx \le c\int _{\varOmega }(1+|Dv|^2)^\frac{q-2}{2} |D v-Q|^2|\nabla \eta |^2\,dx. \end{aligned}$$

(6)

The Lemma 1 is known, see e.g. [7, 18, 28]. Since we did not find a precise reference for estimate (6), we included a prove here following essentially the argument of [18].

Proof of Lemma 1

Without loss of generality, we suppose $\nu =1$ the general case $\nu >0$ follows by replacing f with $f/\nu$ (and thus L with $L/\nu$). Throughout the proof, we write $\lesssim$ if $\le$ holds up to a multiplicative constant depending only on n, N, p and q.

Thanks to the assumption $v\in W^{1,q}_\mathrm{loc}(\varOmega ,{\mathbb {R}}^N)$, the minimizer v satisfies the Euler-Largrange equation

$$\begin{aligned} \int _\varOmega \langle \partial f(D v), D \varphi \rangle \,dx=0\qquad \text{ for } \text{ all } \varphi \in W_0^{1,q}(\varOmega ,{\mathbb {R}}^N) \end{aligned}$$

(7)

(for this we use that the convexity and growth conditions of f imply $|\partial f(z)|\le c(1+|z|^{q-1})$ for some $c=c(L,n,N,q,)<\infty$). Next, we use the difference quotient method, to differentiate the above equation: For $s\in \{1,\dots ,n\}$, we consider the difference quotient operator

$$\begin{aligned} \tau _{s,h}v:=\tfrac{1}{h}(v(\cdot +he_s)-v)\qquad \text{ where } v\in L_\mathrm{loc}^1({\mathbb {R}}^n,{\mathbb {R}}^N). \end{aligned}$$

Fix $\eta \in C_c^1(\varOmega )$. Testing (7) with $\varphi :=\tau _{s,-h}(\eta ^2(\tau _{s,h}(v-\ell _Q)))\in W_0^{1,q}(\varOmega )$, where $\ell _Q(x)=Qx$, we obtain

$$\begin{aligned} (I):=&\int _\varOmega \eta ^2 \langle \tau _{s,h}\partial f(D v),\tau _{s,h}D v\rangle \,dx\\ =&-2\int _\varOmega \eta \langle \tau _{s,h}\partial f(D v), \tau _{s,h}(v-\ell _Q)\otimes \nabla \eta \rangle \,dx=:(II). \end{aligned}$$

Writing $\tau _{s,h}\partial f(D v)=\frac{1}{h} \partial f(D v+th\tau _{s,h}D v)\big |_{t=0}^{t=1}$, the fundamental theorem of calculus yields

$$\begin{aligned}&\int _\varOmega \int _0^1 \eta ^2\langle \partial ^2f(D v+th\tau _{s,h}D v))\tau _{s,h}D v,\tau _{s,h}D v\rangle \,dt\,dx=(I)\nonumber \\ =&(II)=-2\int _\varOmega \int _0^1 \eta \langle \partial ^2f(D v+th\tau _{s,h}D v)\tau _{s,h}D v, (\tau _{s,h}v-Qe_s) \otimes \nabla \eta \rangle \,dt\,dx, \end{aligned}$$

(8)

where we use $\tau _{h,s}\ell _Q=Qe_s$. Youngs inequality yields

$$\begin{aligned} |(II)|\le \tfrac{1}{2}(I)+2(III), \end{aligned}$$

(9)

where

$$\begin{aligned} (III):=\int _\varOmega \int _0^1\langle \partial ^2f(D u+th\tau _{s,h}D u) (\tau _{s,h}v-Qe_s)\otimes \nabla \eta ,(\tau _{s,h}v-Qe_s)\otimes \nabla \eta \rangle \,dt\,dx. \end{aligned}$$

Combining (8), (9) with the assumptions on $\partial ^2f$, see (2), with the elementary estimate

$$\begin{aligned} |\tau _{s,h}(|Dv|^{\frac{p-2}{2}}Dv)|^2\lesssim \int _0^1 |D v+th\tau _{s,h}D v|^\frac{p-2}{2} |\tau _{s,h}D v|^2\,dt\ \end{aligned}$$

for $h>0$ sufficiently small (see e.g. [18, Lemma 3.4]), we obtain

$$\begin{aligned}&\int _\varOmega \eta ^2|\tau _{s,h}(|Dv|^{\frac{p-2}{2}}Dv)|^2\,dx\nonumber \\ \lesssim&\int _\varOmega \int _0^1 \eta ^2|D v+th\tau _{s,h}D v|^\frac{p-2}{2} |\tau _{s,h}D v|^2\,dt\,dx\le (I)\nonumber \\ \le&4(III)\le 4L\int _\varOmega \int _0^1 (1+|Dv+th\tau _{s,h}D v|^{q-2})|\nabla \eta |^2|\tau _{s,h}v-Qe_s|^2\,dt\,dx. \end{aligned}$$

(10)

Estimate (10), the fact $v\in W_\mathrm{loc}^{1,q}(\varOmega )$ and the arbitrariness of $\eta \in C_c^1(\varOmega )$ and $s\in \{1,\dots ,n\}$ yield $|Dv|^{\frac{p-2}{2}}Dv\in W_\mathrm{loc}^{1,2}(\varOmega )$. Sending h to zero in (10), we obtain

$$\begin{aligned} \int _\varOmega \eta ^2|\partial _s(|Dv|^{\frac{p-2}{2}}Dv)|^2\,dx\lesssim L\int _\varOmega (1+|Dv|^{q-2})|\nabla \eta |^2|\partial _sv-Qe_s|^2\,dx \end{aligned}$$

the desired estimate (6) follows by summing over s. $\square$

Next, we state a higher differentiability result under the more restrictive Assumption 3 which will be used in the proof of Theorem 4.

