2.1 Some basic facts on $N$–functions

We recall here some elementary definitions and basic results about Orlicz functions. The following definitions and results can be found, e.g., in [Reference Adams1, Reference Bennett and Sharpley3, Reference Krasnosel'skií and Rutickií28, Reference Kufner, John and Fucik31].

A real-valued function $\varphi \colon \mathbb {R}^{+}_0 \to \mathbb {R}^{+}_0$ is said to be an $N$-function if it is convex and satisfies the following conditions: $\varphi (0)=0$, $\varphi$ admits the derivative $\varphi '$ and this derivative is right continuous, non-decreasing and satisfies $\varphi '(0) = 0$, $\varphi '(t)>0$ for $t>0$, and $\lim _{t\to \infty } \varphi '(t)=\infty$.

We say that $\varphi$ satisfies the $\Delta _2$-condition if there exists $c > 0$ such that for all $t \geq 0$ holds $\varphi (2t) \leq c\, \varphi (t)$. We denote the smallest possible such constant by $\Delta _2(\varphi )$. Since $\varphi (t) \leq \varphi (2t)$, the $\Delta _2$-condition is equivalent to $\varphi (2t) \sim \varphi (t)$, where ‘$\sim$’ indicates the equivalence between $N$-functions.

By $L^{\varphi }$ and $W^{1,\varphi }$ we denote the classical Orlicz and Orlicz-Sobolev spaces, i. e. $f \in L^{\varphi }$ if and only if $\int \varphi (|{f}|)\,\textrm {d}x < \infty$ and $f \in W^{1,\varphi }$ if and only if $f, D f \in L^{\varphi }$. The space $W^{1,\varphi }_0(\Omega )$ will denote the closure of $C^{\infty }_0(\Omega )$ in $W^{1,\varphi }(\Omega )$.

We define the function $(\varphi ')^{-1} \colon \mathbb {R}^{+}_0 \to \mathbb {R}^{+}_0$ as

\begin{align*} (\varphi')^{{-}1}(t) &:= \sup \{ s \in \mathbb{R}^{+}_0\,:\, \varphi'(s) \leq t \} . \end{align*}

If $\varphi '$ is strictly increasing, then $(\varphi ')^{-1}$ is the inverse function of $\varphi '$. Then $\varphi ^{\ast } \colon \mathbb {R}^{+}_0 \to \mathbb {R}^{+}_0$ with

\begin{align*} \varphi^{{\ast}}(t) &:= \int_0^{t} (\varphi')^{{-}1}(s)\,\textrm{d}s \end{align*}

is again an $N$-function and $(\varphi ^{\ast })'(t) = (\varphi ')^{-1}(t)$ for $t>0$. $\varphi ^{\ast }$ is the Young-Fenchel-Yosida conjugate function of $\varphi$. Note that $\varphi ^{*}(t)= \sup _{a \geq 0} (at - \varphi (a))$ and $(\varphi ^{\ast })^{\ast } = \varphi$. When both $\varphi$ and $\varphi ^{*}$ satisfy $\Delta _2$-condition, by elementary convex analysis it is easy to see that for all $\delta >0$ there exists $c_\delta$ (only depending on $\Delta _2(\varphi )$ and $\Delta _2(\varphi ^{\ast })$) such that for all $t, a \geq 0$ it holds that

\begin{equation*} at \leq \delta\, \varphi(t) + c_\delta\, \varphi^{{\ast}}(a). \end{equation*}

In particular, from (iv) it follows that both $\varphi$ and $\varphi ^{*}$ satisfy the $\Delta _2$-condition with constants $\Delta _2(\varphi )$ and $\Delta _2(\varphi ^{*})$ determined by $\mu _1$ and $\mu _2$. We will denote by $\Delta _2({\varphi , \varphi ^{\ast }})$ constants depending on $\Delta _2(\varphi )$ and $\Delta _2(\varphi ^{*})$. Moreover, for $t>0$ we have

\begin{equation*} \varphi(t) \sim \varphi'(t)\,t, \qquad \varphi(t) \sim \varphi''(t)\,t^{2},\qquad \varphi^{{\ast}}\big( \varphi'(t) \big) \sim \varphi^{{\ast}}\big( \varphi(t)/t \big)\sim \varphi(t). \end{equation*}

We recall also that the following inequalities hold for the inverse function $\varphi ^{-1}$:

(2.2)\begin{align} a^{\frac{1}{\mu_1}}\varphi^{{-}1}(t)\leq &\varphi^{{-}1}(at)\leq a^{\frac{1}{\mu_2}}\varphi^{{-}1}(t) \end{align}

for every $t\geq 0$ with $0< a\leq 1$. The same result holds also for $a\geq 1$ by exchanging the role of $\mu _1$ and $\mu _2$.

For given $\varphi$ we define the associated $N$-function $\psi$ by

\begin{equation*} \psi'(t) := \sqrt{ \varphi'(t)\,t\,}. \end{equation*}

Notice that if $\varphi$ satisfies assumption (2.1), then also $\varphi ^{*}$, $\psi$ and $\psi ^{*}$ satisfy this assumption.

Define ${\bf V}\,:\, \mathbb {R}^{N\times n} \to \mathbb {R}^{N\times n}$ in the following way:

(2.3)\begin{equation} {\bf V}({\bf Q})=\psi'(|{\bf Q}|)\frac{{\bf Q}}{|{\bf Q}|}. \end{equation}

It is easy to check that

\begin{equation*} |{{\bf V}({\bf Q})}|^{2} \sim \varphi(|{{\bf Q}}|), \end{equation*}

uniformly in ${\bf Q} \in \mathbb {R}^{N\times n}$.

Another important set of tools are the shifted $N$-functions $\{\varphi _a \}_{a \ge 0}$ (see [Reference Diening and Ettwein12]). We define for $t\geq 0$

\begin{equation*} \varphi_a(t):= \int _0^{t} \varphi_a'(s)\, \mathrm{d}s\qquad\text{with }\quad \varphi'_a(t):=\varphi'(a+t) \frac{t}{a+t}. \end{equation*}

We have the following relations:

(2.4)\begin{align} \varphi_a(t) &\sim \varphi'_a(t)\,t\,; \nonumber\\ \varphi_a(t) &\sim \varphi''(a+t)t^{2}\sim\frac{\varphi(a+t)}{(a+t)^{2}}t^{2}\sim \frac{\varphi'(a+t)}{a+t}t^{2}, \end{align}

(2.5)\begin{align} \varphi(a+t)&\sim [\varphi_a(t)+\varphi(a)]. \end{align}

The families $\{\varphi _a \}_{a \ge 0}$ and $\{(\varphi _a)^{*} \}_{a \ge 0}$ satisfy the $\Delta _2$-condition uniformly in $a \ge 0$. The connection between ${\bf V}$ and $\varphi _{a}$ (see [Reference Diening and Ettwein12]) is the following:

(2.6)\begin{equation} |{ {\bf V}({\bf P}) - {\bf V}({\bf Q})}|^{2} \sim \varphi_{|{{\bf P}}|}(|{{\bf P} - {\bf Q}}|), \end{equation}

uniformly in ${\bf P}, {\bf Q} \in \mathbb {R}^{N\times n}$. The following lemma (see [Reference Diening and Kreuzer14, corollary 26]) deals with the change of shift for $N$-functions.

Lemma 2.2 Let $\varphi$ be an $N$-function with $\Delta _2(\varphi ),\Delta _2(\varphi ^{*})<\infty$. Then for any $\eta >0$ there exists $c_\eta >0$, depending only on $\eta$ and $\Delta _2(\varphi )$, such that for all ${\bf a}, {\bf b}\in \mathbb {R}^{d}$ and $t\geq 0$

(2.7)\begin{equation} \varphi_{|{\bf a}|}(t) \leq c_\eta\varphi_{|{\bf b}|}(t) + \eta \varphi_{|{\bf a}|}(|{\bf a}-{\bf b}|). \end{equation}

We define the function ${\bf V}_a : \mathbb {R}^{N\times n}\to \mathbb {R}^{N\times n}$ for $a\geq 0$ by

(2.8)\begin{equation} {\bf V}_a({\bf Q}):= \sqrt{\varphi'_a(|{\bf Q}|)|{\bf Q}|}\frac{{\bf Q}}{|{\bf Q}|}, \end{equation}

where $\varphi _a$ is the shifted $N$-function of $\varphi$. Since $\varphi _0=\varphi$, we retrieve in (2.8) the function ${\bf V}$ for $a=0$. With the following lemma, we list some properties of functions ${\bf V}_a$ which will be useful in the sequel.

Lemma 2.3 Let $a\geq 0$ and ${\bf V}_a$ be as above. Then for any ${\bf P}, {\bf Q}\in \mathbb {R}^{N\times n}$ a Young-type inequality holds:

(2.9)\begin{equation} {\varphi'_a(|{\bf Q}|)}|{\bf P}| \leq c(|{\bf V}_a({\bf Q})|^{2}+|{\bf V}_a({\bf P})|^{2}), \end{equation}

where the constant $c$ depends only on $\Delta _2(\varphi )$.

Let ${\bf {P}}_0,{\bf {P}}_1\in \mathbb {R}^{N\times n}$, $\theta \in [0,1]$ and define ${\bf {P}}_{\theta }:=(1-\theta ){\bf {P}}_0+\theta {\bf {P}}_1$. Then the following result holds (see [Reference Diening and Ettwein12, lemma 20]).

Lemma 2.4 Let $\varphi$ be a $N$-function with $\Delta _2(\varphi , \varphi ^{*})<\infty .$ Then uniformly for all ${\bf {P}}_0,{\bf {P}}_1\in \mathbb {R}^{N\times n}$ with $|{\bf {P}}_0|+|{\bf {P}}_1|>0$ holds

\begin{equation*} \int_0^{1} \frac{\varphi'(|{\bf {P}}_{\theta}|)}{|{\bf {P}}_{\theta}|}\, \mathrm{d}\theta \sim \frac{\varphi'(|{\bf {P}}_0|+|{\bf {P}}_1|)}{|{\bf {P}}_0|+|{\bf {P}}_1|} \end{equation*}

where the constants only depend on $\Delta _2(\varphi , \varphi ^{*}).$

In view of the previous considerations, the same proposition holds true for the shifted functions, uniformly in $a\ge 0$.

From assumption (F2) we can easily infer an upper bound for $f(x,{\bf u},{\bf P})-f(x,{\bf u},{\bf Q})$, uniformly in $x\in \Omega$ and ${\bf u}\in \mathbb {R}^{N}$, for every ${\bf P}, {\bf Q}\in \mathbb {R}^{N\times n}$; namely,

(2.10)\begin{equation} \begin{split} |f(x,{\bf u},{\bf P})-f(x,{\bf u},{\bf Q})| & \leq |{\bf P}-{\bf Q}|\int_0^{1} |Df(x,{\bf u},{\bf P}+t({\bf Q}-{\bf P}))|\,\mathrm{d}t \\ & \leq L|{\bf P}-{\bf Q}|\int_0^{1}\varphi'(|{\bf P}+t({\bf Q}-{\bf P})|)\,\mathrm{d}t \\ & \leq cL \varphi(|{\bf P}|+|{\bf Q}|). \end{split} \end{equation}

The following estimate is a consequence of (F2) and lemma 2.4 (see [Reference Diening, Lengeler, Stroffolini and Verde17, eq. (2.14)]):

(2.11)\begin{equation} \begin{split} |Df(x,{\bf u},{\bf P})-Df(x,{\bf u},{\bf Q})| & \leq c(\varphi,L) \varphi''(|{\bf P}|+|{\bf Q}|)|{\bf P}-{\bf Q}|\\ & \leq c(\varphi,L) \varphi'_{|{\bf P}|}(|{\bf P}-{\bf Q}|)\\ & = c(\varphi,L) \frac{\varphi'(|{\bf P}|+|{\bf P}-{\bf Q}|)}{|{\bf P}|+|{\bf P}-{\bf Q}|}|{\bf P}-{\bf Q}|, \end{split} \end{equation}

for every ${\bf P}, {\bf Q}\in \mathbb {R}^{N\times n}$.

The following version of Sobolev-Poincaré inequality can be found in [Reference Diening and Ettwein12, lemma 7].

Theorem 2.5 Let $\varphi$ be an $N$-function with $\Delta _2(\varphi ,\varphi ^{*})<+\infty$. Then there exist numbers $\alpha =\alpha (n,\Delta _2(\varphi ,\varphi ^{*}))\in (0,1)$ and $K=K(n,N,\Delta _2(\varphi ,\varphi ^{*}))>0$ such that the following holds. If $B\subset \mathbb {R}^{n}$ is any ball with radius $R$ and ${\bf w}\in W^{1,\varphi }(B,\mathbb {R}^{N})$, then

(2.12)\begin{equation} \mathop {\int\hskip -1,05em -\,} \limits_B \varphi\left(\frac{|{\bf w}-({\bf w})_B|}{R}\right)\,\mathrm{d}x\leq K \left(\mathop {\int\hskip -1,05em -\,} \limits_B \varphi^{\alpha}\left({|D{\bf w}|}\right)\,\mathrm{d}x\right)^{\frac{1}{\alpha}}, \end{equation}

where $\displaystyle ({\bf w})_B:=\mathop {\int\hskip -1,05em -\,} \limits_B {\bf w}(x)\,\mathrm {d}x$. Moreover, if ${\bf w}\in W^{1,\varphi }_0(B,\mathbb {R}^{N})$, then

\begin{equation*} \mathop {\int\hskip -1,05em -\,} \limits_B \varphi\left(\frac{|{\bf w}|}{R}\right)\,\mathrm{d}x\leq K \left(\mathop {\int\hskip -1,05em -\,} \limits_B \varphi^{\alpha}\left({|D{\bf w}|}\right)\,\mathrm{d}x\right)^{\frac{1}{\alpha}}, \end{equation*}

where $K$ and $\alpha$ have the same dependencies as before.

2.2 Some useful lemmas

The following lemma, useful in order to re-absorb certain terms, is a variant of the classical [Reference Giusti25, lemma 6.1] (see [Reference Diening, Lengeler, Stroffolini and Verde17, lemma 3.1]).

Lemma 2.6 Let $\psi$ be an $N$-function with $\psi \in \Delta _2$, let $\varrho >0$ and $h\in L^{\psi }(B_{\varrho }(x_0))$. Let $g:[r,\varrho ]\to \mathbb {R}$ be nonnegative and bounded such that for all $r\leq s< t\leq \varrho$

\begin{equation*} g(s)\leq\theta g(t) + A \int_{B_t(x_0)}\psi\left(\frac{|h(y)|}{t-s}\right)\,\mathrm{d}y+ \frac{B}{(t-s)^{\beta}}+C, \end{equation*}

where $A,B,C\geq 0$, $\beta >0$ and $\theta \in [0,1)$. Then

\begin{equation*} g\left(r\right)\leq c(\theta,\Delta_2(\psi),\beta)\left[A\int_{B_{\varrho}(x_0)}\psi\left(\frac{|h(y)|}{\varrho-r}\right)\,\mathrm{d}y + \frac{B}{(\varrho-r)^{\beta}}+C\right]. \end{equation*}

The following lemma is useful to derive reverse Hölder estimates. It is a variant of the results by Gehring [Reference Gehring24] and Giaquinta-Modica [Reference Giusti25, theorem 6.6].