Lemma 2

Let $\varOmega \subset {\mathbb {R}}^n$, $n\ge 2$, and suppose Assumption 3 is satisfied with $2\le p<q<\infty$. Let $v\in W_\mathrm{loc}^{1,q}(\varOmega ,{\mathbb {R}}^N)$ be a local minimizer of the functional ${\mathcal {F}}$ given in (1). Then, $h:=(1+|Dv|^2)^\frac{p}{4}\in W_\mathrm{loc}^{1,2}(\varOmega )$ and there exists $c=c(\frac{L}{\nu },n,N,p,q)\in [1,\infty )$ such that for every $Q\in {\mathbb {R}}^{N\times n}$

$$\begin{aligned} \int _\varOmega \eta ^2|\nabla h|^2\,dx \le c\int _{\varOmega }(1+|Dv|^2)^\frac{q-2}{2} |D v-Q|^2|\nabla \eta |^2\,dx\quad \text{ for } \text{ all } \eta \in C_c^1(\varOmega ). \end{aligned}$$

(11)

A variation of Lemma 2 can be found in [7] and we only sketch the proof.

Proof of Lemma 2

With the same argument as in the proof of Lemma 1 but using (4) instead of (2), we obtain $v\in W^{2,2}_\mathrm{loc}(\varOmega ,{\mathbb {R}}^N)$ and the Caccioppoli inequality

$$\begin{aligned} \int _\varOmega \eta ^2(1+|Dv|^2)^\frac{p-2}{2}|D^2v|^2\,dx \le c\int _{\varOmega }(1+|Dv|^2)^\frac{q-2}{2} |D v-Q|^2|\nabla \eta |^2\,dx \end{aligned}$$

(12)

for all $\eta \in C_c^1(\varOmega )$, where $c=c(\frac{L}{\nu },n,N,p,q)<\infty$. Formally, the chain-rule implies

$$\begin{aligned} |\nabla h|^2\le c(1+|Dv|^2)^\frac{p-2}{2}|D^2v|^2, \end{aligned}$$

(13)

where $c=c(n,p)<\infty$, and the claimed estimate (11) follows from (12) and (13). In general, we are not allowed to use the chain rule, but the above reasoning can be made rigorous: Consider a truncated version $h_m$ of h, where $h_m:=\varTheta _m(|Dv|)$ with

$$\begin{aligned} \varTheta _m(t):={\left\{ \begin{array}{ll}(1+t^2)^\frac{p}{4}&{}\text{ if } 0\le t\le m\\ (1+m^2)^\frac{p}{4}&{}\text{ if } t\ge m\end{array}\right. }. \end{aligned}$$

For $h_m$ we are allowed to use the chain-rule and (12) together with (13) with h replaced by $h_m$ imply (11) with h replaced by $h_m$. The claimed estimate follows by taking the limit $m\rightarrow \infty$, see [7, Proposition 3.2] for details. $\square$

The following technical lemma is contained in [6] (see also [4, proof of Lemma 2.1, Step 1]) and plays a key role in the proof of Theorem 2

Lemma 3

( [6], Lemma 3) Fix $n\ge 2$. For given $0<\rho<\sigma <\infty$ and $v\in L^1(B_{\sigma })$, consider

$$\begin{aligned} J(\rho ,\sigma ,v):=\inf \left\{ \int _{B_{\sigma }}|v||\nabla \eta |^2\,dx \;|\;\eta \in C_0^1(B_{\sigma }),\,\eta \ge 0,\,\eta =1 \text{ in } B_\rho \right\} . \end{aligned}$$

Then for every $\delta \in (0,1]$

$$\begin{aligned} J(\rho ,\sigma ,v)\le (\sigma -\rho )^{-(1+\frac{1}{\delta })} \biggl (\int _{\rho }^\sigma \left( \int _{\partial B_r} |v|\,d{\mathcal {H}}^{n-1}\right) ^\delta \,dr\biggr )^\frac{1}{\delta }. \end{aligned}$$

(14)

For convenience of the reader we include a short proof of Lemma 3

Proof of Lemma 3

Estimate (14) follows directly by minimizing among radial symmetric cut-off functions. Indeed, we obviously have for every $\varepsilon \ge 0$

$$\begin{aligned}&J(\rho ,\sigma ,v)\\ \le&\inf \left\{ \int _{\rho }^\sigma \eta '(r)^2\left( \int _{\partial B_r}|v|\,d{\mathcal {H}}^{n-1}+\varepsilon \right) \,dr \;|\;\eta \in C^1(\rho ,\sigma ),\,\eta (\rho )=1,\,\eta (\sigma )=0\right\} \\ =:&J_{\mathrm{1d},\varepsilon }. \end{aligned}$$

For $\varepsilon >0$, the one-dimensional minimization problem $J_{\mathrm{1d},\varepsilon }$ can be solved explicitly and we obtain

$$\begin{aligned} J_{\mathrm{1d},\varepsilon }=\biggl (\int _{\rho }^\sigma \biggl (\int _{\partial B_r}|v|\,d{\mathcal {H}}^{n-1}+\varepsilon \biggr )^{-1}\,dr\biggr )^{-1}. \end{aligned}$$

(15)

To see (15), we observe that using the assumption $v\in L^1(B_\sigma )$ and a simple approximation argument we can replace $\eta \in C^1(\rho ,\sigma )$ with $\eta \in W^{1,\infty }(\rho ,\sigma )$ in the definition of $J_{\mathrm{1d},\varepsilon }$. Let ${{\widetilde{\eta }}}:[\rho ,\sigma ]\rightarrow [0,\infty )$ be given by

$$\begin{aligned} {{\widetilde{\eta }}}(r):=1-\biggl (\int _\rho ^\sigma b(r)^{-1}\,dr\biggr )^{-1}\int _{\rho }^rb(r)^{-1}\,dr,\quad \text{ where } b(r):=\int _{\partial B_r}|v|+\varepsilon . \end{aligned}$$