Lemma 2.7 Let $B_0\subset \mathbb {R}^{n}$ be a ball, $f\in L^{1}(B_0)$, and $g\in L^{\sigma _0}(B_0)$ for some $\sigma _0>1$. Assume that for some $\theta \in (0,1)$, $c_1>0$ and all balls $B$ with $2B\subset B_0$

\begin{equation*} \mathop {\int\hskip -1,05em -\,} \limits_B |f|\,\mathrm{d}x\leq c_1 \left(\mathop {\int\hskip -1,05em -\,} \limits_{2B}|f|^{\theta}\,\mathrm{d}x\right)^{1/\theta} + \mathop {\int\hskip -1,05em -\,} \limits_{2B}|g|\,\mathrm{d}x. \end{equation*}

Then there exist $\sigma _1>1$ and $c_2>1$ such that $g\in L^{\sigma _1}_\textrm {loc}(B)$ and for all $\sigma _2\in [1,\sigma _1]$

\begin{equation*} \left(\mathop {\int\hskip -1,05em -\,} \limits_{B}|f|^{\sigma_2}\,\mathrm{d}x\right)^{1/{\sigma_2}}\leq c_2 \mathop {\int\hskip -1,05em -\,} \limits_{2B}|f|\,\mathrm{d}x + c_2 \left(\mathop {\int\hskip -1,05em -\,} \limits_{2B}|g|^{\sigma_2}\,\mathrm{d}x\right)^{1/{\sigma_2}}. \end{equation*}

2.3 $\mathcal {A}$-harmonic and $\varphi$-harmonic functions

Let $\mathcal {A}$ be a bilinear form on $\mathbb {R}^{N\times n}$. We say that $\mathcal {A}$ is strongly elliptic in the sense of Legendre-Hadamard if for all $\boldsymbol \xi \in \mathbb {R}^{N},\boldsymbol \zeta \in \mathbb {R}^{n}$ it holds that

(2.13)\begin{equation} {{\kappa_{\!\mathcal{A}}}} \lvert{\boldsymbol\xi}\rvert^{2} \lvert{\boldsymbol\zeta}\rvert^{2}\leq \langle\mathcal{A}(\boldsymbol\xi\otimes\boldsymbol\zeta)|(\boldsymbol\xi\otimes\boldsymbol\zeta)\rangle\leq L_{\mathcal{A}} \lvert{\boldsymbol\xi}\rvert^{2} \lvert{\boldsymbol\zeta}\rvert^{2} \end{equation}

for some $L_{\mathcal {A}}\geq {{\kappa _{\!\mathcal {A}}}}>0$. We say that a Sobolev function ${\bf w}$ on a ball $B_\varrho (x_0)$ is $\mathcal {A}$-harmonic on $B_\varrho (x_0)$ if it satisfies $-{ {\mathrm {div}}} (\mathcal {A}D {\bf w})=0$ in the sense of distributions; i.e.,

\begin{equation*} \int_{B_\varrho(x_0)} \langle\mathcal{A}D{\bf w}|D\boldsymbol\psi\rangle\,\mathrm{d}x=0,\quad \mbox{ for all }\boldsymbol\psi\in C^{\infty}_0(B_\varrho(x_0),\mathbb{R}^{N}). \end{equation*}

It is well known from the classical theory (see, e.g. [Reference Giusti25, chapter 10]) that ${\bf w}$ is smooth in the interior of $B_\varrho (x_0)$, and it satisfies the estimate

(2.14)\begin{equation} \sup_{B_{\varrho/2}(x_0)}|D {\bf w}|^{2}+\varrho^{2} \sup_{B_{\varrho/2}(x_0)}|D^{2} {\bf w}|^{2}\leq c(n,N,\nu,L) \mathop {\int\hskip -1,05em -\,} \limits_{B_\varrho(x_0)}|D{\bf w}|^{2}\,\mathrm{d}x. \end{equation}

Let $\varphi$ be an Orlicz function. We say that a map ${\bf w}\in W^{1,\varphi }(B_\varrho (x_0),\mathbb {R}^{N})$ is $\varphi$-harmonic on $B_\varrho (x_0)$ (see [Reference Diening, Stroffolini and Verde16]) if and only if

\begin{equation*} \int_{B_\varrho(x_0)} \left\langle\frac{\varphi'(|D{\bf w}|)}{|D{\bf w}|}D{\bf w}\bigg|D\boldsymbol\psi\right\rangle\,\mathrm{d}x=0,\quad \mbox{ for all }\boldsymbol\psi\in C^{\infty}_0(B_\varrho(x_0),\mathbb{R}^{N}). \end{equation*}

More precisely, $D{\bf w}$ and ${\bf V}(D{\bf w})$ are Hölder continuous due to the following decay estimate, see [Reference Diening, Stroffolini and Verde15].

Proposition 2.8 Let $\varphi$ be a convex function complying with (φ1), (φ2) and

1. (φ3)

   \begin{align*} \hspace{0.5cm} & \varphi' \text{ is H}\unicode{x04E7}\text{lder continuous off the diagonal:} \hspace{1cm}\\ & \hspace{2cm} \left|\varphi''(s+t)-\varphi''(t)\right|\leq c_0\, \varphi'' (t)\, \bigg( \frac{\left|s\right|}{t} \bigg)^{\beta_0}, \quad \beta_0>0, \hspace{1cm}\\ & \text{ for all } t>0 \text{ and } s \in \mathbb{R} \text{ with } \left|s\right| < \frac{1}{2} t. \end{align*}

Then there exist a constant $c\geq 1$ and an exponent $\gamma _0 \in (0,1)$ depending only on $n,N$ and the characteristics of $\varphi$, such that the following statement holds true: whenever ${\bf w} \in W^{1,\varphi }(B_{R}(x_0),\mathbb {R}^{N})$ is a weak solution of the system

\begin{equation*} \, {\rm div}\left( \frac{\varphi'({\vert{D {\bf u}}\vert})}{|D {\bf u}|} \, D {\bf u}\right) = \, 0 \qquad \text{in }B_{R}(x_0),\ \end{equation*}

then for every $\tau \in (0,1)$ there hold

\begin{align*} &\sup_{B_{\tau R/2}(x_0)} \varphi({\vert{D {\bf w}}\vert})\leq c \mathop{\int\hskip -1,05em -\,}\nolimits_{\!\!\!\!{B_{\tau R}(x_0)}} \varphi({\vert{D {\bf w}}\vert})\,\mathrm{d}x, \\ &\mathop{\int\hskip -1,05em -\,}\nolimits_{\!\!\!\!{B_{\tau R}(x_0)}} {\vert{{\bf V}(D{\bf w})-({\bf V}(D {\bf w}))_{x_0,{\tau R}}}\vert}^{2} \, \mathrm{d}x \\ &\quad \leq c \tau^{2 \gamma_0} \, \mathop{\int\hskip -1,05em -\,}\nolimits_{\!\!\!\!{B_R(x_0)}} {\vert{{\bf V}(D{\bf w})-({\bf V}(D {\bf w}))_{x_0,R}}\vert}^{2} \, \mathrm{d}x. \end{align*}

This result can be viewed as the Orlicz version of the milestone theorem of Uhlenbeck [Reference Uhlenbeck41] for differential forms solving a $p$-harmonic system, see also [Reference Beck and Stroffolini2].

2.4 Harmonic type approximation results

We recall here two different harmonic type approximation results. The first one is the $\mathcal {A}$-harmonic approximation: given a Sobolev function ${\bf u}$ on a ball $B$, we want to find an $\mathcal {A}$-harmonic function ${\bf w}$ which is ‘close’ to the function ${\bf u}$. It will be the $\mathcal {A}$-harmonic function with the same boundary values as ${\bf u}$; i.e., a Sobolev function ${\bf w}$ which satisfies

(2.15)\begin{equation} \begin{cases} -{{\mathrm{div}}} (\mathcal{A} D {\bf w})= 0 & \qquad\text{on } B \\ {\bf w}= {\bf u} & \qquad\text{on } \partial B \end{cases} \end{equation}

in the sense of distributions.

Setting ${\bf z} := {\bf w} - {\bf u}$, then (2.15) is equivalent to finding a Sobolev function ${\bf z}$ which satisfies

(2.16)\begin{equation} \begin{cases} -{{\mathrm{div}}} (\mathcal{A} D {\bf z}) ={-}{{\mathrm{div}}}(\mathcal{A} D {\bf u}) & \qquad\text{on } B \\ {\bf z}= {{\mathbf{0}}} & \qquad\text{on } \partial B \end{cases} \end{equation}

in the sense of distributions.

The following $\mathcal {A}$-harmonic approximation result in the setting of Orlicz spaces has been proved in [Reference Diening, Lengeler, Stroffolini and Verde17, theorem 14].

Theorem 2.9 Let $B \subset \subset \Omega$ be a ball with radius $r_B$ and let $\widetilde {B} \subset \Omega$ denote either $B$ or $2B$. Let $\mathcal {A}$ be a strongly elliptic (in the sense of Legendre-Hadamard) bilinear form on $\mathbb {R}^{N\times n}$. Let $\psi$ be an N-function with $\psi \in \Delta _2(\psi , \psi ^{*})$ and let $s>1$. Then for every $\varepsilon >0$, there exists $\delta >0$ only depending on $n$, $N$, $\kappa _A$, $\lvert {\mathcal {A}}\rvert$, $\Delta _2(\psi ,\psi ^{*})$ and $s>1$ such that the following holds. Let ${\bf u} \in W^{1,\psi }(\widetilde {B},\mathbb {R}^{N})$ be almost $\mathcal {A}$-harmonic on $B$ in the sense that

(2.17)\begin{align} \bigg\lvert{\mathop {\int\hskip -1,05em -\,} \limits_B \langle\mathcal{A}D {\bf u} | D \boldsymbol\eta\rangle\,\mathrm{d}x}\bigg\rvert \leq \delta \mathop {\int\hskip -1,05em -\,} \limits_{\widetilde{B}} \lvert{D {\bf u}}\rvert\,\mathrm{d}x \lVert{D \boldsymbol\eta}\rVert_{L^{\infty}(B)} \end{align}

for all $\boldsymbol \eta \in C^{\infty }_0(B,\mathbb {R}^{N})$. Then the unique solution ${\bf z} \in W^{1, \psi }_0(B,\mathbb {R}^{N})$ of (2.16) satisfies

\begin{equation*} \mathop {\int\hskip -1,05em -\,} \limits_B \psi\bigg(\frac{\lvert{{\bf z}}\rvert}{r_B}\bigg)\,\mathrm{d}x + \mathop {\int\hskip -1,05em -\,} \limits_B \psi(\lvert{D {\bf z}}\rvert)\,\mathrm{d}x \leq \varepsilon \left(\bigg(\mathop {\int\hskip -1,05em -\,} \limits_{\widetilde{B}} \big(\psi(\lvert{D {\bf u}}\rvert)\big)^{s} \,\mathrm{d}x\bigg)^{\frac 1s}+\mathop {\int\hskip -1,05em -\,} \limits_{\widetilde{B}} \psi(\lvert{D {\bf u}}\rvert)\,\mathrm{d}x\right). \end{equation*}

Remark 2.10 We will exploit the previous approximation result in a slightly modified version. Indeed, following [Reference Celada and Ok7, lemma 2.7], under the additional assumption

\begin{equation*} \mathop {\int\hskip -1,05em -\,} \limits_{\tilde{B}} \psi(|D{\bf u}|)\,\mathrm{d}x \leq \left(\mathop {\int\hskip -1,05em -\,} \limits_{\tilde{B}} [\psi(|D{\bf u}|)]^{s}\,\mathrm{d}x\right)^{\frac{1}{s}}\leq \psi(\mu) \end{equation*}

for some exponent $s>1$ and for a constant $\mu >0$, and (2.17) replaced by

\begin{equation*} \bigg\lvert{\mathop {\int\hskip -1,05em -\,} \limits_B \langle\mathcal{A}D {\bf u} | D \boldsymbol\eta\rangle\,\mathrm{d}x}\bigg\rvert \leq \delta \mu \lVert{D \boldsymbol\eta}\rVert_{L^{\infty}(B)}, \end{equation*}

it can be seen with minor changes in the proof that the unique solution ${\bf z} \in W^{1, \psi }_0(B,\mathbb {R}^{N})$ of (2.16) satisfies

\begin{equation*} \mathop {\int\hskip -1,05em -\,} \limits_B \psi\bigg(\frac{\lvert{{\bf z}}\rvert}{r_B}\bigg)\,dx + \mathop {\int\hskip -1,05em -\,} \limits_B \psi(\lvert{D {\bf z}}\rvert)\,dx \leq \varepsilon \psi(\mu). \end{equation*}

Now, moving on to $\varphi$-harmonic functions, the following $\varphi$-harmonic approximation lemma ([Reference Diening, Stroffolini and Verde16, lemma 1.1]) is the extension to general convex functions of the $p$-harmonic approximation lemma [Reference Duzaar and Mingione21], [Reference Duzaar and Mingione22, lemma 1], and allows to approximate ‘almost $\varphi$-harmonic’ functions by $\varphi$-harmonic functions.

Lemma 2.11 Let $\varphi$ satisfy assumption (2.1). For every $\varepsilon >0$ and $\theta \in (0,1)$ there exists $\delta >0$ which only depends on $\varepsilon$, $\theta$, and the characteristics of $\varphi$ such that the following holds. Let $B\subset \mathbb {R}^{n}$ be a ball and let $\tilde {B}$ denote either $B$ or $2B$. If ${\bf u}\in W^{1,\varphi }(\tilde {B},\mathbb {R}^{N})$ is almost $\varphi$-harmonic on a ball $B\subset \mathbb {R}^{n}$ in the sense that

(2.18)\begin{equation} \mathop {\int\hskip -1,05em -\,} \limits_B \left\langle\frac{\varphi'(|D{\bf u}|)}{|D{\bf u}|}D{\bf u}\biggl|D\boldsymbol\eta\right\rangle\,\mathrm{d}x \leq \delta\left(\mathop {\int\hskip -1,05em -\,} \limits_{\tilde{B}}\varphi(|D{\bf u}|)\,\mathrm{d}x+\varphi(\|D\boldsymbol\eta\|_\infty)\right) \end{equation}

for all $\boldsymbol \eta \in C^{\infty }_0(B,\mathbb {R}^{N})$, then the unique $\varphi$-harmonic ${\bf w}\in W^{1,\varphi }(B,\mathbb {R}^{N})$ with ${\bf w}={\bf u}$ on $\partial B$ satisfies

(2.19)\begin{equation} \left(\mathop {\int\hskip -1,05em -\,} \limits_B |{\bf V}(D{\bf u})-{\bf V}(D{\bf w})|^{2\theta}\,\mathrm{d}x\right)^{\frac{1}{\theta}}\leq \varepsilon \mathop {\int\hskip -1,05em -\,} \limits_{\tilde{B}}\varphi(|D{\bf u}|)\,\mathrm{d}x, \end{equation}

where ${\bf V}$ is as in (2.3).

The estimate (2.19) can be improved when $\varphi (|D{\bf u}|)$ satisfies a reverse Hölder inequality as follows (see [Reference Celada and Ok7, corollary 2.10]).

Lemma 2.12 Let $B\subset \mathbb {R}^{n}$ be a ball. Let ${\bf u}\in W^{1,\varphi }(2B,\mathbb {R}^{N})$ be such that

\begin{equation*} \left(\mathop {\int\hskip -1,05em -\,} \limits_{B}\varphi^{s_1}(|D{\bf u}|)\,\mathrm{d}x\right)^{\frac{1}{s_1}}\leq \tilde{c}_0 \mathop {\int\hskip -1,05em -\,} \limits_{2B}\varphi(|D{\bf u}|)\,\mathrm{d}x \end{equation*}

for $s_1>1$ and $\tilde {c}_0>0$. Then for every $\varepsilon \in (0,1)$ there exists $\delta _0=\delta _0(n,N,\mu _1,\mu _2,s_1, \tilde {c}_0,\varepsilon )>0$ such that the following holds: if ${\bf u}$ is almost $\varphi$-harmonic as in (2.18) with $\delta _0$ in place of $\delta$, then the unique $\varphi$-harmonic function ${\bf w}\in W^{1,\varphi }(B, \mathbb {R}^{N})$ such that ${\bf w}={\bf u}$ on $\partial B$ satisfies

\begin{equation*} \mathop {\int\hskip -1,05em -\,} \limits_{B} |{\bf V}(D{\bf u})-{\bf V}(D{\bf w})|^{2}\,\mathrm{d}x\leq \varepsilon \mathop {\int\hskip -1,05em -\,} \limits_{2B}\varphi(|D{\bf u}|)\,\mathrm{d}x. \end{equation*}

3.1 Caccioppoli inequalities and higher integrability results

As usual, the first step in proving a regularity theorem for the minimizers of integral functionals is to establish suitable Caccioppoli-type inequalities.

First, we state a ‘zero order’ Caccioppoli inequality. The proof is an adaptation to the $\varphi$-setting of [Reference Bögelein4, lemma 3.1], we then omit the details (see also [Reference Celada and Ok7, theorem 2.4]).

Lemma 3.1 Let ${\bf u}\in W^{1,\varphi }(\Omega ,\mathbb {R}^{N})$ be a minimizer of the functional (1.1), under assumptions (F1)–(F2). Then, for every ${\bf u}_0\in \mathbb {R}^{N}$ and $x_0\in \Omega$ and all $0<\varrho <\textrm {dist}(x_0,\partial \Omega )$ and $r\in [\varrho /2,\varrho )$ there holds

\begin{equation*} \mathop {\int\hskip -1,05em -\,} \limits_{B_r(x_0)} \varphi(|D{\bf u}|)\,\mathrm{d}x\leq c \mathop {\int\hskip -1,05em -\,} \limits_{B_\varrho(x_0)}\varphi\left(\frac{|{\bf u} - {\bf u}_0|}{\varrho-r}\right)\,\mathrm{d}x \end{equation*}

for some constant $c=c(\varphi , L,\nu )>0$.

From lemma 3.1 together with the Sobolev-Poincaré inequality (theorem 2.5) and Gehring's Lemma (lemma 2.7), one can infer in a standard way the following higher integrability result (see, e.g., [Reference Celada and Ok7, theorem 2.5]).

Lemma 3.2 There exist an exponent $s_0=s_0(n,N,\varphi ,L,\nu )>1$ and a constant $c$ depending only on $n,N,\varphi ,L,\nu$ such that, if ${\bf u}\in W^{1,\varphi }(\Omega ;\mathbb {R}^{N})$ is a minimizer of the functional (1.1), complying with (F1)–(F2), then the following holds: for every $s\in (1,s_0]$, for any $x_0\in \Omega$, any radius $0<\varrho <{\rm dist}(x_0,\partial \Omega )$ and $r\in [\varrho /2,\varrho )$, one has

\begin{equation*} \mathop {\int\hskip -1,05em -\,} \limits_{B_r(x_0)} \varphi^{s}(|D{\bf u}|)\,\mathrm{d}x \leq {c}\left(\frac{\varrho}{\varrho-r}\right)^{n(s-1)}\left(\mathop {\int\hskip -1,05em -\,} \limits_{B_\varrho(x_0)}\varphi(|D{\bf u}|)\,\mathrm{d}x\right)^{s}. \end{equation*}

Another useful tool will be the following global higher integrability result on balls for minimizers of (1.1), which has been proven in the Orlicz setting for more general integrands in [Reference De Filippis10, lemma 4.3].

Lemma 3.3 Let ${\bf u}\in W^{1,\varphi }(B_r(x_0),\mathbb {R}^{N})$ be such that $\varphi (|D{\bf u}|)\in L^{s_0}(B_r(x_0), \mathbb {R}^{N})$ for some $s_0>1$. Then there exists an exponent $s=s(n,N,\varphi ,L,\nu ,s_0)\in (1,s_0]$ and a constant $c=c(n,N,\varphi ,L,\nu )$ such that, if ${\bf v}\in {\bf u}+W^{1,\varphi }_0(B_r(x_0),\mathbb {R}^{N})$ is a minimizer of the functional $\mathcal {G}[{\bf v}]:=\int _{B_r(x_0)}g(D{\bf v})\,\mathrm {d}x$ with a $C^{1}$-integrand $g:\mathbb {R}^{N\times n}\to \mathbb {R}$ complying with the growth assumptions

\begin{equation*} \nu \varphi(|\boldsymbol\xi|)\leq g(\boldsymbol\xi) \leq L \varphi(1+|\boldsymbol\xi|) \quad \mbox{ and }\quad |Dg(\boldsymbol\xi)|\leq L \varphi'(|\boldsymbol\xi|) \end{equation*}

for all $\boldsymbol \xi \in \mathbb {R}^{nN}$, then we have $\varphi (|D{\bf v}|)\in L^{s}(B_r(x_0),\mathbb {R}^{N})$ and

\begin{equation*} \left(\mathop {\int\hskip -1,05em -\,} \limits_{B_r(x_0)} \varphi^{s}(|D{\bf v}|)\,\mathrm{d}x\right)^{\frac{1}{s}}\leq c\left(\mathop {\int\hskip -1,05em -\,} \limits_{B_r(x_0)} \varphi^{s_0}(|D{\bf u}|)\,\mathrm{d}x\right)^{\frac{1}{s_0}}. \end{equation*}

We have the following Caccioppoli inequality of second type for local minimizers of (1.1), involving affine functions.

Lemma 3.4 There exists a constant $c=c(n,N,\Delta _2(\varphi ),\nu ,L)>0$ such that, if ${\bf u}\in W^{1,\varphi }(\Omega ;\mathbb {R}^{N})$ is a minimizer of the functional (1.1) under assumptions (F1)–(F7), and $\boldsymbol \ell :\mathbb {R}^{n}\to \mathbb {R}^{N}$ is an affine function, say $\boldsymbol \ell (x):={\bf u}_0+{\bf Q}(x-x_0)$ for some ${\bf u}_0\in \mathbb {R}^{N}$ and ${\bf Q}\in \mathbb {R}^{N\times n}$, then for any ball $B_\varrho (x_0)\subseteq \Omega$ with $\varrho \leq \varrho _0$ there holds

\begin{align*} & \mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}(x_0)} \varphi_{|{\bf Q}|}(|D{\bf u}-{\bf Q}|)\,\mathrm{d}x\\ & \quad \leq c \mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho}(x_0)} \varphi_{|{\bf Q}|}\left(\frac{|{\bf u}-{\boldsymbol\ell}|}{\varrho}\right)\,\mathrm{d}x \\ &\qquad + c\varphi(|{\bf Q}|)\left[\omega\left(\mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho}(x_0)}|{\bf u}-{\bf u}_0|+|{\bf u}-\boldsymbol\ell|\,\mathrm{d}x\right)^{1-\frac{1}{s}}+ [{\mathcal{V}}(\varrho)]^{1-\frac{1}{s}}\right] \end{align*}

for every $s\in (1,s_0]$ where $s_0$ is that of lemma 3.2.

Proof. We follow the argument of [Reference Bögelein4, lemma 3.5] for functionals with $p$-growth, just mentioning how to obtain the analogous main estimates therein. We assume, without loss of generality, that $x_0=0$. For radii $\frac {\varrho }{2}\leq r<\tau < t\leq \frac {3\varrho }{4}$ with $\tau :=\frac {r+t}{2}$ we consider a cut-off function $\eta \in C_0^{\infty }(B_\tau ;[0,1])$ such that $\eta \equiv 1$ on $B_r$ and $|D\eta |\leq \frac {4}{t-r}$ on $B_\tau$. Correspondingly, we define the functions $\boldsymbol \xi :=\eta ({\bf u} - \boldsymbol \ell )\in W^{1,\varphi }(B_\tau ;\mathbb {R}^{N})$ and $\boldsymbol \psi :=(1-\eta )({\bf u} - \boldsymbol \ell )\in W^{1,\varphi }(B_\tau ;\mathbb {R}^{N})$. Note that $\boldsymbol \ell +\boldsymbol \xi ={\bf u}-\boldsymbol \psi$. From the quasi-convexity assumption (F3), (2.4) and simple manipulations we obtain

(3.1)\begin{equation} \begin{split} \int_{B_\tau} \varphi_{|{\bf Q}|}(|D\boldsymbol\xi|)\,\mathrm{d}x & \leq c(\nu,\varphi)\int_{B_\tau} \varphi''(|{\bf Q}|+|D\boldsymbol\xi|)|D\boldsymbol\xi|^{2}\,\mathrm{d}x \\ & \leq c\int_{B_\tau}[(f({\cdot}, {\bf u}_0, {\bf Q}+D\boldsymbol\xi(x)))_\tau - (f({\cdot}, {\bf u}_0, {\bf Q}))_\tau]\,\mathrm{d}x \\ & = c(J_1+J_2+J_3+J_4+J_5+J_6+J_7), \end{split} \end{equation}

where

\begin{equation*} \begin{split} J_1 & := \int_{B_\tau}[(f({\cdot}, {\bf u}_0, D{\bf u}(x)-D\boldsymbol\psi(x)))_\tau - (f({\cdot}, {\bf u}_0, D{\bf u}(x)))_\tau]\,\mathrm{d}x,\\ J_2 & := \int_{B_\tau}[(f({\cdot}, {\bf u}_0, D{\bf u}(x)))_\tau - (f({\cdot}, {\bf u}(x), D{\bf u}(x)))_\tau]\,\mathrm{d}x, \\ J_3 & := \int_{B_\tau}[(f({\cdot}, {\bf u}(x), D{\bf u}(x)))_\tau - f(x, {\bf u}(x), D{\bf u}(x))]\,\mathrm{d}x, \\ J_4 & := \int_{B_\tau}[ f(x, {\bf u}(x), D{\bf u}(x)) - f(x, {\bf u}(x)-\boldsymbol\xi(x), D{\bf u}(x)-D\boldsymbol\xi(x))]\,\mathrm{d}x, \\ J_5 & := \int_{B_\tau}[f(x, {\bf u}(x)-\boldsymbol\xi(x), {\bf Q}+D\boldsymbol\psi(x))-f(x, {\bf u}_0, {\bf Q}+D\boldsymbol\psi(x))]\,\mathrm{d}x, \\ J_6 & := \int_{B_\tau}[f(x, {\bf u}_0, {\bf Q}+D\boldsymbol\psi(x))-(f({\cdot}, {\bf u}_0, {\bf Q}+D\boldsymbol\psi(x)))_\tau]\,\mathrm{d}x,\\ J_7 & := \int_{B_\tau}[(f({\cdot}, {\bf u}_0, {\bf Q}+D\boldsymbol\psi(x)))_\tau- (f({\cdot}, {\bf u}_0, {\bf Q}))_\tau]\,\mathrm{d}x. \end{split} \end{equation*}

Now, we proceed to estimate each term above separately. From the minimizing property of ${\bf u}$ we infer that $J_4\leq 0$, and by assumptions (F5) and (F4) we obtain the estimates

\begin{equation*} \begin{split} J_2 & \leq \int_{B_\tau} \omega(|{\bf u}-{\bf u}_0|)\varphi(|D{\bf u}|)\,\mathrm{d}x,\\ J_3 & \leq \int_{B_\tau} {v}_0({\cdot},\tau)\varphi(|D{\bf u}|)\,\mathrm{d}x, \end{split} \end{equation*}

respectively. Again by exploiting property (F5), the monotonicity of $\omega$ and $\varphi$, and the fact that

\begin{equation*} |\boldsymbol\xi|\leq|{\bf u}-\boldsymbol\ell|, \mbox{ and } |D\boldsymbol\psi|\leq |D{\bf u}-{\bf Q}|+4\left|\frac{{\bf u}-\boldsymbol\ell}{t-\tau}\right|, \end{equation*}

we can estimate $J_5$ as

\begin{equation*} \begin{split} J_5 & \leq c(\varphi)\int_{B_\tau} \omega(|{\bf u}-{\bf u}_0|+|{\bf u}-\boldsymbol\ell|)\varphi\left(|{\bf Q}|+|D{\bf u}|+\left|\frac{{\bf u}-\boldsymbol\ell}{t-\tau}\right|\right)\,\mathrm{d}x\\ & \leq c(\varphi) \int_{B_\tau} \omega(|{\bf u}-{\bf u}_0|+|{\bf u}-\boldsymbol\ell|)\left[\varphi(|{\bf Q}|+|D{\bf u}|)+\varphi\left(\left|\frac{{\bf u}-\boldsymbol\ell}{t-\tau}\right|\right)\right]\,\mathrm{d}x, \end{split} \end{equation*}

whence, taking into account that by virtue of (2.5),

(3.2)\begin{equation} \begin{split} \varphi\left(\left|\frac{{\bf u}-\boldsymbol\ell}{t-\tau}\right|\right) & \leq c\varphi_{|{\bf Q}|}\left(\left|\frac{{\bf u}-\boldsymbol\ell}{t-\tau}\right|\right)+c\varphi(|{\bf Q}|)\\ & \leq c(\varphi)\left|{\bf V}_{|{\bf Q}|}\left(\left|\frac{{\bf u}-\boldsymbol\ell}{t-\tau}\right|\right)\right|^{2}+c\varphi(|{\bf Q}|) \end{split} \end{equation}

and recalling that $\omega \leq 1$, we get

\begin{align*} J_5&\leq c(\varphi) \left(\int_{B_\tau} \left|{\bf V}_{|{\bf Q}|}\left(\left|\frac{{\bf u}-\boldsymbol\ell}{t-\tau}\right|\right)\right|^{2}\,\mathrm{d}x \right.\\ &\quad \left. + \int_{B_\tau} \omega(|{\bf u}-{\bf u}_0|+|{\bf u}-\boldsymbol\ell|)\varphi\left(|{\bf Q}|+|D{\bf u}|\right)\,\mathrm{d}x\right).\end{align*}

For what concerns $J_6$, an analogous computation as for the estimate of $J_5$ based on (3.2) and the VMO assumption (F4) gives

\begin{equation*} \begin{split} J_6 & \leq \int_{B_\tau} {v}_0({\cdot},\tau)\varphi(|{\bf Q}+D\boldsymbol\psi|)\,\mathrm{d}x \\ & \leq c(\Delta_2(\varphi)) \left(\int_{B_\tau} \left|{\bf V}_{|{\bf Q}|}\left(\left|\frac{{\bf u}-\boldsymbol\ell}{t-\tau}\right|\right)\right|^{2}\,\mathrm{d}x + \int_{B_\tau} {v}_0({\cdot},\tau)\varphi\left(|{\bf Q}|+|D{\bf u}|\right)\,\mathrm{d}x\right). \end{split} \end{equation*}

The terms $J_1$ and $J_7$ can be combined together as

\begin{align*} J_7+J_1&= \int_{B_\tau}\mathop {\int\hskip -1,05em -\,} \limits_{B_\tau}\int_0^{1}\langle Df(y,{\bf u}_0,{\bf Q}+\theta D\boldsymbol\psi(x)) - Df(y,{\bf u}_0,{\bf Q})|D\boldsymbol\psi(x)\rangle\,\mathrm{d}\theta\,\mathrm{d}y\,\mathrm{d}x \\ & \quad + \int_{B_\tau}\mathop {\int\hskip -1,05em -\,} \limits_{B_\tau}\int_0^{1}\langle Df(y,{\bf u}_0,{\bf Q}) - Df(y,{\bf u}_0,D{\bf u}(x)\\ &\quad -(1-\theta)D\boldsymbol\psi(x))| D\boldsymbol\psi(x)\rangle\,\mathrm{d}\theta\,\mathrm{d}y\,\mathrm{d}x\\ & =: J'_7+J'_1. \end{align*}

From the Cauchy-Schwarz inequality, (2.11) and the fact that $D\boldsymbol \psi ={\bf 0}$ on $B_r$ we infer

\begin{equation*} \begin{split} J'_7 & \leq \int_{B_\tau}\mathop {\int\hskip -1,05em -\,} \limits_{B_\tau}\int_0^{1}|Df(y,{\bf u}_0,{\bf Q}+\theta D\boldsymbol\psi(x)) - Df(y,{\bf u}_0,{\bf Q})||D\boldsymbol\psi(x)|\,\mathrm{d}\theta\,\mathrm{d}y\mathrm{d}x \\ & \leq c\int_{B_\tau}\int_0^{1}\varphi'_{|{\bf Q}|}(\theta |D\boldsymbol\psi(x)|)|D\boldsymbol\psi(x)|\,\mathrm{d}\theta\,\mathrm{d}x \\ & \leq c\int_{B_\tau}\varphi_{|{\bf Q}|}(|D\boldsymbol\psi(x)|)\,\mathrm{d}x \leq c\int_{B_\tau\backslash B_r}|{\bf V}_{|{\bf Q}|}(D\boldsymbol\psi(x))|^{2}\,\mathrm{d}x. \end{split} \end{equation*}

We can estimate $J'_1$ analogously, by recalling that $D{\bf u}-(1-\theta )D\boldsymbol \psi ={\bf Q}+D\boldsymbol \xi +\theta D\boldsymbol \psi$, $D\boldsymbol \psi ={\bf 0}$ on $B_r$ and applying the triangle inequality for $\varphi '_{|{\bf Q}|}$, (2.11) and the Young's inequality (2.9). In this way we get

\begin{align*} J'_1 & \leq \int_{B_\tau}\mathop {\int\hskip -1,05em -\,} \limits_{B_\tau}\int_0^{1}|Df(y,{\bf u}_0,{\bf Q}) - Df(y,{\bf u}_0,D{\bf u}(x)\\ &\quad -(1-\theta)D\boldsymbol\psi(x))||D\boldsymbol\psi(x)|\,\mathrm{d}\theta\,\mathrm{d}y\,\mathrm{d}x \\ & \leq c\int_{B_\tau}\int_0^{1}\varphi'_{|{\bf Q}|}(|D\boldsymbol\xi+\theta D\boldsymbol\psi|)|D\boldsymbol\psi|\,\mathrm{d}\theta\,\mathrm{d}x \\ & \leq c \int_{B_\tau}\varphi'_{|{\bf Q}|}(|D\boldsymbol\psi|)|D\boldsymbol\psi|\,\mathrm{d}x + c \int_{B_\tau}\varphi'_{|{\bf Q}|}(|D\boldsymbol\xi|)|D\boldsymbol\psi|\,\mathrm{d}x\\ & \leq c\int_{B_\tau\backslash B_r}(|{\bf V}_{|{\bf Q}|}(D\boldsymbol\psi)|^{2}+|{\bf V}_{|{\bf Q}|}(D\boldsymbol\xi)|^{2})\,\mathrm{d}x. \end{align*}