Clearly, ${{\widetilde{\eta }}}\in W^{1,\infty }(\rho ,\sigma )$ (since $b\ge \varepsilon >0$), ${{\widetilde{\eta }}}(\rho )=1$, ${{\widetilde{\eta }}}(\sigma )=0$, and thus

$$\begin{aligned} J_{\mathrm{1d},\varepsilon }\le \int _{\rho }^\sigma {{\widetilde{\eta }}}'(r)^2b(r)\,dr=\biggl (\int _{\rho }^\sigma b(r)^{-1}\,dr\biggr )^{-1}. \end{aligned}$$

The reverse inequality follows by Hölder’s inequality. Next, we deduce (14) from (15): For every $s>1$, we obtain by Hölder inequality $\sigma -\rho =\int _\rho ^\sigma (\frac{b}{b})^\frac{s-1}{s}\le \left( \int _\rho ^\sigma b^{s-1}\right) ^\frac{1}{s}\left( \int _\rho ^\sigma \frac{1}{b}\right) ^\frac{s-1}{s}$ with b as above, and by (15) that

$$\begin{aligned} J_{\mathrm{1d},\varepsilon }\le (\sigma -\rho )^{-\frac{s}{s-1}}\biggl (\int _{\rho }^\sigma \left( \int _{\partial B_r}|v|+\varepsilon \right) ^{s-1}\,dr\biggr )^{\frac{1}{s-1}}. \end{aligned}$$

Sending $\varepsilon$ to zero, we obtain (14) with $\delta =s-1>0$. $\square$

3 Higher integrability - Proof of Theorem 2

In this section, we prove the following higher integrability and differentiability result which clearly contains Theorem 2

Theorem 5

Let $\varOmega \subset {\mathbb {R}}^n$, $n\ge 2$, and suppose Assumption 1 is satisfied with $2\le p<q<\infty$ such that $\frac{q}{p}<1+\min \{\frac{2}{n-1},1\}$. Let $u\in W_\mathrm{loc}^{1,1}(\varOmega ,{\mathbb {R}}^N)$ be a local minimizer of the functional ${\mathcal {F}}$ given in (1). Then, $u\in W_\mathrm{loc}^{1,q}(\varOmega ,{\mathbb {R}}^N)$ and $|Du|^\frac{p-2}{2}Du\in W_\mathrm{loc}^{1,2}(\varOmega ,{\mathbb {R}}^{N\times n})$. Moreover, for

$$\begin{aligned} \chi =\frac{n-1}{n-3}\quad \text{ if } n\ge 4\quad \chi \in (\frac{1}{2-\frac{q}{p}},\infty )\quad \text{ if } n=3 \text{ and } \quad \chi :=\infty \quad \text{ if } n=2. \end{aligned}$$

(16)

there exists $c=c(\frac{L}{\nu },n,N,p,q,\chi )\in [1,\infty )$ such that for every $B_{R}(x_0)\Subset \varOmega$

(17)

where

$$\begin{aligned} \alpha :=\frac{1-\frac{q}{\chi p}}{2-\frac{q}{p}-\frac{1}{\chi }}. \end{aligned}$$

(18)

Proof of Theorem 5

Without loss of generality, we suppose $\nu =1$ the general case $\nu >0$ follows by replacing f with $f/\nu$. Throughout the proof, we write $\lesssim$ if $\le$ holds up to a multiplicative constant depending only on L, n, N, p and q.

Following, e.g., [7, 18, 19], we consider the perturbed integral functionals

$$\begin{aligned} {\mathcal {F}}_\lambda (w):=\int _\varOmega f_\lambda (Dw)\,dx,\qquad \text{ where }\quad f_\lambda (z):=f(z)+\lambda |z|^q\quad \text{ with } \lambda \in (0,1). \end{aligned}$$

(19)

We then derive suitable a priori higher differentiability and integrability estimates for local minimizers of ${\mathcal {F}}_\lambda$ that are independent of $\lambda \in (0,1)$. The claim then follows with help of a by now standard double approximation procedure in spirit of [18].

Step 1. One-step improvement.

Let $v\in W^{1,1}_\mathrm{loc}(\varOmega ,{\mathbb {R}}^N)$ be a local minimizer of the functional ${\mathcal {F}}_\lambda$ defined in (19), $B_1\Subset \varOmega$, and let $\chi >1$ be defined in (16). We claim that there exists $c=c(L,n,N,p,q,\chi )\in [1,\infty )$ such that for all $\frac{1}{2}\le \rho <\sigma \le 1$ and every $\lambda \in (0,1]$

$$\begin{aligned}&\int _{B_1}1+f_\lambda (Dv)+\int _{B_\rho }|D(|Dv|^\frac{p-2}{2}Dv)|^2\,\,dx\nonumber \\ \le&\frac{c\biggl (\int _{B_1}1+f_\lambda (Dv)\biggr )^{\frac{\chi }{\chi -1}(1-\frac{q}{\chi p})}}{(\sigma -\rho )^{1+\frac{q}{p}}}\nonumber \\&\times \biggl (\int _{B_1}1+f_\lambda (Dv)+\int _{B_{\sigma }}|D(|Dv|^\frac{p-2}{2}Dv)|^2\,\,dx\biggr )^{\frac{\chi }{\chi -1}(\frac{q}{p}-1)} \end{aligned}$$

(20)

with the understanding $\frac{\infty }{\infty -1}=1$ and

$$\begin{aligned} \int _{B_\rho }|D(|Dv|^\frac{p-2}{2}Dv)|^2\,\,dx \lesssim \frac{1}{(\sigma -\rho )^2}\frac{1}{\lambda }\int _{B_{\sigma }}1+ f_\lambda (Dv)\,dx. \end{aligned}$$

(21)

The growth conditions of $f_\lambda$ and the minimality of v imply $v\in W^{1,q}_\mathrm{loc}(\varOmega ,{\mathbb {R}}^N)$ and thus by Lemma 1

$$\begin{aligned} \int _\varOmega |D(|Dv|^\frac{p-2}{2}Dv)|^2\eta ^2\,\,dx \lesssim \int _{\varOmega }(1+|Dv|^2)^\frac{q-2}{2} |D v|^2|\nabla \eta |^2\,dx \end{aligned}$$

(22)

for all $\eta \in C_c^1(\varOmega )$. Estimate (21) follows directly from (22) for $\eta \in C_c^1(B_\sigma )$ with $0\le \eta \le 1$, $\eta \equiv 1$ on $B_\rho$ and $|\nabla \eta |\le \frac{2}{\sigma -\rho }$, combined with $|z|^q\le \frac{1}{\lambda }f_\lambda (z)$ and $\lambda \in (0,1]$.

Hence, it is left to show (20). For this, we use a technical estimate which follows from Lemma 3 and Hölders inequality: For given $0<\rho<\sigma <\infty$ and $w\in L^q(B_{\sigma })$ it holds

$$\begin{aligned} J(\rho ,\sigma ,|w|^q)\le \frac{\biggl (\int _{B_{\sigma }\setminus B_\rho } |w|^p\biggr )^{\frac{\chi }{\chi -1}(1-\frac{q}{\chi p})}}{(\sigma -\rho )^{1+\frac{q}{p}}} \biggl (\int _{\rho }^{\sigma }\Vert w\Vert _{L^{\chi p}(\partial B_r)}^p\,dr\biggr )^{\frac{\chi }{\chi -1}(\frac{q}{p}-1)}, \end{aligned}$$

(23)

where J is defined as in Lemma 3. We postpone the derivation of (23) to the end of this step.

Combining (22) with $(1+|Dv|^2)^\frac{q-2}{2} |D v|^2\le (1+|Dv|)^q$ and estimate (23) with $w=1+|Dv|$, we obtain

$$\begin{aligned}&\int _{B_\rho }|D(|Dv|^\frac{p-2}{2}Dv)|^2\,\,dx\nonumber \\ \lesssim&\frac{\biggl (\int _{B_{\sigma }\setminus B_\rho }(1+|D v|)^p\,dx\biggr )^{\frac{\chi }{\chi -1}(1-\frac{q}{\chi p})}}{(\sigma -\rho )^{1+\frac{q}{p}}}\biggl (\int _{\rho }^{\sigma }\Vert 1+|Dv|\Vert _{L^{\chi p}(\partial B_r)}^p\,dr\biggr )^{\frac{\chi }{\chi -1}(\frac{q}{p}-1)}. \end{aligned}$$

(24)

Next, we use the Sobolev inequality on spheres to estimate the second factor on the right-hand side in (24): For $n\ge 2$ there exists $c=c(n,N,\chi )\in [1,\infty )$ such that for all $r>0$

$$\begin{aligned}&\Vert D v\Vert _{L^{\chi p}(\partial B_r)}^p\nonumber \\ \le&c r^{(n-1)(\frac{1}{\chi }-1)}\biggl (\int _{\partial B_r}|D v|^p\,d{\mathcal {H}}^{n-1}+r^2\int _{\partial B_r}|D(|Dv|^{\frac{p-2}{2}}Dv)|^2\,d{\mathcal {H}}^{n-1}\biggr ). \end{aligned}$$

(25)

Combining (25) with elementary estimates and assumption $\frac{1}{2}\le \rho <\sigma \le 1$, we obtain

$$\begin{aligned} \int _{\rho }^{\sigma }\Vert 1+|Dv|\Vert _{L^{\chi p}(\partial B_r)}^p\,dr\lesssim&\int _{\rho }^{\sigma }1+\Vert Dv\Vert _{L^{\chi p}(\partial B_r)}^p\,dr\nonumber \\ \lesssim&\int _{\rho }^{\sigma }1+\biggl (\int _{\partial B_r}|D v|^p+|D(|Dv|^{\frac{p-2}{2}}Dv)|^2\,d{\mathcal {H}}^{n-1}\biggr )\,dr\nonumber \\ \lesssim&\int _{B_{\sigma }\setminus B_\rho }1+|D v|^p+|D(|Dv|^{\frac{p-2}{2}}Dv)|^2\,dx. \end{aligned}$$

(26)

Combining (24) and estimate (26), we obtain

$$\begin{aligned}&\int _{B_\rho }|D(|Dv|^\frac{p-2}{2}Dv)|^2\,\,dx\\ \le&\frac{c\biggl (\int _{B_{1}}(1+|D v|)^p\,dx\biggr )^{\frac{\chi }{\chi -1}(1-\frac{q}{\chi p})}}{(\sigma -\rho )^{1+\frac{q}{p}}}\biggl (\int _{B_{\sigma }}1+|D v|^p+|D(|Dv|^{\frac{p-2}{2}}Dv)|^2\,dx\biggr )^{\frac{\chi }{\chi -1}(\frac{q}{p}-1)}, \end{aligned}$$

The claimed estimate (20) now follows since $|z|^p\le f(z)\le f_\lambda (z)$, $\frac{\chi }{\chi -1}(1-\frac{q}{\chi p}+\frac{q}{p}-1)=\frac{q}{p}\ge 1$ and $\int _{B_1}1+f_\lambda (Dv)\,dx\ge |B_1|$.