Recalling the definitions of $\boldsymbol \xi$ and $\boldsymbol \psi$, by a simple computation we find that

\begin{align*} D\boldsymbol\psi &=(1-\eta)(D{\bf u}-{\bf Q})-D\eta\otimes({\bf u}-\boldsymbol\ell), \\ D\boldsymbol\xi &=\eta (D{\bf u}-{\bf Q}) + D\eta\otimes({\bf u}-\boldsymbol\ell), \end{align*}

whence

\begin{align*} &\int_{B_\tau\backslash B_r} (\varphi_{|{\bf Q}|}(|D\boldsymbol\psi|) + \varphi_{|{\bf Q}|}(|D\boldsymbol\xi|))\,\mathrm{d}x\\ &\quad \leq c \int_{B_\tau\backslash B_r} \varphi_{|{\bf Q}|}(|D{\bf u}-{\bf Q}|)\,\mathrm{d}x + c\int_{B_\tau}\varphi_{|{\bf Q}|}\left(\frac{{\bf u}-\boldsymbol\ell}{t-r}\right)\,\mathrm{d}x, \end{align*}

so that combining with the previous estimates we get

\begin{equation*} J_1+J_7\leq c\left(\int_{B_\tau\backslash B_r}|{\bf V}_{|{\bf Q}|}(D{\bf u}-{\bf Q})|^{2}\,\mathrm{d}x+\int_{B_\tau}\left|{\bf V}_{|{\bf Q}|}\left(\frac{{\bf u}-\boldsymbol\ell}{t-r}\right)\right|^{2}\,\mathrm{d}x\right). \end{equation*}

Since $\boldsymbol \xi ={\bf u}-\boldsymbol \ell$ on $B_r$ and $\tau \leq \varrho$, from (3.1) and the estimates for $J_1-J_7$ we obtain

\begin{align*} & \int_{B_r}|{\bf V}_{|{\bf Q}|}(D{\bf u}-{\bf Q})|^{2}\,\mathrm{d}x \\ & \quad \leq \tilde{c}\left(\int_{B_\tau\backslash B_r}|{\bf V}_{|{\bf Q}|}(D{\bf u}-{\bf Q})|^{2}\,\mathrm{d}x+\int_{B_\varrho}\left|{\bf V}_{|{\bf Q}|}\left(\frac{{\bf u}-\boldsymbol\ell}{t-r}\right)\right|^{2}\,\mathrm{d}x\right)\\ & \qquad + \tilde{c} \int_{B_\tau} \left(\omega(|{\bf u}-{\bf u}_0|+|{\bf u}-\boldsymbol\ell|)+{v}_0({\cdot},\tau)\right)\varphi(|{\bf Q}|+|D{\bf u}|)\,\mathrm{d}x. \end{align*}

Now, in a standard way we ‘fill the hole’ thus obtaining

(3.3)\begin{align} & \int_{B_r}|{\bf V}_{|{\bf Q}|}(D{\bf u}-{\bf Q})|^{2}\,\mathrm{d}x \notag\\ & \quad \leq \sigma\int_{B_\tau}|{\bf V}_{|{\bf Q}|}(D{\bf u}-{\bf Q})|^{2}\,\mathrm{d}x+\int_{B_\varrho}\left|{\bf V}_{|{\bf Q}|}\left(\frac{{\bf u}-\boldsymbol\ell}{t-r}\right)\right|^{2}\,\mathrm{d}x\notag\\ & \qquad + \int_{B_\tau} \left(\omega(|{\bf u}-{\bf u}_0|+|{\bf u}-\boldsymbol\ell|)+{v}_0({\cdot},\tau)\right)\varphi(|{\bf Q}|+|D{\bf u}|)\,\mathrm{d}x, \end{align}

where $\sigma :=\frac {\tilde {c}}{\tilde {c}+1}<1$. In order to bound the latter term further, we exploit the higher integrability result of lemma 3.2. Thus, with fixed $s\in (1,s_0]$, as a consequence of Hölder's inequality, the concavity of $\omega$, the bounds $\omega \leq 1$ and ${v}_0\leq 2L$, and Jensen's inequality also we obtain

\begin{align*} & \int_{B_\tau} \left(\omega(|{\bf u}-{\bf u}_0|+|{\bf u}-\boldsymbol\ell|)+{v}_0({\cdot},\tau)\right)\varphi(|{\bf Q}|+|D{\bf u}|)\,\mathrm{d}x\\ & \quad \leq c |B_\tau|\left(\mathop {\int\hskip -1,05em -\,} \limits_{B_\varrho}\omega(|{\bf u}-{\bf u}_0|+|{\bf u}-\boldsymbol\ell|)^{\frac{s}{s-1}}\,\mathrm{d}x+\mathop {\int\hskip -1,05em -\,} \limits_{B_\tau}{v}_0({\cdot},\tau)^{\frac{s}{s-1}}\,\mathrm{d}x\right)^{1-\frac{1}{s}}\\ &\qquad \times\left(\mathop {\int\hskip -1,05em -\,} \limits_{B_\tau}\varphi^{s}(|{\bf Q}|)+\varphi^{s}(|D{\bf u}|)\,\mathrm{d}x\right)^{\frac{1}{s}} \\ & \quad \leq c \tau^{n}\left(\frac{t}{t-r}\right)^{n(s-1)}\left[\omega\left(\mathop {\int\hskip -1,05em -\,} \limits_{B_\varrho}|{\bf u}-{\bf u}_0|+|{\bf u}-\boldsymbol\ell|\,\mathrm{d}x\right)^{1-\frac{1}{s}}+{\mathcal{V}}(\tau)^{1-\frac{1}{s}}\right]\\ &\qquad \times \mathop {\int\hskip -1,05em -\,} \limits_{B_t}\varphi(|{\bf Q}|)+\varphi(|D{\bf u}|)\,\mathrm{d}x \\ & \quad \leq c\left(\frac{\varrho}{t-r}\right)^{n(s-1)}\left[\omega\left(\mathop {\int\hskip -1,05em -\,} \limits_{B_\varrho}|{\bf u}-{\bf u}_0|+|{\bf u}-\boldsymbol\ell|\,\mathrm{d}x\right)^{1-\frac{1}{s}}+{\mathcal{V}}(\tau)^{1-\frac{1}{s}}\right]\\ &\qquad \times \mathop {\int\hskip -1,05em -\,} \limits_{B_{3\varrho/4}}\varphi(|{\bf Q}|)+\varphi(|D{\bf u}|)\,\mathrm{d}x, \end{align*}

where $c=c(n,N,\Delta _2(\varphi ),\nu ,L)$. This estimate, combined with (3.3) gives

\begin{align*} & \int_{B_r}|{\bf V}_{|{\bf Q}|}(D{\bf u}-{\bf Q})|^{2}\,\mathrm{d}x \\ & \quad \leq \sigma\int_{B_t}|{\bf V}_{|{\bf Q}|}(D{\bf u}-{\bf Q})|^{2}\,\mathrm{d}x+c\int_{B_\varrho}\left|{\bf V}_{|{\bf Q}|}\left(\frac{{\bf u}-\boldsymbol\ell}{t-r}\right)\right|^{2}\,\mathrm{d}x\\ & \qquad + c\left(\frac{\varrho}{t-r}\right)^{n(s-1)}\left[\omega\left(\mathop {\int\hskip -1,05em -\,} \limits_{B_\varrho}|{\bf u}-{\bf u}_0|+|{\bf u}-\boldsymbol\ell|\,\mathrm{d}x\right)^{1-\frac{1}{s}}+{\mathcal{V}}(\tau)^{1-\frac{1}{s}}\right]\\ &\qquad \times \mathop {\int\hskip -1,05em -\,} \limits_{B_{3\varrho/4}}\varphi(|{\bf Q}|)+\varphi(|D{\bf u}|)\,\mathrm{d}x, \\ & \quad =: \sigma\int_{B_t}|{\bf V}_{|{\bf Q}|}(D{\bf u}-{\bf Q})|^{2}\,\mathrm{d}x+c\int_{B_\varrho}\left|{\bf V}_{|{\bf Q}|}\left(\frac{{\bf u}-\boldsymbol\ell}{t-r}\right)\right|^{2}\,\mathrm{d}x + c\left(\frac{\varrho}{t-r}\right)^{n(s-1)}\mathcal{U}. \end{align*}

Now, since the previous estimate holds for arbitrary radii $r,t$ such that $\varrho /2\leq r< t\leq 3\varrho /4$, the constant $c$ depends only on $n,N,\Delta _2(\varphi ),\nu ,L$ and $\sigma <1$, as a consequence of lemma 2.6 applied with $\beta :=n(s-1)$ we obtain

(3.4)\begin{equation} \int_{B_{\varrho/2}}|{\bf V}_{|{\bf Q}|}(D{\bf u}-{\bf Q})|^{2}\,\mathrm{d}x \leq c \int_{B_\varrho}\left|{\bf V}_{|{\bf Q}|}\left(\frac{{\bf u}-\boldsymbol\ell}{\varrho}\right)\right|^{2}\,\mathrm{d}x + c\mathcal{U}. \end{equation}

In view of lemma 3.1 applied with $\varrho$ in place of $t-s$ and from (3.2) we get

\begin{equation*} \begin{split} \int_{B_{3\varrho/4}} \varphi(|D{\bf u}|)\,\mathrm{d}x & \leq c \int_{B_{\varrho}}\varphi\left(\left|\frac{{\bf u}-{\bf u}_0}{\varrho}\right|\right)\,\mathrm{d}x \\ & \leq c \int_{B_{\varrho}}\varphi\left(\left|\frac{{\bf u}-{\boldsymbol\ell}}{\varrho}\right|\right)\,\mathrm{d}x + c \varphi(|{\bf Q}|)\\ & \leq c\left[\int_{B_{\varrho}}\left|{\bf V}_{|{\bf Q}|}\left(\frac{{\bf u}-\boldsymbol\ell}{\varrho}\right)\right|^{2}\,\mathrm{d}x + \varphi(|{\bf Q}|)\right], \end{split} \end{equation*}

which combined with (3.4) and using the fact that $\omega \leq 1$ as well as ${\mathcal {V}}(\varrho )\leq 2L$ gives

\begin{align*} &\int_{B_{\varrho/2}}|{\bf V}_{|{\bf Q}|}(D{\bf u}-{\bf Q})|^{2}\,\mathrm{d}x \\ & \quad \leq c \int_{B_\varrho}\left|{\bf V}_{|{\bf Q}|}\left(\frac{{\bf u}-\boldsymbol\ell}{\varrho}\right)\right|^{2}\,\mathrm{d}x \\ & \qquad + c\varrho^{n} \varphi(|{\bf Q}|)\left[\omega\left(\mathop {\int\hskip -1,05em -\,} \limits_{B_\varrho}|{\bf u}-{\bf u}_0|+|{\bf u}-\boldsymbol\ell|\,\mathrm{d}x\right)^{1-\frac{1}{s}}+{\mathcal{V}}(\tau)^{1-\frac{1}{s}}\right], \end{align*}

where $c=c(n,N,\Delta _2(\varphi ),\nu ,L)$. The Caccioppoli inequality then follows by taking means on both sides of the latter inequality.

We can apply lemma 3.4 to affine functions $\boldsymbol \ell _{x_0,r}(x):=({\bf u})_{x_0,\varrho }+{\bf Q}(x-x_0)$ for some ${\bf Q}\in \mathbb {R}^{N\times n}$, and the resulting Caccioppoli inequality can be compared with that of [Reference Celada and Ok7, theorem 3.1]. We notice that, apart from an extra VMO term due to assumption (F4), the dependence of the integrand $f$ also on ${\bf u}$ implies that the remainder term inside $\omega$; i.e.,

(3.5)\begin{equation} R(x_0,\varrho,{\bf u},{\bf Q}):=\displaystyle \mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho}(x_0)}|{\bf u}-({\bf u})_{x_0,\varrho}|+|{\bf u}-\boldsymbol\ell_{x_0,\varrho}|\,\mathrm{d}x \end{equation}

is, in general, non-monotone in the radius $\varrho$. Indeed, it can be estimated from above by the Morrey-type excess

(3.6)\begin{equation} \Theta(x_0,\varrho):=\varrho\varphi^{{-}1}\left(\mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho}(x_0)}\varphi(|D{\bf u}|)\,\mathrm{d}x\right), \end{equation}

which fails to be monotone for small $\varrho$ (lemma 3.5(i)). This does not allow, in general, for an application of Gehring's lemma in order to infer an higher integrability result: for this purpose, a suitable ‘smallness’ regime (3.9) has to be imposed (lemma 3.5(ii)).

Proof. (i) First, from Poincaré inequality and Jensen's inequality we obtain

\begin{equation*} \varphi\left(\mathop {\int\hskip -1,05em -\,} \limits_{B_\varrho(x_0)} \frac{|{\bf u}-({\bf u})_{x_0,\varrho}|}{\varrho}\,\mathrm{d}x\right) \leq \mathop {\int\hskip -1,05em -\,} \limits_{B_\varrho(x_0)} \varphi\left(\frac{|{\bf u}-({\bf u})_{x_0,\varrho}|}{\varrho}\right)\,\mathrm{d}x \leq c\mathop {\int\hskip -1,05em -\,} \limits_{B_\varrho(x_0)} \varphi(|D{\bf u}|)\,\mathrm{d}x, \end{equation*}

whence

\begin{equation*} \mathop {\int\hskip -1,05em -\,} \limits_{B_\varrho(x_0)} |{\bf u}-({\bf u})_{x_0,\varrho}|\,\mathrm{d}x \leq c \Theta(x_0,\varrho). \end{equation*}

Then, recalling the definition of $\boldsymbol \ell _{x_0,r}$, it is immediate to infer the estimate (3.7). As for (3.8), it follows from (3.7) since $\varrho |(D{\bf u})_{x_0,\varrho }|\leq c \Theta (x_0,\varrho )$.

(ii) We note from (φ2) that $\varphi (t^{1/\mu _1})$ is convex for $t\ge 0$. Applying Jensen's inequality, the Poincaré type estimate in theorem 2.5 and the change-shift formula (2.7) with ${\bf a}={\bf 0}$, and using assumption (3.9) , we obtain

\begin{align*} &\varphi\left(\left(\mathop {\int\hskip -1,05em -\,} \limits_{B_\varrho(x_0)} \left[\frac{|{\bf u}-({\bf u})_{x_0,\varrho}|}{\varrho}\right]^{\mu_1}\,\mathrm{d}x\right)^{\frac{1}{\mu_1}}\right) \\ & \quad \leq \mathop {\int\hskip -1,05em -\,} \limits_{B_\varrho(x_0)} \varphi\left(\frac{|{\bf u}-({\bf u})_{x_0,\varrho}|}{\varrho}\right)\,\mathrm{d}x \leq c\mathop {\int\hskip -1,05em -\,} \limits_{B_\varrho(x_0)} \varphi(|D{\bf u}|)\,\mathrm{d}x \\ & \quad \leq c \mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho}(x_0)} \varphi(|D{\bf u}-{\bf Q}|)\,\mathrm{d}x + c \varphi(|{\bf Q}|) \\ & \quad \leq c\mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho}(x_0)} \varphi_{|{\bf Q}|}(|D{\bf u}-{\bf Q}|)\,\mathrm{d}x + c\varphi(|{\bf Q}|) \leq \varphi(c(|{\bf Q}|)), \end{align*}

which yields (3.10) up to applying $\varphi ^{-1}$ to both sides.

Now, we are in position to establish a ‘conditioned’ higher integrability result for $\varphi _{|{\bf Q}|}(|D{\bf u}-{\bf Q}|)$, under the smallness assumption (3.9). The result follows as a consequence of Gehring's lemma with increasing supports (lemma 2.7):

Corollary 3.6 If ${\bf u}\in W^{1,\varphi }(\Omega ;\mathbb {R}^{N})$ is a minimizer of the functional (1.1) under assumptions (F1)–(F7), and ${\bf Q}\in \mathbb {R}^{N\times n}$ is such that (3.9) holds for some $\Lambda \in (0,1]$, then there exist a constant $c=c(n,N,\Delta _2(\varphi ),\nu , L)>0$ and $\sigma >1$ such that

(3.11)\begin{align} & \left(\mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}(x_0)} \varphi_{|{\bf Q}|}^{\sigma}(|D{\bf u}-{\bf Q}|)\,\mathrm{d}x\right)^{\frac{1}{\sigma}} \notag\\ & \quad \leq c \mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho}(x_0)} \varphi_{|{\bf Q}|}(|D{\bf u}-{\bf Q}|)\,\mathrm{d}x+ c\varphi(|{\bf Q}|)\left[\omega\left(\varrho|{\bf Q}|\right)^{1-\frac{1}{s}}+ [{\mathcal{V}}(\varrho)]^{1-\frac{1}{s}}\right] \end{align}

holds for every $s\in (1,s_0]$ where $s_0$ is that of lemma 3.2.