Finally, we present the computations regarding (23): Lemma 3 yields

$$\begin{aligned} J(\sigma ,\rho ,|w|^q)\le \frac{\biggl (\int _{\rho }^{\sigma }\Vert w\Vert _{L^q(\partial B_r)}^{q\delta }\,dr\biggr )^\frac{1}{\delta }}{(\sigma -\rho )^{1+\frac{1}{\delta }}}\qquad \text{ for } \text{ every } \delta >0. \end{aligned}$$

Using two times the Hölder inequality, we estimate

$$\begin{aligned} \biggl (\int _{\rho }^{\sigma }\Vert w\Vert _{L^q(\partial B_r)}^{q\delta }\,dr\biggr )^\frac{1}{\delta }\le&\biggl (\int _{\rho }^{\sigma }\Vert w\Vert _{L^p(\partial B_r)}^{\theta q\delta }\Vert w\Vert _{L^{\chi p}(\partial B_r)}^{(1-\theta ) q\delta }\,dr\biggr )^\frac{1}{\delta }\quad \text{ where } \frac{\theta }{p}+\frac{1-\theta }{\chi p}=\frac{1}{q}\\ \le&\biggl (\int _{\rho }^{\sigma }\Vert w\Vert _{L^p(\partial B_r)}^{\theta q\delta \frac{s}{s-1}}\,dr\biggr )^\frac{s-1}{s\delta }\biggl (\int _{\rho }^{\sigma }\Vert w\Vert _{L^{\chi p}(\partial B_r)}^{(1-\theta ) q\delta s}\,dr\biggr )^\frac{1}{\delta s}\quad \text{ for } \text{ every } s>1. \end{aligned}$$

Inequality (23) follows with the admissible choice

$$\begin{aligned} \delta =\frac{p}{q}\quad \text{ and }\quad s=\frac{1}{1-\theta }\qquad \biggl (\text{ recall } 1-\theta =\frac{\frac{1}{p}-\frac{1}{q}}{\frac{1}{p}-\frac{1}{\chi p}}\quad \text{ and } p<q\biggr ) \end{aligned}$$

which ensures $\theta q\delta \frac{s}{s-1}=(1-\theta ) q\delta s=p$.

Step 2. Iteration.

We claim that there exists $c=c(L,n,N,p,q,\chi )\in [1,\infty )$ such that

$$\begin{aligned} \int _{B_\frac{1}{2}}|Dv|^p+|D(|Dv|^\frac{p-2}{2}Dv)|^2\,\,dx\le&c\biggl (\int _{B_1}1+f_\lambda (Dv)\,dx\biggr )^\alpha , \end{aligned}$$

(27)

where $\alpha$ is defined in (18). For $k\in {\mathbb {N}}\cup \{0\}$, we set

$$\begin{aligned} \rho _k=\frac{3}{4}- \frac{1}{4^{1+k}}\quad \text{ and }\quad J_k:=\int _{B_1}1+f_\lambda (Dv)+\int _{B_{\rho _k}}|D(|Dv|^\frac{p-2}{2}Dv)|^2\,\,dx. \end{aligned}$$

Estimate (21) and the choice of $\rho _k$ imply for $\lambda \in (0,1]$

$$\begin{aligned} \sup _{k\in {\mathbb {N}}}J_k\le \int _{B_1}1+f_\lambda (Dv)+\int _{B_{\frac{3}{4}}}|D(|Dv|^\frac{p-2}{2}Dv)|^2\,\,dx\lesssim \frac{1}{\lambda }\int _{B_1}1+f_\lambda (Dv)\,dx<\infty .\nonumber \\ \end{aligned}$$

(28)

From (20) we deduce the existence of $c=c(L,n,N,p,q,\chi )\in [1,\infty )$ such that for every $k\in {\mathbb {N}}$

$$\begin{aligned} J_{k-1}\le c 4^{(1+\frac{q}{p}) k}\biggl (\int _{B_1}1+f_\lambda (Dv)\biggr )^{\frac{\chi }{\chi -1}(1-\frac{q}{\chi p})}J_k^{\frac{\chi }{\chi -1}\frac{q-p}{p}}. \end{aligned}$$

(29)

Assumption $\frac{q}{p}<1+\min \{1,\frac{2}{n-1}\}$ and the choice of $\chi$ yield

$$\begin{aligned} {\frac{\chi }{\chi -1}\frac{q-p}{p}}{\mathop {=}\limits ^{(16)}}{\left\{ \begin{array}{ll}\frac{q}{p}-1&{}\text{ if } n=2\\ {\frac{\chi }{\chi -1}\frac{q-p}{p}}&{}\text{ if } n=3\\ \frac{n-1}{2}(\frac{q}{p}-1)&{}\text{ if } n\ge 4\end{array}\right. }<1, \end{aligned}$$

where we use for $n=3$ that $\chi {\mathop {>}\limits ^{(16)}}\frac{1}{2-\frac{q}{p}}>0$ and

$$\begin{aligned} \frac{\chi }{\chi -1}\frac{q-p}{p}<1\quad \Leftrightarrow \quad \frac{q-p}{p}<1-\frac{1}{\chi }\quad \Leftrightarrow \quad \frac{1}{\chi }<2-\frac{q}{p}. \end{aligned}$$

Hence, iterating (29) we obtain (using the uniform bound (28) on $J_k$ and ${\frac{\chi }{\chi -1}\frac{q-p}{p}}<1$)

$$\begin{aligned}&\int _{B_{\frac{1}{2}}}|Dv|^p+|D(|Dv|^\frac{p-2}{2}Dv)|^2\,\,dx\le J_0\lesssim \biggl (\int _{B_1}1+f_\lambda (Dv)\biggr )^{\frac{\chi }{\chi -1}(1-\frac{q}{\chi p})\sum _{k=0}^\infty ({\frac{\chi }{\chi -1}\frac{q-p}{p}})^k}\nonumber \\ \end{aligned}$$

(30)

and the claimed estimate (27) follow from

$$\begin{aligned} \alpha = \frac{\chi }{\chi -1}(1-\frac{q}{\chi p})\sum _{k=0}^\infty ({\frac{\chi }{\chi -1}\frac{q-p}{p}})^k. \end{aligned}$$

Step 3. Conclusion.

We assume $B_1\Subset \varOmega$ and show that there exists $c=c(L,n,N,p,q,\chi )\in [1,\infty )$

$$\begin{aligned} \int _{B_\frac{1}{8}}|D u|^{q}\,dx\le c\left( \int _{B_1}1+f(D u)\,dx\right) ^{\frac{\alpha q}{p}}, \end{aligned}$$

(31)

where $\alpha$ is given as in (18) above. Clearly, standard scaling, translation and covering arguments yield

for all $B_R(x_0)\Subset \varOmega$ and $c=c(L,n,N,p,q,\chi )\in [1,\infty )$. The claimed estimate (17) then follows from Lemma 1.