Proof. Let $y\in \Omega$ and $r>0$ be such that $B_{2r}(y)\subset \subset B_\varrho (x_0)$. In view of lemma 3.4 applied with $\varrho =2r$, $x_0=y$, ${\bf u}_0=({\bf u})_{y,2r}$ and an arbitrary ${\bf Q}$, we obtain

(3.12)\begin{equation} \begin{aligned} & \mathop {\int\hskip -1,05em -\,} \limits_{B_{r}(y)} \varphi_{|{\bf Q}|}(|D{\bf u}-{\bf Q}|)\,\mathrm{d}x \leq c \mathop {\int\hskip -1,05em -\,} \limits_{B_{2r}(y)} \varphi_{|{\bf Q}|}\left(\frac{|{\bf u}-({\bf u})_{y,2r}-{\bf Q}(x-y)|}{2r}\right)\,\mathrm{d}x\\ & \qquad + c\varphi(|{\bf Q}|)\Bigg[\omega\left(\mathop {\int\hskip -1,05em -\,} \limits_{B_{2r}(y)}|{\bf u}-({\bf u})_{y,2r}|+ |{\bf Q}||x-y|\, \textrm{d} x\right)^{1-\frac{1}{s}}+ {\mathcal{V}}(2r)^{1-\frac{1}{s}}\Bigg]. \end{aligned} \end{equation}

Here, we observe that

\begin{equation*} \begin{split} \mathop {\int\hskip -1,05em -\,} \limits_{B_{2r}(y)}|{\bf u}-({\bf u})_{y,2r}|+ |{\bf Q}||x-y|\, \textrm{d} x & \le c \mathop {\int\hskip -1,05em -\,} \limits_{B_{2r}(y)}|{\bf u}-({\bf u})_{y,2r}|\, \textrm{d} x + c |{\bf Q}|r \\ & \le c \left(\frac{1}{\varrho|{\bf Q}|}\mathop {\int\hskip -1,05em -\,} \limits_{B_{2r}(y)}|{\bf u}-({\bf u})_{y,2r}|\, \textrm{d} x+1 \right) \varrho |{\bf Q}|, \end{split} \end{equation*}

which, recalling that $\omega (ct)\le c\omega (t)$ when $c\ge 1$ since $\omega$ is concave and $\omega (0)=0$, yields

\begin{align*} &\omega\left(\mathop {\int\hskip -1,05em -\,} \limits_{B_{2r}(y)}|{\bf u}-({\bf u})_{y,2r}|+|{\bf Q}||x-y|\, \textrm{d} x\right)^{1-\frac{1}{s}}\\ &\quad \le c \left(\frac{1}{\varrho|{\bf Q}|}\mathop {\int\hskip -1,05em -\,} \limits_{B_{2r}(y)}|{\bf u}-({\bf u})_{y,2r}|\, \textrm{d} x+1 \right) \omega(\varrho|{\bf Q}|)^{1-\frac{1}{s}}. \end{align*}

Moreover, as $\unicode{x2A0D} _{B_{2r}(y)}{\bf u}-({\bf u})_{y,2r}-{\bf Q}(x-y)\,\textrm{d} x={\bf 0}$, by the Sobolev–Poincaré type inequality (2.12),

\[ \mathop {\int\hskip -1,05em -\,} \limits_{B_{2r}(y)} \varphi_{|{\bf Q}|}\left(\frac{|{\bf u}-({\bf u})_{y,2r}-{\bf Q}(x-y)|}{r}\right)\,\mathrm{d}x \leq c \left(\mathop {\int\hskip -1,05em -\,} \limits_{B_{2r}(y)} \varphi_{|{\bf Q}|}^{\alpha}(|D{\bf u}-{\bf Q}|)\,\mathrm{d}x\right)^{\frac{1}{\alpha}} \]

for some $\alpha \in (0,1)$. Therefore, plugging the preceding two estimates into (3.12) and taking into account that

\begin{equation*} \mathop {\int\hskip -1,05em -\,} \limits_{B_{2r}(y)}|{\bf u}-({\bf u})_{y,2r}|\, \textrm{d} x \leq 2 \mathop {\int\hskip -1,05em -\,} \limits_{B_{2r}(y)}|{\bf u}-({\bf u})_{x_0,\varrho}|\, \textrm{d} x, \end{equation*}

we obtain

\begin{align*} &\mathop {\int\hskip -1,05em -\,} \limits_{B_{r}(y)} \varphi_{|{\bf Q}|}(|D{\bf u}-{\bf Q}|)\,\mathrm{d}x \\ &\quad \leq c \left(\mathop {\int\hskip -1,05em -\,} \limits_{B_{2r}(y)} \varphi_{|{\bf Q}|}^{\alpha}(|D{\bf u}-{\bf Q}|)\,\mathrm{d}x\right)^{\frac{1}{\alpha}}\\ &\qquad + c\frac{\varphi(|{\bf Q}|)\omega(\varrho|{\bf Q}|)^{1-\frac{1}{s}}}{\varrho|{\bf Q}|} \mathop {\int\hskip -1,05em -\,} \limits_{B_{2r}(y)}|{\bf u}-({\bf u})_{x_0,\varrho}|\, \textrm{d} x+c\varphi(|{\bf Q}|)\\ &\qquad \times \left[\omega\left(\varrho|{\bf Q}|\right)^{1-\frac{1}{s}}+ {\mathcal{V}}(\varrho)^{1-\frac{1}{s}}\right].\end{align*}

Now, since ${\bf u} -({\bf u})_{x_0,\varrho }\in L^{\mu _1}(B_{\varrho }(x_0))$, as a consequence of Gehring's lemma there exists $\sigma =\sigma (n,N,\mu _1,\mu _2,\nu ,L)\in (1,\mu _1)$ such that

\[ \begin{aligned} & \left( \mathop {\int\hskip -1,05em -\,} \limits_{B_{\rho/2}(x_0)} \varphi_{|{\bf Q}|}^{\sigma}(|D{\bf u} -{\bf Q}|)\,\mathrm{d}x \right)^{\frac{1}{\sigma}} \leq c \mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho}(x_0)} \varphi_{|{\bf Q}|}(|D{\bf u}-{\bf Q}|)\,\mathrm{d}x\\ & \qquad + c\frac{ \varphi(|{\bf Q}|)\omega(\varrho|{\bf Q}|)^{1-\frac{1}{s}}}{\varrho|{\bf Q}|}\left(\mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho}(x_0)}|{\bf u}-({\bf u})_{x_0,\varrho}|^{\sigma}\, \textrm{d} x\right)^{\frac{1}{\sigma}}\\ &\qquad +c\varphi(|{\bf Q}|)\left[\omega\left(\varrho|{\bf Q}|\right)^{1-\frac{1}{s}}+ {\mathcal{V}}(\varrho)^{1-\frac{1}{s}}\right]. \end{aligned} \]

Finally, applying lemma 3.5 (ii), we obtain (3.11).

We conclude this section by introducing the excess functional and other tools useful in the sequel. Let $\boldsymbol L_{x_0,\varrho }:\mathbb {R}^{n}\to \mathbb {R}^{N}$ be the affine function associated to ${\bf u}$ defined as

(3.13)\begin{equation} \boldsymbol L_{x_0,\varrho}(x):=({\bf u})_{x_0,\varrho} + {\bf Q}_{x_0,\varrho}(x-x_0), \end{equation}

where ${\bf Q}_{x_0,\varrho }:=(D{\bf u})_{x_0,\varrho }$. For $x_0\in \Omega$ and $\varrho \in (0,\textrm {dist}(x_0,\partial \Omega ))$, $\varrho \leq 1$, we define the excess functional as

(3.14)\begin{equation} \Phi(x_0,\varrho)\equiv\Phi(x_0,\varrho, \boldsymbol L_{x_0,\varrho}):=\mathop {\int\hskip -1,05em -\,} \limits_{B_\varrho(x_0)}\varphi_{|({D{\bf u}})_{x_0,\varrho}|}(|D{\bf u}-({D{\bf u}})_{x_0,\varrho}|)\,\mathrm{d}x \end{equation}

and

(3.15)\begin{equation} \Psi(x_0,\varrho):=\mathop {\int\hskip -1,05em -\,} \limits_{B_\varrho(x_0)}\varphi\left(\frac{|{\bf u}-({\bf u})_{x_0,\varrho}|}{\varrho}\right)\,\mathrm{d}x. \end{equation}

Moreover, we define also

(3.16)\begin{equation} H(x_0,\varrho):=\frac{1}{1+(2L)^{1-\frac{1}{s}}}\left([\omega(\varrho|(D{\bf u})_{x_0,\varrho}|)]^{1-\frac{1}{s}} + [{\mathcal{V}}(\varrho)]^{1-\frac{1}{s}}\right), \end{equation}

and

(3.17)\begin{equation} \widetilde{H}(x_0,\varrho):=\frac{1}{1+(2L)^{1-\frac{1}{s}}}\left([\omega(\Theta(x_0,\varrho))]^{1-\frac{1}{s}} + [{\mathcal{V}}(\varrho)]^{1-\frac{1}{s}}\right), \end{equation}

where $s\in (1,s_0]$ is the exponent of lemma 3.3 and $\Theta (x_0,\varrho )$ is the excess defined in (3.6). Since $\omega \leq 1$ and ${\mathcal {V}}(\varrho )\leq 2L$, we have that $H(x_0,\varrho ),\widetilde {H}(x_0,\varrho )\leq 1$, and

\begin{equation*} H(x_0,\varrho)\leq c\widetilde{H}(x_0,\varrho) \end{equation*}

as a consequence of lemma 3.5(i). Under the smallness assumption $\Phi (x_0,\varrho )\leq \Lambda \varphi (|(D{\bf u})_{x_0,\varrho }|)$, by virtue of lemma 3.5(ii) there exists a constant $\tilde {c}=\tilde {c}(\varphi )$ such that

\begin{equation*} \frac{1}{\tilde{c}}\widetilde{H}(x_0,\varrho)\leq H(x_0,\varrho)\leq c\widetilde{H}(x_0,\varrho). \end{equation*}

We can rewrite the Caccioppoli inequality (3.11) as

(3.18)\begin{equation} \Phi(x_0,\varrho/2) \leq c \Phi(x_0,\varrho) + c\varphi(|(D{\bf u})_{x_0,\varrho}|)H(x_0,\varrho). \end{equation}

Note also that by (2.6) and, e.g., [Reference Diening, Kaplický and Schwarzacher13, lemma A.2] we have the following equivalence:

\begin{equation*} \Phi(x_0,\varrho) \sim \mathop {\int\hskip -1,05em -\,} \limits_{B_\varrho(x_0)}|{\bf V}(D{\bf u})-{\bf V}((D{\bf u})_{x_0,\varrho})|^{2}\,\mathrm{d}x \sim \mathop {\int\hskip -1,05em -\,} \limits_{B_\varrho(x_0)}|{\bf V}(D{\bf u})-({\bf V}(D{\bf u}))_{x_0,\varrho}|^{2}\,\mathrm{d}x. \end{equation*}

In the case $x_0=0$, we will use the shorthands $\Phi (\varrho )$, $\Psi (\varrho )$, $\Theta (\varrho )$, $H(\varrho )$ and $\widetilde {H}(\varrho )$ in place of $\Phi (0,\varrho )$, $\Psi (0,\varrho )$, $\Theta (0,\varrho )$, $H(0,\varrho )$ and $\widetilde {H}(0,\varrho )$, respectively.

3.2 Comparison maps via Ekeland's variational principle

The proof of the main results will require suitable comparison functions, which will be constructed with a freezing argument in the variables $(x,{\bf u})$ based on Ekeland's variational principle. We recall below a version of this classical tool, whose proof can be found, e.g., in [Reference Giusti25, theorem 5.6].

Lemma 3.7 (Ekeland's principle)

Let $(X,d)$ be a complete metric space, and assume that $F : X\to [0,\infty ]$ be not identically $\infty$ and lower semicontinuous with respect to the metric topology on $X$. If for some $u\in X$ and some $\kappa >0$, there holds

\begin{equation*} F(u)\leq \inf_XF + \kappa, \end{equation*}

then there exists $v\in X$ with the properties

\begin{equation*} d(u,v)\leq 1 \mbox{ and } F(v)\leq F(w)+\kappa d(v,w) \quad \forall w\in X. \end{equation*}

Although a similar analysis in the Orlicz setting, for integrands $f=f(x,\boldsymbol \xi )$, has been performed in [Reference Celada and Ok7, theorem 3.3], we will follow a quite different argument, which refers to the case of $p$-growth as in [Reference Bögelein4, lemma 3.7]. We will also specify the appropriate complete metric space $X$, which is not explicitly mentioned in [Reference Celada and Ok7, theorem 3.3].

To this aim, let $B_\varrho (x_0)\subseteq \Omega$ with $\varrho \leq \varrho _0$ and set

(3.19)\begin{equation} g(\boldsymbol\xi) \equiv g_{{x}_0,\varrho}(\boldsymbol\xi) := (f({\cdot}, ({\bf u})_{{x}_0,\varrho},\boldsymbol\xi))_{{x}_0,\varrho} \quad \mbox{ for all} \boldsymbol\xi\in\mathbb{R}^{N\times n}, \end{equation}

and

(3.20)\begin{equation} K(x_0,\varrho):=\widetilde{H}(x_0,\varrho)\Psi(x_0,\varrho) \end{equation}

where $\widetilde {H}(x_0,\varrho )$ and $\Psi (x_0,\varrho )$ are defined as in (3.17) and (3.15), respectively.

As for the complete metric space $(X,d)$, following [Reference Ok36, lemma 4.4] we consider

\begin{equation*} X:=\left\{{\bf w}\in {\bf u}+W^{1,1}_0(B_{\varrho/2}(x_0)):\, \mathop {\int\hskip -1,05em -\,} \limits_{B_\varrho/2(x_0)}\varphi(|D{\bf w}|)\,\mathrm{d}x\leq \mathop {\int\hskip -1,05em -\,} \limits_{B_\varrho/2(x_0)}\varphi(|D{\bf u}|)\,\mathrm{d}x\right\} \end{equation*}

with the metric

\begin{align*} &d({\bf w}_1,{\bf w}_2):= \frac{1}{c_*\varphi^{{-}1}(K(\varrho))}\mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}(x_0)}|D{\bf w}_1-D{\bf w}_2|\,\mathrm{d}x, \end{align*}

for ${\bf w}_1,{\bf w}_2\in{\bf u}+W_0^{1,1}(B_{\varrho/2(x_0)},\mathbb{R}^{N})$, and note that the functional

(3.21)\begin{equation} \mathcal{G}[{\bf w}]:= \mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}(x_0)}g(D{\bf w})\,\mathrm{d}x \quad \mbox{ in } {\bf u}+W^{1,1}_0(B_{\varrho/2}(x_0),\mathbb{R}^{N}), \end{equation}

is lower semicontinuous in the metric topology. We would get a comparison map ${\bf v}\in {\bf u} + W^{1,1}_0(B_{\varrho /2}({x}_0),\mathbb {R}^{N})$ by proving the following lemma.

Lemma 3.8 Assume that ${\bf u}\in W^{1,\varphi }(\Omega ,\mathbb {R}^{N})$ is a minimizer of the functional (1.1), under assumptions (F1)–(F6). Then there exists a minimizer ${\bf v}\in {\bf u} + W^{1,1}_0(B_{\varrho /2}({x}_0),\mathbb {R}^{N})$ of the functional

\begin{equation*} \widetilde{\mathcal{G}}[{\bf w}]:= \mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}(x_0)} g(D{\bf w})\,\mathrm{d}x + \frac{K(x_0,\varrho)}{\varphi^{{-}1}(K(x_0,\varrho))}\mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}(x_0)}|D{\bf v}-D{\bf w}|\,\mathrm{d}x, \end{equation*}

that satisfies

(3.22)\begin{equation} \mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}(x_0)}|D{\bf v}-D{\bf u}|\,\mathrm{d}x \leq c_*\varphi^{{-}1}(K(x_0,\varrho)) \end{equation}

for some constant $c_*=c_*(n,N,\Delta _2(\varphi ),\nu ,L)$. Moreover, ${\bf v}$ fulfils the following Euler-Lagrange variational inequality:

(3.23)\begin{equation} \left|\mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}(x_0)}\langle Dg(D{\bf v})|D{\boldsymbol\eta}\rangle\,\mathrm{d}x\right|\leq \frac{K(x_0,\varrho)}{\varphi^{{-}1}(K(x_0,\varrho))}\mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}(x_0)}|D\boldsymbol\eta|\,\mathrm{d}x \end{equation}

for every $\boldsymbol \eta \in C^{\infty }_0(B_{\varrho /2}(x_0),\mathbb {R}^{N})$.

Proof. We may assume, without loss of generality, that $x_0=0$ and, correspondingly, we use the shorthand $K(\varrho )$ for $K(0,\varrho )$. As a first remark, we recall that from lemma 3.1 with $r=\frac {3}{4}\varrho$ we have

(3.24)\begin{equation} \mathop {\int\hskip -1,05em -\,} \limits_{B_{3\varrho/4}} \varphi(|D{\bf u}|)\,\mathrm{d}x\leq c \mathop {\int\hskip -1,05em -\,} \limits_{B_\varrho}\varphi\left(\frac{|{\bf u} - ({\bf u})_\varrho|}{\varrho}\right)\,\mathrm{d}x = c\Psi(\varrho), \end{equation}

where $c=c(\Delta _2(\varphi ),L,\nu )$. We then denote by ${\tilde {\boldsymbol {v}}{\in }} X$ a minimizer of the functional (3.21) whose existence is ensured by the direct method under assumptions (F1)–(F2). From the minimality of $\tilde {\boldsymbol {v}}$, assumption (F1) and (2.10) we get

(3.25)\begin{equation} \begin{split} \mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}}\varphi(|D\tilde{\boldsymbol{v}}|)\,\mathrm{d}x & \leq\frac{1}{\nu}\mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}}g(D\tilde{\boldsymbol{v}})-g({\bf 0})\,\mathrm{d}x\\ & \leq\frac{1}{\nu}\mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}}g(D{\bf u})-g({\bf 0})\,\mathrm{d}x\leq\frac{c(\varphi)L}{\nu}\mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}}\varphi(|D{\bf u}|)\,\mathrm{d}x. \end{split} \end{equation}

By the sublinearity of $\varphi$, the Poincaré inequality (theorem 2.5), Jensen's inequality and (3.24) this gives

\begin{align*} & \varphi\left(\mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}} \frac{|\tilde{\boldsymbol{v}}-({\bf u})_\varrho|}{\varrho}\,\mathrm{d}x\right) \\ & \quad \leq c \left(\mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}} \varphi\left(\frac{|\tilde{\boldsymbol{v}}-{\bf u}|}{\varrho}\right)\,\mathrm{d}x + \mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}} \varphi\left(\frac{|{\bf u}-({\bf u})_\varrho|}{\varrho}\right)\,\mathrm{d}x\right) \\ & \quad \leq c\left[\left(\mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}} \varphi^{\alpha}(|D\tilde{\boldsymbol{v}}-D{\bf u}|)\,\mathrm{d}x\right)^{\frac{1}{\alpha}} + \left(\mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}} \varphi^{\alpha}(|D{\bf u}|)\,\mathrm{d}x\right)^{\frac{1}{\alpha}}\right] \\ & \quad \leq c \mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}} \varphi(|D\tilde{\boldsymbol{v}}|) + \varphi(|D{\bf u}|)\,\mathrm{d}x \\ & \quad \leq c \mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}}\varphi(|D{\bf u}|)\,\mathrm{d}x, \end{align*}

whence

(3.26)\begin{equation} \mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}}|\tilde{\boldsymbol{v}}-({\bf u})_\varrho|\,\mathrm{d}x\leq c \Theta(\varrho) \end{equation}

where $c=c(\varphi ,n,L,\nu )$. Moreover, as a consequence of the higher integrability results of both lemmas 3.2 and 3.3, together with (3.25) and (3.24), we infer the higher integrability result

(3.27)\begin{equation} \left(\mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}} \varphi^{s}(|D\tilde{\boldsymbol{v}}|)\,\mathrm{d}x\right)^{\frac{1}{s}} \leq c \mathop {\int\hskip -1,05em -\,} \limits_{B_{3\varrho/4}}\varphi(|D{\bf u}|)\,\mathrm{d}x\leq c\Psi(\varrho), \end{equation}

where $c=c(n,N,\varphi ,\nu ,L)$ and $s=s(n,N,\varphi , \nu , L)\in (1,s_0]$.

Now we prove that ${\bf u}$ is an almost minimizer of the functional $\mathcal {G}$. Indeed, from the minimality of ${\bf u}$ and assumptions (F4), (F3) we get

\begin{align*} \mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}}f(x,{\bf u},D{\bf u})\,\mathrm{d}x - \mathcal{G}[\tilde{\boldsymbol{v}}] & \leq \mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}}f(x,\tilde{\boldsymbol{v}},D\tilde{\boldsymbol{v}})\,\mathrm{d}x - \mathcal{G}[\tilde{\boldsymbol{v}}] \\ & = \mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}}f(x,\tilde{\boldsymbol{v}},D\tilde{\boldsymbol{v}})-(f({\cdot},\tilde{\boldsymbol{v}},D\tilde{\boldsymbol{v}}))_\varrho\,\mathrm{d}x \\ & \quad + \mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}} ( f({\cdot},\tilde{\boldsymbol{v}},D\tilde{\boldsymbol{v}}))_\varrho - (f({\cdot},({\bf u})_\varrho,D\tilde{\boldsymbol{v}}))_\varrho\,\mathrm{d}x \\ & \leq c(L)\mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}}[{v}_0({\cdot},\varrho)+\omega(|\tilde{\boldsymbol{v}}-({\bf u})_\varrho|)] \varphi(|D\tilde{\boldsymbol{v}}|)\,\mathrm{d}x. \end{align*}

Then, by using Jensen's inequality, the concavity and sub-linearity of $\omega$, (3.26) and (3.27), from the previous estimate we obtain

(3.28)\begin{align} & \mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}}f(x,{\bf u},D{\bf u})\,\mathrm{d}x - \mathcal{G}[\tilde{\boldsymbol{v}}] \notag\\ & \quad \leq c\left[\omega\left(\mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}}|\tilde{\boldsymbol{v}}-({\bf u})_\varrho|\,\mathrm{d}x\right)^{1-\frac{1}{s}}+[{\mathcal{V}}(\varrho)]^{1-\frac{1}{s}}\right]\left(\mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}}\varphi^{s}(|D\tilde{\boldsymbol{v}}|)\,\mathrm{d}x\right)^{\frac{1}{s}}\notag\\ & \quad \leq c \left[\omega(\Theta(\varrho))^{1-\frac{1}{s}}+[{\mathcal{V}}(\varrho)]^{1-\frac{1}{s}}\right]\Psi(\varrho) = c K(\varrho), \end{align}

where $c=c(n,N,\Delta _2(\varrho ), \nu , L)$. Arguing similarly, we can estimate

(3.29)\begin{align} \mathcal{G}[{\bf u}]-\mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}}f(x,{\bf u},D{\bf u})\,\mathrm{d}x & = \mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}}\left [(f({\cdot},({\bf u})_\varrho,D{\bf u}))_\varrho - f(x,({\bf u})_\varrho,D{\bf u})\right]\,\mathrm{d}x\notag\\ &\quad + \mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}} \left[f(x,({\bf u})_\varrho,D{\bf u}) - f(x,{\bf u},D{\bf u})\right]\,\mathrm{d}x \notag\\ & \leq c \left[\omega(\Theta(\varrho))^{1-\frac{1}{s}}+[{\mathcal{V}}(\varrho)]^{1-\frac{1}{s}}\right]\Psi(\varrho) = c K(\varrho), \end{align}

where the constant $c$ has the same dependencies as before. Adding term by term (3.28)–(3.29) and taking into account the minimality of $\tilde {\boldsymbol {v}}$, we infer

\begin{equation*} \mathcal{G}[{\bf u}]\leq \mathcal{G}[\tilde{\boldsymbol{v}}] + c_*K(\varrho) = \min_{{\bf u}+W_0^{1,1}(B_{\varrho/2},\mathbb{R}^{N})} \mathcal{G} + c_*K(\varrho), \end{equation*}

for a constant $c_*=c_*(n, N, \Delta _2(\varphi ), \nu , L)$. Finally, Ekeland's variational principle (lemma 3.7) with the choice $\kappa =c_*K(\varrho )$ provides the existence of a function ${{\bf v}\in X}$ with the desired property of minimality for the functional $\widetilde {\mathcal {G}}$ and such that ${d({\bf u},{\bf v})\leq 1}$, which corresponds to (3.22). The inequality (3.23) follows from the validity of the associated Euler-Lagrange variational inequality for ${\bf v}$ in a standard way.

3.3 Approximate $\mathcal {A}$-harmonicity and $\varphi$-harmonicity

In this section, we provide two different linearization strategies for the minimization problem, along the lines of [Reference Bögelein4, section 3.2], where an analogous analysis has been performed for functionals with $p$-growth. On the one hand, with lemma 3.9 we will show that the minimizer ${\bf u}$ of $\mathcal {F}$ is an almost $\mathcal {A}$-harmonic function for a suitable elliptic bilinear form $\mathcal {A}$. On the other hand, this ${\bf u}$ turns out to be an almost $\varphi$-harmonic function (see lemma 3.10). These results will allow us to apply the $\mathcal {A}$-harmonic approximation lemma, respectively the $\varphi$-harmonic approximation lemma. The proof will require, in both cases, the comparison maps obtained with lemma 3.8.

We start by proving the approximate $\mathcal {A}$-harmonicity of a minimizer to (1.1). To this aim, only assumptions (F1)–(F6) are required on $f$.

Let $\boldsymbol L_{x_0,\varrho }$ be the affine function associated to ${\bf u}$ as in (3.13), which complies with $\boldsymbol L_{x_0,\varrho }(x_0)=({\bf u})_{x_0,\varrho }$ and $D\boldsymbol L_{x_0,\varrho }=(D{\bf u})_{x_0,\varrho }=:{\bf Q}_{x_0,\varrho }$. We set

\begin{equation*} \mathcal{A}:=\frac{D^{2}g((D{\bf u})_{x_0,\varrho})}{\varphi''(|(D{\bf u})_{x_0,\varrho}|)}\equiv \frac{\left(D^{2}f({\cdot}, ({\bf u})_{x_0,\varrho},(D{\bf u})_{x_0,\varrho})\right)_{x_0,\varrho}}{\varphi''(|(D{\bf u})_{x_0,\varrho}|)}. \end{equation*}

We point out that $\mathcal {A}$ defined above is a bilinear form on $\mathbb {R}^{N\times n}$, satisfying the ellipticity assumption (2.13) by virtue of (F2) and (F3).

Lemma 3.9 Let ${\bf u}\in W^{1,\varphi }(\Omega ,\mathbb {R}^{N})$ be a minimizer of the functional (1.1), under assumptions (F1)–(F6), and assume that for a ball $B_\varrho (x_0)\subseteq \Omega$ the non-degeneracy assumptions

\begin{equation*} \Phi(x_0,\varrho)\leq \varphi(|(D{\bf u})_{x_0,\varrho}|)\quad \mbox{ and }\quad \varrho\leq 1, \end{equation*}

are satisfied. Then, ${\bf u}$ is approximately $\mathcal {A}$-harmonic on the ball $B_{\varrho /2}(x_0)$, in the sense that there exists $\beta _1=\beta _1(n,N,\mu _1,\mu _2,\nu ,L,\beta _0)\in (0,\frac {1}{2})$ such that

(3.30)\begin{align} & \left|\mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}(x_0)} \langle\mathcal{A}(D{\bf u}-(D{\bf u})_{x_0,\varrho})|D\boldsymbol\eta\rangle\,\mathrm{d}x\right| \notag\\ & \quad \leq c |(D{\bf u})_{x_0,\varrho}|\|D\boldsymbol\eta\|_\infty\left\{[{H}(x_0,\varrho)]^{\beta_1}+\frac{\Phi(x_0,\varrho)}{\varphi(|(D{\bf u})_{x_0,\varrho}|)}+\left(\frac{\Phi(x_0,\varrho)}{\varphi(|(D{\bf u})_{x_0,\varrho}|)}\right)^{\frac{1+\beta_0}{2}}\right\} \end{align}

holds for every $\boldsymbol \eta \in C^{\infty }_c(B_{\varrho /2}(x_0),\mathbb {R}^{N})$ for some constant $c = c(n,N,\mu _1, \mu _2,\nu , c_0, L)>0$, where $\mu _1,\mu _2$ are the characteristics of $\varphi$ and $c_0,\beta _0$ are the constants of assumption (F6).

If, in addition, $f$ complies also with (F7), we can show that each local minimizer of the functional $\mathcal {F}({\bf u})$ (eq. (1.1)) is almost $\varphi$-harmonic.

For this, we preliminarily note (see [Reference Celada and Ok7, eq. (4.19)–(4.20)]) that assumption (F7) implies the following:

(3.31)\begin{align} \hbox{for all}\ \delta>0, \mbox{exists } \sigma=\sigma(\delta)>0\ \mbox{such that }\left|Dg({\bf P})-\frac{{\bf P}}{|{\bf P}|}\varphi'(|{\bf P}|)\right|\leq \delta \varphi'(|{\bf P}|), \end{align}

for every ${\bf P}\in \mathbb {R}^{N\times n}$ with $0<|{\bf P}|\leq \sigma$, where the function $g$ has been introduced in (3.19).

We then have the following result.

Lemma 3.10 Let ${\bf u}\in W^{1,\varphi }_\textrm {loc}(\Omega ,\mathbb {R}^{N})$ be a local minimizer of the functional (1.1), and assume that $f$ complies also with (F7). Then there exists $\beta _2=\beta _2(n,N,\mu _1,\mu _2,c_0,L)\in (0,\frac {1}{2})$ such that, for every $\delta >0$ and for $\sigma =\sigma (\delta )>0$ given by (3.31), the inequality

\begin{align*} & \left|\mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}(x_0)}\left\langle\frac{\varphi'(|D{\bf u}|)}{|D{\bf u}|}D{\bf u}\bigg|D\boldsymbol\eta\right\rangle\,\mathrm{d}x\right| \\ & \quad \leq c\left(\delta+[\widetilde{H}(x_0,\varrho)]^{\beta_2}+\frac{\varphi^{{-}1}(\Psi(x_0,\varrho))}{\sigma}\right) \left(\mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho}(x_0)}\varphi(|D{\bf u}|)\,\mathrm{d}x+\varphi(\|D\boldsymbol\eta\|_\infty)\right) \end{align*}

holds for every $\boldsymbol \eta \in C^{\infty }_c(B_{\varrho /2}(x_0),\mathbb {R}^{N})$ for some constant $c=c(n,N,\mu _1,\mu _2,c_0, \nu ,L)>0$.

3.4 Excess decay estimates: the non-degenerate regime

We start by establishing excess improvement estimates in the non-degenerate regime characterized by (3.33) below, i.e. the fact that $\Phi (x_0,\varrho )\leq c\varphi (|(D{\bf u})_{x_0,\varrho }|)$. The strategy of the proof is to exploit lemma 3.9 to approximate the given minimizer by $\mathcal {A}$-harmonic functions, for which suitable decay estimates are available from theorem 2.9.

We introduce the hybrid excess functional

(3.32)\begin{equation} \Phi_*(x_0,\varrho):=\Phi(x_0,\varrho)+\varphi(|({D{\bf u}})_{x_0,\varrho}|)[H(x_0,\varrho)]^{\beta_1}, \end{equation}

where $\beta _1$ is the exponent of lemma 3.9. Since $\beta _1<1/2$ and $H(x_0,\varrho )\leq 1$, we deduce, in particular, that $H(x_0,\varrho )\leq [H(x_0,\varrho )]^{\beta _1}$. Thus, the Caccioppoli inequality (3.18) can be re-read as

\begin{equation*} \Phi(x_0,\varrho/2)\leq c\Phi_*(x_0,\varrho), \end{equation*}

where $c=c(n,N,\mu _1,\mu _2,\nu ,L)$.

Lemma 3.11 For every $\varepsilon \in (0,1)$ there exist $\delta _1,\delta _2\in (0,1]$, where $\delta _i=\delta _i(n,N,\mu _1,\mu _2, \beta _0, \nu ,L,\varepsilon )$, $i=1,2$, with the following property: if

(3.33)\begin{align} &\frac{\Phi(x_0,\varrho)}{\varphi(|(D{\bf u})_{x_0,\varrho}|)}\leq \delta_1 \end{align}

(3.34)\begin{align} &[H(x_0,\varrho)]^{\beta_1}\leq \delta_2 \end{align}

then the excess improvement estimate

(3.35)\begin{equation} \Phi(x_0,\vartheta\varrho)\leq c_\textrm{dec}\vartheta^{2}\left[1+\frac{\varepsilon}{\vartheta^{n+2}}\right]\Phi_*(x_0,\varrho) \end{equation}

holds for every $\vartheta \in (0,1)$ for some constant $c_\textrm {dec}=c_\textrm {dec}(n,N,\mu _1,\mu _2,\nu ,L,c_1)>0$, where $\Phi _*$ is defined in (3.32).

Proof. The proof follows the argument of [Reference Celada and Ok7, lemma 4.2]. We emphasize that corollary 3.6 is crucial in order to obtain the estimate

\begin{align*} &\left(\mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}}\left[\frac{\varphi_{|{\bf Q}_{\varrho}|}(|D{\bf u}-{\bf Q}_{\varrho}|)}{\varphi(|{\bf Q}_{\varrho}|)}\right]^{s_0}\,\mathrm{d}x\right)^{\frac{1}{s_0}} \\ & \quad \leq \frac{c}{\varphi(|{\bf Q}_{\varrho}|)}\mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho}}\varphi_{|{\bf Q}_{\varrho}|}(|D{\bf u}-{\bf Q}_{\varrho}|)\,\mathrm{d}x + c[H(\varrho)]^{\beta_1}\\ & \quad \leq {c} \frac{\Phi_*(\varrho)}{\varphi(|{\bf Q}_{\varrho}|)}, \end{align*}

which comes into play in applying the $\mathcal {A}$-harmonic approximation theorem in the modified version of remark 2.10.