Following [18], we introduce in addition to $\lambda \in (0,1)$ a second small parameter $\varepsilon >0$ which is related to a suitable regularization of u. For $\varepsilon \in (0,\varepsilon _0)$, where $0<\varepsilon _0\le 1$ is such that $B_{1+\varepsilon _0}\Subset \varOmega$, we set $u_\varepsilon :=u*\varphi _\varepsilon$ with $\varphi _\varepsilon :=\varepsilon ^{-n}\varphi (\frac{\cdot }{\varepsilon })$ and $\varphi$ being a non-negative, radially symmetric mollifier, i.e. it satisfies

$$\begin{aligned} \varphi \ge 0,\quad \mathrm{supp}\; \varphi \subset B_1,\quad \int _{{\mathbb {R}}^n}\varphi (x)\,dx=1,\quad \varphi (\cdot )={{\widetilde{\varphi }}}(|\cdot |)\quad \text{ for } \text{ some } {{\widetilde{\varphi }}}\in C^\infty ({\mathbb {R}}). \end{aligned}$$

Given $\varepsilon ,\lambda \in (0,\varepsilon _0)$, we denote by $v_{\varepsilon ,\lambda }\in u_\varepsilon +W_0^{1,q}(B_1)$ the unique function satisfying

$$\begin{aligned} \int _{B_1}f_\lambda (D v_{\varepsilon ,\lambda })\,dx\le \int _{B_1}f_\lambda (D v)\,dx\qquad \text{ for } \text{ all } v\in u_\varepsilon +W_0^{1,q}(B_1). \end{aligned}$$

(32)

Combining Sobolev inequality with the assumption $\frac{q}{p}<1+\frac{2}{n-2}$ and estimate (27), we have

$$\begin{aligned} \biggl (\int _{B_\frac{1}{8}}|D v_{\varepsilon ,\lambda }|^{q}\,dx\biggr )^\frac{p}{q}&\lesssim \int _{B_\frac{1}{8}}|Dv_{\varepsilon ,\lambda }|^p+|D(|Dv_{\varepsilon ,\lambda }|^\frac{p-2}{2}Dv_{\varepsilon ,\lambda })|^2\,\,dx\nonumber \\&{\mathop {\lesssim }\limits ^{(27)}}\biggl (\int _{B_1}1+f_\lambda (Dv_{\varepsilon ,\lambda })\,dx\biggr )^\alpha \nonumber \\&{\mathop {\le }\limits ^{(19),(32)}}\left( \int _{B_{1}}1+f(D u_\varepsilon )+\lambda |D u_\varepsilon |^q\,dx\right) ^\alpha \nonumber \\&\le \left( |B_1|+\int _{B_{1+\varepsilon }}f(D u)\,dx+\lambda \int _{B_1}|D u_\varepsilon |^q\,dx\right) ^\alpha , \end{aligned}$$

(33)

where we used Jensen’s inequality and the convexity of f in the last step. Similarly,

$$\begin{aligned} \int _{B_1}|D v_{\varepsilon ,\lambda }|^p\,dx{\mathop {\le }\limits ^{(2)}}&\int _{B_1}f(D v_{\varepsilon ,\lambda })\,dx{\mathop {\le }\limits ^{(19)(32)}} \int _{B_1}f(D u_\varepsilon )+\lambda |D u_\varepsilon |^q\,dx\nonumber \\ \le&\int _{B_{1+\varepsilon }}f(D u)\,dx+\lambda \int _{B_1}|D u_\varepsilon |^q\,dx. \end{aligned}$$

(34)

Fix $\varepsilon \in (0,\varepsilon _0)$. In view of (33) and (34), we find $w_\varepsilon \in u_\varepsilon +W_0^{1,p}(B_1)$ such that as $\lambda \rightarrow 0$, up to subsequence,

$$\begin{aligned} v_{\varepsilon ,\lambda }\rightharpoonup w_\varepsilon \qquad \text{ weakly } \text{ in } W^{1,p}(B_1),\\ D v_{\varepsilon ,\lambda }\rightharpoonup Dw_\varepsilon \qquad \text{ weakly } \text{ in } L^{q}(B_\frac{1}{8}). \end{aligned}$$

Hence, a combination of (33), (34) with the weak lower-semicontinuity of convex functionals yield

$$\begin{aligned} \Vert D w_\varepsilon \Vert _{L^{q}(B_\frac{1}{8})}\le&\liminf _{\lambda \rightarrow 0}\Vert D v_{\varepsilon ,\lambda }\Vert _{L^{\kappa p}(B_\frac{1}{8})}\lesssim \left( \int _{B_{1+\varepsilon }}f(D u)\,dx+1\right) ^\frac{\alpha }{p} \end{aligned}$$

(35)

$$\begin{aligned} \int _{B_1}|D w_\varepsilon |^p\,dx\le&\int _{B_{1}}f(D w_\varepsilon )\,dx\le \int _{B_{1+\varepsilon }}f(D u)\,dx. \end{aligned}$$

(36)

Since $w_\varepsilon \in u_\varepsilon +W_0^{1,q}(B_1)$ and $u_\varepsilon \rightarrow u$ in $W^{1,p}(B_1)$, we find by (36) a function $w\in u+W_0^{1,p}(B_1)$ such that, up to subsequence,

$$\begin{aligned} D w_{\varepsilon }\rightharpoonup D w\quad \text{ weakly } \text{ in } L^p(B_1). \end{aligned}$$

Appealing to the bounds (35), (36) and lower semicontinuity, we obtain

$$\begin{aligned} \Vert D w\Vert _{L^{q}(B_\frac{1}{8})}\lesssim&\left( \int _{B_{1}}f(D u)\,dx+1\right) ^\frac{\alpha }{p} \end{aligned}$$

(37)

$$\begin{aligned} \int _{B_{1}}f(D w)\,dx\le&\int _{B_{1}}f(D u)\,dx. \end{aligned}$$

(38)

Inequality (38), strict convexity of f and the fact $w\in u+W_0^{1,p}(B_1)$ imply $w=u$ and thus the claimed estimate (31) is a consequence of (37). $\square$

4 Partial regularity - Proof of Theorem 4

Theorem 4 follows from, the higher integrability statement Theorem 2, the $\varepsilon$-regularity statement of Lemma 4 below and a well-known iteration argument.