Lemma 3.12 Let $\vartheta \in (0,1)$, and assume that

(3.36)\begin{equation} \frac{\Phi(x_0,\varrho)}{\varphi(|(D{\bf u})_{x_0,\varrho}|)}\leq \frac{\vartheta^{n}}{2^{\mu_2+1} c_{\mu_2}}, \end{equation}

where $c_{\mu _2}$ is the constant of the change of shift formula (2.7) with $\eta =\frac {1}{2^{\mu _2+1}}$. Then it holds that

(3.37)\begin{equation} |(D{\bf u})_{x_0,\varrho}|\leq 2|(D{\bf u})_{x_0,\vartheta\varrho}|. \end{equation}

Proof. As a consequence of (2.7) for $\eta =\frac {1}{2^{\mu _2+1}}$ and with (3.36) we get

\begin{equation*} \begin{split} \varphi(|(D{\bf u})_{x_0,\varrho}-(D{\bf u})_{x_0,\vartheta\varrho}|) & \leq \mathop {\int\hskip -1,05em -\,} \limits_{B_{\vartheta\varrho}(x_0)} \varphi(|D{\bf u} - (D{\bf u})_{x_0,\varrho}|)\,\mathrm{d}x \\ & \leq c_{\mu_2}\vartheta^{{-}n}\Phi(x_0,\varrho) + \frac{1}{2^{\mu_2+1}}\varphi(|(D{\bf u})_{x_0,\varrho}|) \\ & \leq \frac{1}{2^{\mu_2}} \varphi(|(D{\bf u})_{x_0,\varrho}|), \end{split} \end{equation*}

whence, passing to $\varphi ^{-1}$ and taking into account (2.2), we obtain

\begin{equation*} |(D{\bf u})_{x_0,\varrho}-(D{\bf u})_{x_0,\vartheta\varrho}| \leq \frac{1}{2}|(D{\bf u})_{x_0,\varrho}|. \end{equation*}

Now,

\begin{equation*} \begin{split} |(D{\bf u})_{x_0,\varrho}| \leq |(D{\bf u})_{x_0,\varrho}-(D{\bf u})_{x_0,\vartheta\varrho}|+|(D{\bf u})_{x_0,\vartheta\varrho}| \leq \frac{1}{2}|(D{\bf u})_{x_0,\varrho}| + |(D{\bf u})_{x_0,\vartheta\varrho}|, \end{split} \end{equation*}

whence (3.37) follows by re-absorbing the first term of the right-hand side into the left.

The excess-decay estimate (3.35) can be iterated, as the non-degeneracy conditions (3.33)–(3.34) are also satisfied on any smaller ball $B_{\vartheta ^{m}\varrho }(x_0)$, $m\in \mathbb {N}$, $\vartheta <1$.

Lemma 3.13 Let $\Phi (x_0,\varrho )$ and $\Theta (x_0,\varrho )$ be defined as in (3.14) and (3.6), respectively. Then there exist constants $\delta _*$, $\varepsilon _*$, $\varrho _*\in (0,1]$ and $\vartheta$ such that the following holds: if the conditions

(3.38)\begin{equation} \frac{\Phi(x_0,\varrho)}{\varphi(|(D{\bf u})_{x_0,\varrho}|)}\leq \varepsilon_* \quad \mbox{ and }\quad \Theta(x_0,\varrho)\leq\delta_*. \end{equation}

hold on $B_\varrho (x_0)\subseteq \Omega$ for $\varrho \in (0,\varrho _*]$, then

(3.39)\begin{equation} \frac{\Phi(x_0,\vartheta^{m}\varrho)}{{\varphi(|(D{\bf u})_{x_0,\vartheta^{m}\varrho}|)}}\leq \varepsilon_* \quad \mbox{ and }\quad \Theta(x_0,\vartheta^{m}\varrho)\leq\delta_* \end{equation}

for every $m=0,1,\dots .$. As a consequence, for any $\alpha \in (0,1)$ the following Morrey-type estimate holds:

(3.40)\begin{equation} \Theta(y,r)\leq c\delta_*\left(\frac{r}{\varrho}\right)^{\alpha} \end{equation}

for all $y\in B_{\varrho /2}(x_0)$ and $r\in (0,\varrho /2]$.

Proof. As usual, we omit the explicit dependence on $x_0$. Let $\vartheta \in (0,1)$ be such that

(3.41)\begin{equation} \vartheta\leq \min\left\{(6c_\textrm{dec}2^{\mu_2})^{-\frac{1}{2}},\frac{1}{2^{\mu_2}},\frac{1}{2^{\frac{\mu_2}{\mu_1(1-\alpha)}}}\right\}, \end{equation}

where $c_\textrm {dec}$ is the constant of lemma 3.11 depending only on $n,N,\mu _1,\mu _2,\nu ,L,c_0$. Correspondingly, let $\delta _i=\delta _i(n,N,\mu _1,\mu _2,\beta _0,\nu ,L,\vartheta )$, $i=1,2$ be the constants of lemma 3.11, applied with the choice $\varepsilon =\vartheta ^{n+2}$. We choose $\varepsilon _*>0$ such that

(3.42)\begin{equation} \varepsilon_*\leq\min\left\{\frac{\delta_1}{3}, \frac{\delta_2}{2}, \frac{\vartheta^{n}}{\max\{2c_{\frac{1}{2}},2^{\mu_2+1} c_{\mu_2}\}}\right\}, \end{equation}

where $c_{\frac {1}{2}}$ is the constant in the change-shift formula (2.7) with $\eta =\frac {1}{2}$, and we fix the constant $\delta _*>0$ so small that

(3.43)\begin{equation} \left(\frac{\omega(\delta_*)^{1-\frac{1}{s}}}{1+(2L)^{1-\frac{1}{s}}}\right)^{\beta_1}<\varepsilon_*. \end{equation}

Moreover, we choose a radius $\varrho _*>0$ such that

(3.44)\begin{equation} \varrho_*\leq 1 \qquad \mbox{and} \qquad \left(\frac{\mathcal{V}(\varrho_*)^{1-\frac{1}{s}}}{1+(2L)^{1-\frac{1}{s}}}\right)^{\beta_1}<\varepsilon_*. \end{equation}

As a consequence, $\varepsilon _*$, $\delta _*$ and $\varrho _*$ have the same dependencies as $\delta _1$, $\delta _2$. In addition, $\delta _*$ depends also on $\omega$, while $\varrho _*$ also on $\omega$ and $\mathcal {V}$.

We argue by induction on $m$. Since (3.39) are trivially true for $m=0$ by assumption (3.38), our aim is to show that if (3.39) holds for some $m\geq 1$, then the corresponding inequalities hold with $m+1$ in place of $m$. Setting

\begin{equation*} E(B_{\vartheta^{m}\varrho}):=\mathop {\int\hskip -1,05em -\,} \limits_{B_{\vartheta^{m}\varrho}}\varphi(|D{\bf u}|)\,\mathrm{d}x, \end{equation*}

in order to prove the second inequalities in (3.39) it will suffice to show that

(3.45)\begin{equation} E(B_{\vartheta^{m}\varrho})\leq \varphi\left(\frac{\delta_*}{\vartheta^{m}\varrho}\right). \end{equation}

We have, with (3.39) at step $m$, the shift-change formula (2.7) with $\eta =\frac {1}{2}$ and (3.42), the estimate

(3.46)\begin{equation} \begin{split} E(B_{\vartheta^{m+1}\varrho}) & \leq 2^{\mu_2-1}\left(c_{\frac{1}{2}}\vartheta^{{-}n} \Phi(\vartheta^{m}\varrho) + \frac{1}{2}\varphi(|(D{\bf u})_{\vartheta^{m}\varrho}|)+ \varphi(|(D{\bf u})_{\vartheta^{m}\varrho}|)\right)\\ & \leq 2^{\mu_2-1}\left(c_{\frac{1}{2}}\vartheta^{{-}n} \Phi(\vartheta^{m}\varrho) +\frac{3}{2} E(B_{\vartheta^{m}\varrho})\right) \\ & \leq 2^{\mu_2-1}\left(c_{\frac{1}{2}}\vartheta^{{-}n}\varepsilon_*+\frac{3}{2}\right) E(B_{\vartheta^{m}\varrho})\\ & \leq 2^{\mu_2-1}\left(c_{\frac{1}{2}}\vartheta^{{-}n}\varepsilon_*+\frac{3}{2}\right)\vartheta\varphi\left(\frac{\delta_*}{\vartheta^{m+1}\varrho}\right)\\ & \leq \varphi\left(\frac{\delta_*}{\vartheta^{m+1}\varrho}\right). \end{split} \end{equation}

Now, we prove by induction the first inequality in (3.39) for $m+1$. From (3.39) at step $k$ and the choices of $\delta _*$ and $\varrho _*$ as in (3.43)–(3.44), we have

\begin{align*} &\frac{\Phi(\vartheta^{m}\varrho)}{{\varphi(|(D{\bf u})_{\vartheta^{m}\varrho}|)}}\leq \varepsilon_*<3\varepsilon_*\leq\delta_1,\\ &[H(\vartheta^{m}\varrho)]^{\beta_1}<2\varepsilon_*\leq \delta_2, \end{align*}

and

\begin{equation*} \begin{split} \Phi_*(\vartheta^{m}\varrho)=\Phi(\vartheta^{m}\varrho) + \varphi(|(D{\bf u})_{\vartheta^{m}\varrho}|) [H(\vartheta^{m}\varrho)]^{\beta_1} & \leq 3\varepsilon_*\varphi(|(D{\bf u})_{\vartheta^{m}\varrho}|). \end{split} \end{equation*}

Then, by virtue of lemma 3.11 and lemma 3.12 applied with radius $\vartheta ^{m}\varrho$ in place of $\varrho$, and recalling the choice of $\vartheta$ (3.41), we get

\begin{equation*} \begin{split} \Phi(\vartheta^{m+1}\varrho)\leq 2c_\textrm{dec}\vartheta^{2}\Phi_*(\vartheta^{m}\varrho) & \leq 6c_\textrm{dec}\varepsilon_*\vartheta^{2}\varphi(|(D{\bf u})_{\vartheta^{m}\varrho}|) \\ & \leq \varepsilon_* \varphi(|(D{\bf u})_{\vartheta^{m+1}\varrho}|). \end{split} \end{equation*}

Finally, since the iteration starting from $m\,{=}\,0$ of the estimate $\varphi ^{-1}(E(B_{\vartheta ^{m+1}\varrho }))\leq 2^{\frac {\mu _2}{\mu _1}}\varphi ^{-1}(E(B_{\vartheta ^{m}\varrho }))$, obtained by (3.46) and (2.2), with (3.41) yields

\begin{equation*} (\vartheta^{m}\varrho)^{1-\alpha}\varphi^{{-}1}(E(B_{\vartheta^{m}\varrho}))\leq \varrho^{1-\alpha}\varphi^{{-}1}(E(B_{\varrho}))\leq \delta_*\varrho^{-\alpha}, \end{equation*}

and this estimate a fortiori holds if we consider $E(B_{\vartheta ^{m}\varrho }(y))$ for $y\in B_{\varrho /2}$ in place of $E(B_{\vartheta ^{m}\varrho })$, we deduce the Morrey-type estimate

\begin{equation*} r^{1-\alpha}\varphi^{{-}1}(E(B_{r}(y)))\leq c\delta_*\varrho^{-\alpha} \end{equation*}

for all $y\in B_{\varrho /2}$ and $r\leq \varrho /2$, which is equivalent to (3.40). The proof is now concluded.

3.5 Excess decay estimate: the degenerate regime

In this section, with lemma 3.14 we will establish an excess improvement estimate for the degenerate case which is characterized by the fact that $\Phi (x_0,\varrho )$ is ‘large’ compared to $\varphi (|(D{\bf u})_{x_0,\varrho }|)$.

In view of lemma 3.10, this will be achieved via the $\varphi$-harmonic approximation lemma (lemma 2.11) which allows to approximate the original minimizer by a $\varphi$-harmonic function. In this way, one can transfer the a priori estimates for $\varphi$-harmonic functions (proposition 2.8) to the minimizer.

Lemma 3.14 Let $\gamma _0>0$ be the exponent of proposition 2.8. Then, for every $0<\gamma <\gamma _0$ and every $\kappa ,\mu \in (0,1)$ there exist $\varepsilon _\#, \tau \in (0,1)$ and $\varrho _\#\in (0,1]$ depending on $n,N,\mu _1,\mu _2, c_0, \beta _0$, $L,\nu ,\gamma ,\gamma _0,\mu$ and $\kappa$ ($\varepsilon _\#$ also depends on $\tau$ and $\sigma (\delta )$, where $\delta$ satisfies (3.51) below, and $\varrho _\#$ also depends on $\omega$ and $\mathcal {V}$) with the following property: if

(3.47)\begin{align} \kappa & \varphi(|(D{\bf u})_{x_0,\varrho}|) \leq \Phi(x_0,\varrho)\leq \varepsilon_\# \end{align}

for $B_\varrho (x_0)\subseteq \Omega$ with $\varrho \in (0,\varrho _\#]$, then

(3.48)\begin{equation} \Phi(x_0,\tau\varrho) \leq \tau^{2\gamma} \Phi(x_0,\varrho)\quad \mbox{ and }\quad \Theta(x_0,\tau\varrho)<\mu. \end{equation}

Proof. Without loss of generality, we assume that $x_0=0$ and, correspondingly, we use the abbreviations $\Phi (\varrho )=\Phi (0,\varrho )$, $\Psi (\varrho )=\Psi (0,\varrho )$, $\Theta (\varrho )=\Theta (0,\varrho )$. Let $0<\gamma <\gamma _0$ be fixed, $\tau \in (0,\frac {1}{2^{\mu _2}}]$ to be specified later, and we set $\varepsilon :=\tau ^{2\gamma _0+n}$. Furthermore, let $\delta _0=\delta _0(n,N,\varphi ,\nu ,L,\varepsilon )\in (0,1]$ be the constant according to the $\varphi$-harmonic approximation (lemma 2.11) with $\theta$ the exponent of higher integrability as in (2.19).