Lemma 4

Let $\varOmega \subset {\mathbb {R}}^n$, $n\ge 3$, and suppose Assumption 3 is satisfied with $2\le p<q<\infty$ such that $\frac{q}{p}<1+\frac{2}{n-1}$. Fix $M>0$. There exists $C^*=C^*(n,N,p,q,\frac{L}{\nu },M)\in [1,\infty )$ such that for every $\tau \in (0,\frac{1}{4})$ there exists $\varepsilon =\varepsilon (M,\tau )>0$ such that the following is true: Let $u\in W_\mathrm{loc}^{1,1}(\varOmega ,{\mathbb {R}}^N)$ be a local minimizer of the functional ${\mathcal {F}}$ given in (1). Suppose for some ball $B_r(x)\Subset \varOmega$

$$\begin{aligned} |(Du)_{x,r}|\le M, \end{aligned}$$

where we use the shorthand , and

then

$$\begin{aligned} E(x,\tau r)\le C^*\tau ^2E(x,r). \end{aligned}$$

With the higher integrability of Theorem 5 and the Caccioppoli inequality of Lemma 2 at hand, we can prove Lemma 4 following almost verbatim the proof of the corresponding result [7, Lemma 4.1], which contain the statement of Lemma 4 under the assumption $\frac{q}{p}<1+\frac{2}{n}$ (note that in [7] somewhat more general growth conditions including also the case $1<p<q$ are considered). Thus, we only sketch the argument.

Proof of Lemma 4

Fix $M>0$. Suppose that Lemma 4 is wrong. Then there exists $\tau \in (0,\frac{1}{4})$, a local minimizer $u\in W^{1,1}_\mathrm{loc}(\varOmega ,{\mathbb {R}}^N)$, which in view of Theorem 2 satisfies $u\in W^{1,q}_\mathrm{loc}(\varOmega ,{\mathbb {R}}^N)$, and a sequence of balls $B_{r_m}(x_m)\Subset B_R$ satisfying

$$\begin{aligned}&|(Du)_{x_m,r_m}|\le M,\quad E(x_m,r_m)=:\lambda _m\quad \text{ with }\quad \lim _{m\rightarrow \infty }\lambda _m=0, \end{aligned}$$

(39)

$$\begin{aligned}&E(x_m,\tau r_m)>C^*\tau ^2\lambda _m^2, \end{aligned}$$

(40)

where $C^*$ is chosen below. We consider the sequence of rescaled functions given by

$$\begin{aligned}&v_m(z):=\frac{1}{\lambda _m r_m}(u(x_m+r_mz)-a_m-r_mA_mz), \end{aligned}$$

where $a_m:=(u)_{x_m,r_m}$ and $A_m:=(Du)_{x_m,r_m}$. Assumption (39) implies $\sup _m|A_m|\le M$ and thus, up to subsequence,

$$\begin{aligned} A_m\rightarrow A\in {\mathbb {R}}^{N\times n}. \end{aligned}$$

The definition of $v_m$ yields

$$\begin{aligned} Dv_m(z)=\lambda _m^{-1}(Du(x_m+r_mz)-A_m),\quad (v_m)_{0,1}=0,\quad (Dv_m)_{0,1}=0 \end{aligned}$$

(41)

Assumptions (39) and (40) imply

(42)

(43)

The bound (42) together with (41) imply the existence of $v\in W^{1,2}(B_1,{\mathbb {R}}^N)$ such that, up to extracting a further subsequence,

$$\begin{aligned}&v_m\rightharpoonup v&\text{ in } W^{1,2}(B_1,{\mathbb {R}}^{ N}),\\&\lambda _m Dv_m\rightarrow 0&\text{ in } L^2(B_1,{\mathbb {R}}^{N\times n}) \text{ and } \text{ almost } \text{ everywhere }\\&\lambda _m^{1-\frac{2}{q}}v_m\rightharpoonup 0\quad&\text{ in } W^{1,q}(B_1,{\mathbb {R}}^N). \end{aligned}$$

The function v satisfies the linear equation with constant coefficients

$$\begin{aligned} \int _{B_1}\langle \partial ^2 f(A) Dv,D\varphi \rangle \,dz=0\qquad \text{ for } \text{ all } \varphi \in C^1_0(B_1), \end{aligned}$$

see, e.g., [21] or [7, Proposition 4.2]. Standard estimates for linear elliptic systems with constant coefficients imply $v\in C_\mathrm{loc}^\infty (B_1,{\mathbb {R}}^N)$ and existence of $C^{**}<\infty$ depending only on n, N and the ellipticity contrast of $\partial ^2 f(A)$ (and thus on $\frac{L}{\nu },p,q,$ and M) such that

(44)

Choosing $C^*=2C^{**}$ we obtain a contradiction between (43) and (44) provided we have as $m\rightarrow \infty$

$$\begin{aligned}&Dv_m\rightarrow Dv\quad&\text{ in } L^2_\mathrm{loc}(B_1),\end{aligned}$$

(45)

$$\begin{aligned}&\lambda _m^{1-\frac{2}{q}}Dv_m\rightarrow 0\quad&\text{ in } L^q_\mathrm{loc}(B_1). \end{aligned}$$

(46)

Exanctly as in [7, Proposition 4.3] (with $\mu =2-p$, see also [9, Section 3.4.3.2] for a more detailed presentation of the proof), we have for all $\rho \in (0,1)$,

$$\begin{aligned} \lim _{m\rightarrow \infty }\int _{B_\rho }\int _0^1(1-s)\biggl (1+|A_m+\lambda _m(Dv+sDw_m)|^2\biggr )^\frac{p-2}{2}|Dw_m|^2\,dz=0, \end{aligned}$$

(47)

where $w:=v_m-v$, and thus the local $L^2$-convergence (45) follows. It is left to prove (46). For this, we introduce for $\rho \in (0,1)$ and $T>0$ the sequence of subsets

$$\begin{aligned} U_m:=U_m(\rho ,T):=\{\,z\in B_\rho \,:\,\lambda _m|Dv_m|\le T\,\}. \end{aligned}$$

The local Lipschitz regularity of v, $q>2$ and (45) imply for all $\rho \in (0,1)$ and $T>0$

$$\begin{aligned} \limsup _{m\rightarrow \infty }\int _{U_m(\rho ,T)}\lambda _m^{q-2}|Dv_m|^q\,dz\lesssim&\limsup _{m\rightarrow \infty }\int _{U_m(\rho ,T)}\lambda _m^{q-2}|Dw_m|^q\,dz\\ \lesssim&\limsup _{m\rightarrow \infty }\int _{B_\rho }(M^{q-2}+\lambda _m^{q-2}|Dv|^{q-2})|Dw_m|^2\,dz\\ =&0, \end{aligned}$$

where here and for the rest of the proof $\lesssim$ means $\le$ up to a multiplicative constant depending only on L, n, N, p and q. Hence, it is left to show that there exists $T>0$ such that

$$\begin{aligned} \limsup _{m\rightarrow \infty }\int _{B_\rho \setminus U_m(\rho ,T)}\lambda _m^{q-2}|Dv_m|^q\,dz\le 0\quad \text{ for } \text{ all } \rho \in (0,1). \end{aligned}$$

As in [7], we introduce a sequence of auxiliary functions

$$\begin{aligned} \psi _m:=\lambda _m^{-1}\biggl [(1+|A_m+\lambda _m Dv_m|^2)^\frac{p}{4}-(1+|A_m|^2)^{\frac{p}{4}}\biggr ], \end{aligned}$$

which satisfy

$$\begin{aligned} \limsup _{m\rightarrow \infty }\Vert \psi _m\Vert _{W^{1,2}(B_\rho )}\lesssim c(\rho )\in [1,\infty )\qquad \text{ for } \text{ all } \rho \in (0,1). \end{aligned}$$

(48)

Indeed, by Theorem 2 and Lemma 2, we have for every $\rho \in (0,1)$ and every $Q\in {\mathbb {R}}^{N\times n}$

$$\begin{aligned} \int _{B_{\rho r_m}(x_m)}|\nabla (1+|D u(x)|^2)^\frac{p}{4}|^2\,dx\lesssim r_m^{-2}c(\rho )\int _{B_{r_m}(x_m)}(1+|\nabla u(x)|)^{q-2}|Du(x)-Q|^2\,dx \end{aligned}$$

and thus by rescaling and setting $Q=A_m$

$$\begin{aligned} \int _{B_\rho }|\nabla \psi _m|^2\,dz\lesssim c(\rho )\int _{B_1}(1+|A|^{q-2}+|\lambda _m Dv_m|^{q-2}))|Dv_m|^2\,dz{\mathop {\lesssim }\limits ^{(42)}} c(\rho )(1+M^{q-2}). \end{aligned}$$

The identity $\psi _m=\lambda _m^{-1}\int _0^1\frac{d}{dt}\varTheta (A_m+t\lambda _mv_m)\,dt$ with $\varTheta (F):=(1+|F|^2)^\frac{p}{4}$ implies

$$\begin{aligned} |\psi _m|\le c(|Dv_m|+\lambda _m^{\frac{p-2}{2}}|Dv_m|^{\frac{p}{2}}) \end{aligned}$$

(see [7, p. 555] for details) and thus with help of (47), we obtain

$$\begin{aligned} \limsup _{m\rightarrow \infty }\int _{B_\rho }|\psi _m|^2\,dz\lesssim c(\rho ). \end{aligned}$$

For T sufficiently large (depending on M) there exists $c>0$ such that for all $z\in B_\rho \setminus U_m(\rho ,T)$

$$\begin{aligned} \psi _m(z)\ge c\lambda _m^{-1}\lambda _m^{\frac{p}{2}}|Dv_m(z)|^\frac{p}{2}\quad \text{ and } \text{ thus } \quad \lambda _m^{2(1+\frac{q}{p})}\psi _m^{\frac{2q}{p}}(z)\ge c^{\frac{2q}{p}}\lambda _m^{q-2}|Dv_m(z)|^q \end{aligned}$$

Estimate (48) and Sobolev embedding imply $\limsup _{m\rightarrow \infty }\Vert \psi _m\Vert _{L^{\frac{2n}{n-2}}(B_\rho )}\lesssim c(\rho )\in [1,\infty )$. Hence, using assumption $\frac{q}{p}<1+\frac{2}{n-1}$ (and thus $\frac{2q}{p}<\frac{2n}{n-2}$), we obtain for every $\rho \in (0,1)$

$$\begin{aligned} \limsup _{m\rightarrow \infty }\int _{B_\rho \setminus U_m(\rho ,T)}\lambda _m^{q-2}|Dv_m|^q\,dz\lesssim \lambda _m^{2(1+\frac{q}{p})}\int _{B_\rho }\psi _m^{\frac{2q}{p}}(z)\,dz\lesssim c(\rho )\limsup _{m\rightarrow \infty }\lambda _m^{2(1+\frac{q}{p})}=0, \end{aligned}$$

which finishes the proof. $\square$