Now, we have to check that ${\bf u}$ complies with (2.18), in order to apply lemma 2.11. From the Poincaré inequality, the shift change formula (2.7) with $\eta =\kappa$ and (3.47) we have

(3.49)\begin{equation} \Psi(\varrho)\leq c_{P}\mathop {\int\hskip -1,05em -\,} \limits_{B_\varrho} \varphi(|D{\bf u}|)\,\mathrm{d}x\leq 2^{\mu_2-1}c_{P} (c_{\kappa}+1+{\kappa}^{{-}1})\Phi(\varrho)\leq c_{P}c(\kappa,\mu_2)\varepsilon_\#, \end{equation}

where $c(\kappa ,\mu _2):=2^{\mu _2-1}(c_{\kappa }+1+{\kappa }^{-1})>1$. An analogous computation and the concavity of $\varphi ^{-1}$ give

\begin{equation*} \begin{split} E(B_\varrho):=\mathop {\int\hskip -1,05em -\,} \limits_{B_\varrho} \varphi(|D{\bf u}|)\,\mathrm{d}x & \leq \varphi\left(\varphi^{{-}1}\left(\frac{c(\kappa,\mu_2)\varepsilon_\#}{\varrho}\right)\right) \\ & \leq \varphi\left(\frac{(c(\kappa,\mu_2))^{\frac{1}{\mu_1}}\varphi^{{-}1}(\varepsilon_\#)}{\varrho}\right), \end{split} \end{equation*}

whence

(3.50)\begin{equation} \Theta(\varrho)\leq (c(\kappa,\mu_2))^{\frac{1}{\mu_1}}\varphi^{{-}1}(\varepsilon_\#)=: \tilde{c}(\kappa,\mu_1,\mu_2)\varphi^{{-}1}(\varepsilon_\#). \end{equation}

In applying lemma 3.10 we choose $\delta >0$ (which, in turn, determines $\sigma =\sigma (\delta )>0$ such that (3.31) holds) in such a way that

(3.51)\begin{equation} c_*\delta\leq \frac{\delta_0}{2}, \end{equation}

where $c_*$ is the constant of lemma 3.10. Then, we choose $\varepsilon _\#<1$ such that

\begin{equation*} \varepsilon_\#\leq \min\left\{\frac{1}{c_{P}c(\kappa,\mu_2)}\varphi\left(\frac{\delta_0\sigma c_*}{4}\right), \varphi\left(\frac{\mu}{\tilde{c}(\kappa,\mu_1,\mu_2)}\right), \frac{\tau^{n}}{2 c_{\frac{1}{2}}}\varphi(\mu)\right\}, \end{equation*}

so that, with (3.49)–(3.50), we have

\begin{equation*} \frac{\varphi^{{-}1}(\Psi(\varrho))}{\sigma}\leq\frac{\delta_0}{4} \quad \mbox{ and }\quad\Theta(x_0,\varrho)\leq\mu. \end{equation*}

We also determine a radius $\varrho _\#\in (0,1]$ according to

\begin{equation*} \left[\omega(\mu)^{1-\frac{1}{s}}+\mathcal{V}(\varrho_\#)^{1-\frac{1}{s}}\right]^{\beta_2}\leq\frac{\delta_0}{4}, \end{equation*}

where $\beta _2$ is defined in lemma 3.10. Recalling the definition of $\widetilde {H}(\varrho )$, for $\varrho \leq \varrho _\#$ we then have

\begin{equation*} [\widetilde{H}(\varrho)]^{\beta_2}\leq\frac{\delta_0}{4}. \end{equation*}

For such choice of $\varepsilon _\#$ and $\varrho _\#$ it holds that

\begin{equation*} c_*\left(\delta+[\widetilde{H}(\varrho)]^{\beta_2}+\frac{\varphi^{{-}1}(\Psi(\varrho))}{\sigma}\right) \leq \delta_0, \end{equation*}

so that, by lemma 2.11, there exists a unique $\varphi$-harmonic ${\bf w}\in W^{1,\varphi }(B_{\varrho /2},\mathbb {R}^{N})$ with ${\bf w}={\bf u}$ on $\partial B_{\varrho /2}$ that satisfies

\begin{equation*} \left(\mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}}|{\bf V}(D{\bf u})-{\bf V}(D{\bf w})|^{2\theta}\,\mathrm{d}x\right)^{\frac{1}{\theta}}\leq \tau^{2\gamma_0+n} \mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho}}\varphi(|D{\bf u}|)\,\mathrm{d}x, \end{equation*}

and

\begin{equation*} \mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}}\varphi(|D{\bf w}|)\,\mathrm{d}x\leq c \mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}}\varphi(|D{\bf u}|)\,\mathrm{d}x. \end{equation*}

Taking into account the higher integrability result of lemma 3.2, lemma 2.12 implies that

(3.52)\begin{equation} \mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho/2}} |{\bf V}(D{\bf u})-{\bf V}(D{\bf w})|^{2}\,\mathrm{d}x\leq \tau^{2\gamma_0+n} \mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho}}\varphi(|D{\bf u}|)\,\mathrm{d}x, \end{equation}

and since $\tau <\frac {1}{2}$, from proposition 2.8 we also have

(3.53)\begin{equation} \mathop{\int\hskip -1,05em -\,}\nolimits_{\!\!\!\!{B_{\tau\varrho}}} {\vert{{\bf V}(D{\bf w})-({\bf V}(D {\bf w}))_{\tau\varrho}}\vert}^{2} \, \mathrm{d}x \, \leq \, c \, \tau^{2 \gamma_0} \, \mathop{\int\hskip -1,05em -\,}\nolimits_{\!\!\!\!{B_{\varrho/2}}} {\vert{{\bf V}(D{\bf w})-({\bf V}(D {\bf w}))_{\varrho/2}}\vert}^{2} \, \mathrm{d}x. \end{equation}

Thus, with (3.52) and (3.53) we infer

\begin{equation*} \begin{split} \Phi(\tau\varrho) & \leq 4 \mathop{\int\hskip -1,05em -\,}\nolimits_{\!\!\!\!{B_{\tau\varrho}}} {\vert{{\bf V}(D{\bf u})-({\bf V}(D {\bf w}))_{\tau\varrho}}\vert}^{2} \, \mathrm{d}x \\ & \leq 8 \mathop{\int\hskip -1,05em -\,}\nolimits_{\!\!\!\!{B_{\tau\varrho}}} {\vert{{\bf V}(D{\bf u})-{\bf V}(D {\bf w})}\vert}^{2} \, \mathrm{d}x + 8 \mathop{\int\hskip -1,05em -\,}\nolimits_{\!\!\!\!{B_{\tau\varrho}}} {\vert{{\bf V}(D{\bf w})-({\bf V}(D {\bf w}))_{\tau\varrho}}\vert}^{2} \, \mathrm{d}x \\ & \leq c\tau^{{-}n}(\tau^{2\gamma_0+n}) \mathop {\int\hskip -1,05em -\,} \limits_{B_\varrho}\varphi(|D{\bf u}|)\,\mathrm{d}x + c \tau^{2 \gamma_0} \, \mathop{\int\hskip -1,05em -\,}\nolimits_{\!\!\!\!{B_{\varrho/2}}} {\vert{{\bf V}(D{\bf w})-({\bf V}(D {\bf w}))_{\varrho/2}}\vert}^{2} \, \mathrm{d}x \\ & \leq \tilde{c}_1\tau^{2\gamma_0}\mathop {\int\hskip -1,05em -\,} \limits_{B_\varrho}\varphi(|D{\bf u}|)\,\mathrm{d}x \end{split} \end{equation*}

for some constant $\tilde {c}_1=\tilde {c}_1(n,N,\mu _1,\mu _2,c_1)>0$. Now, by virtue of the computation in (3.49) we conclude that

\begin{equation*} \Phi(\tau\varrho) \leq \tilde{c}_1c(\kappa,\mu_2)\tau^{2\gamma_0} \Phi(\varrho), \end{equation*}

whence (3.48) follows if we choose $\tau$ such that

\begin{equation*} \tau\leq \left(\frac{1}{\tilde{c}_1c(\kappa,\mu_2)}\right)^{\frac{1}{2(\gamma_0-\gamma)}}. \end{equation*}

As for the second assertion in (3.48), the shift change formula (2.7) for $\eta =\frac {1}{2}$, with (3.47), (3.50) and the choice of $\varepsilon _\#$ shows that

\begin{equation*} \begin{split} E(B_{\tau\varrho})&:=\mathop {\int\hskip -1,05em -\,} \limits_{B_\tau\varrho} \varphi(|D{\bf u}|)\,\mathrm{d}x\\ & \leq 2^{\mu_2-1}\left(c_{\frac{1}{2}}\tau^{{-}n}\mathop {\int\hskip -1,05em -\,} \limits_{B_{\varrho}} \varphi_{|(D{\bf u})_{\varrho}|}(|D{\bf u}-(D{\bf u})_{\varrho}|)\,\mathrm{d}x + \frac{3}{2}\varphi(|(D{\bf u})_{\varrho}|)\right) \\ & \leq 2^{\mu_2-1}\left(c_\frac{1}{2}\tau^{{-}n}\Phi(\varrho) +\frac{3}{2}E(B_{\varrho})\right)\\ & \leq 2^{\mu_2-1}\left(c_\frac{1}{2} \tau^{{-}n}\varepsilon_\#+\frac{3}{2}E(B_{\varrho})\right)\\ & \leq 2^{\mu_2}\tau\varphi\left(\frac{\mu}{\tau\varrho}\right) \end{split} \end{equation*}

whence the assertion follows since $\tau \leq \frac {1}{2^{\mu _2}}$.

3.6 Proof of theorem 1.1

Lemma 3.15 Under the assumptions of theorem 1.1, let $\alpha \in (0,1)$. Then there exist constants $\varepsilon _\#$, $\delta _*$ and $\tilde {\varrho }$ such that the conditions

(3.54)\begin{equation} \Phi(x_0,\varrho)<\varepsilon_\# \quad \mbox{ and } \quad \Theta(x_0,\varrho)<\delta_*, \end{equation}

for $B_\varrho (x_0)\subseteq \Omega$ with $\varrho \in (0,\tilde {\varrho }]$ imply

(3.55)\begin{equation} {\bf u} \in C^{0,\alpha}(\overline{B_{\varrho/2}(x_0)}). \end{equation}

Proof. Without loss of generality, we assume that $x_0=0$ and, correspondingly, we omit the dependence on it. Let $\varepsilon _*,\delta _*$, $\vartheta \in (0,1)$ and $\varrho _*\in (0,1]$ be the constants of lemma 3.13. We then choose $\mu =\delta _*$ in lemma 3.14 leaving $\kappa$ unchanged. This fixes the constants $\varepsilon _\#, \tau$ and $\varrho _\#$. We set $\tilde {\varrho }:=\min \{\varrho _*,\varrho _\#\}$.

We introduce the set of integers

\begin{equation*} \mathbb{S}:=\left\{k\in\mathbb{N}_0:\, \kappa \varphi(|(D{\bf u})_{\varrho}|) \leq \Phi(\tau^{k}\varrho)\right\}, \end{equation*}

and we distinguish between the cases $\mathbb {S}=\mathbb {N}_0$ and $\mathbb {S}\neq \mathbb {N}_0$.

The case $\mathbb {S}=\mathbb {N}_0$. We prove by induction that the bounds

(3.56)\begin{equation} \Phi(\tau^{k}\varrho)<\varepsilon_\# \quad \mbox{ and }\quad \Theta(\tau^{k}\varrho)<\delta_* \end{equation}

hold for every $k\in \mathbb {N}_0$. The case $k=0$ is trivial from the assumption (3.54). Now, since $k\in \mathbb {S}=\mathbb {N}_0$, the assumption (3.47) of lemma 3.14 hold with $\tau ^{k}\varrho$ in place of $\varrho$. Then, an application of lemma 3.14 gives (3.56) for $k+1$ (recall that $\tau <1$). The validity of (3.56) implies, as in lemma 3.13, that the Morrey-type estimate

(3.57)\begin{equation} \Theta(y,r)\leq c\delta_*\left(\frac{r}{\varrho}\right)^{\alpha} \end{equation}

holds for every $\alpha \in (0,1)$, for all $y\in B_{\varrho /2}(x_0)$ and $r\in (0,\varrho /2]$. For $y,z\in B_{\varrho /2}$, with $|y-z|\leq \varrho /4$ we estimate the telescopic sum

\begin{equation*} \frac{|{\bf u}(y)-{\bf u}(z)|}{|y-z|}\leq \sum_{j\in\mathbb{Z}}\frac{1}{2^{j}}\mathop {\int\hskip -1,05em -\,} \limits_{B_{r_j}}\frac{|{\bf u}(x)-({\bf u})_j|}{|y-z|}\,\mathrm{d}x \leq \sum_{j\in\mathbb{Z}}\frac{1}{2^{j}}\mathop {\int\hskip -1,05em -\,} \limits_{B_{r_j}}\frac{|{\bf u}(x)-({\bf u})_j|}{r_j}\,\mathrm{d}x, \end{equation*}

where $B_{r_j}:=B_{2^{1-j}|y-z|(y)}$ for $j\geq 0$ and $B_{r_j}:=B_{2^{1+j}|y-z|(z)}$ for $j<0$. Now, with the Poincaré inequality we get

\begin{equation*} \begin{split} \frac{|{\bf u}(y)-{\bf u}(z)|}{|y-z|}&\leq c\sum_{j\in\mathbb{Z}}\mathop {\int\hskip -1,05em -\,} \limits_{B_{r_j}}|D{\bf u}|\,\mathrm{d}x \\ & \leq c \sum_{j\in\mathbb{Z}}\varphi^{{-}1}\left(\mathop {\int\hskip -1,05em -\,} \limits_{B_{r_j}}\varphi(|D{\bf u}|)\,\mathrm{d}x\right)\\ & \leq c \sum_{j\geq 0} \frac{\Theta(y,2^{1-j}|y-z|)}{2^{1-j}|y-z|} + c \sum_{j<0} \frac{\Theta(z,2^{1+j}|y-z|)}{2^{1+j}|y-z|}. \end{split} \end{equation*}

Finally, with the estimate (3.57) we infer

(3.58)\begin{equation} \begin{split} |{\bf u}(y)-{\bf u}(z)| & \leq c \delta_* |y-z|^{\alpha}\varrho^{-\alpha} \left(\sum_{j\geq 0}2^{(\alpha-1)(1-j)}+\sum_{j<0}2^{(\alpha-1)(1+j)}\right)\\ & \leq c \delta_* |y-z|^{\alpha}\varrho^{-\alpha}, \end{split} \end{equation}

which, in particular, implies that ${\bf u}\in C^{0,\alpha }(\overline {B_{\varrho /2}})$.

The case $\mathbb {S}\neq \mathbb {N}_0$. In this case, there exists $k_0:=\min \mathbb {N}\backslash \mathbb {S}$. Since $k\in \mathbb {S}$ for any integer $k< k_0$ we can iterate as in the case $\mathbb {S}=\mathbb {N}_0$ for $k=0,1,\dots ,k_0-1$ to infer that (3.56) holds for any $k\leq k_0$. By the definition of $\mathbb {S}$ we have

\begin{equation*} \Phi(\tau^{k_0}\varrho)<\kappa\varphi(|(D{\bf u})_{\varrho}|), \end{equation*}

which together with the second inequality in (3.56) with $k=k_0$ ensures that the assumptions (3.38) of lemma 3.13 are satisfied for $\varrho$ replaced by $\tau ^{k_0}\varrho$. Then, by virtue of this lemma, we have

(3.59)\begin{equation} \frac{\Phi(\vartheta^{m}\tau^{k_0}\varrho)}{{\varphi(|(D{\bf u})_{\vartheta^{m}\tau^{k_0}\varrho}|)}}\leq \kappa \quad \mbox{ and }\quad \Theta(\vartheta^{m}\tau^{k_0}\varrho)\leq\delta_* \end{equation}

for every $m\in \mathbb {N}_0$.

Now, we consider an arbitrary radius $r\in (0,\varrho ]$. If $r\in (\tau ^{k_0}\varrho /2, \varrho ]$ we find $0\leq k \leq k_0$ such that $\tau ^{k+1}\leq r \leq \theta ^{k}$ and then we can argue as in the case $\mathbb {S}=\mathbb {N}_0$. In the case $r\in (0,\tau ^{k_0}\varrho /2]$, instead, we find $m\in \mathbb {N}_0$ such that $\vartheta ^{m+1}\tau ^{k_0}\varrho < r\leq \vartheta ^{m}\tau ^{k_0}\varrho$. Then arguing as in the proof of (3.40) and taking into account the second estimate in (3.59), we have

\begin{equation*} r^{1-\alpha}\varphi^{{-}1}(E(B_r(y)))\leq c (\vartheta^{m}\tau^{k_0}\varrho)^{1-\alpha} \varphi^{{-}1}(E(B_{\vartheta^{m}\tau^{k_0}\varrho})) \leq \frac{c\delta_*}{(\vartheta^{m}\tau^{k_0}\varrho)^{\alpha}} \end{equation*}

for every $y\in B_{\vartheta ^{m}\tau ^{k_0}\varrho /2}\subseteq B_{\varrho /2}$ whence

\begin{equation*} \Theta(y,r)\leq c\frac{\delta_*}{(\vartheta^{m}\tau^{k_0})^{\alpha}}\left(\frac{r}{\varrho}\right)^{\alpha}. \end{equation*}

At this point, we can argue as in the case $\mathbb {S}=\mathbb {N}_0$ for the proof of (3.58), whence (3.55) follows thus concluding the proof.

Proof of theorem 1.1. Let $\varepsilon _\#$, $\delta _*$ and $\tilde {\varrho }$ be the constants of lemma 3.15. Let $\Sigma _1$ and $\Sigma _2$ be defined as in the statement of theorem 1.1. Note that, by Lebesgue's differentiation theorem, it holds that $|\Sigma _1\cup \Sigma _2|=0$. Thus, we are reduced to show that each $x_0\in \Omega \backslash (\Sigma _1\cup \Sigma _2)$ belongs to the set

\begin{equation*} \Omega_0:=\left\{z_0\in\Omega:\, {\bf u}\in C^{0,\alpha}(U_{z_0},\mathbb{R}^{N})\,\mbox{ for every }\alpha\in(0,1)\,\mbox{ and for some }U_{z_0}\subset\Omega\right\}, \end{equation*}

where $U_{z_0}$ is an open neighbourhood of $z_0$. For this, let $x_0\in \Omega$ be such that both the conditions

(3.60)\begin{align} &\mathop{\lim\inf}_{\varrho\searrow 0}\mathop {\int\hskip -1,05em -\,} \limits_{B_\varrho(x_0)}|{\bf V}_{|(D{\bf u})_{x_0,\varrho}|}(D{\bf u}-(D{\bf u})_{x_0,\varrho})|^{2}\,\mathrm{d}x=0\,\mbox{ and }\notag\\ & m_{x_0}:=\mathop{\lim\sup}_{\varrho\searrow 0}|(D{\bf u})_{x_0,\varrho}|<{+}\infty \end{align}

hold.

We set

(3.61)\begin{equation} \sigma:=\min\left\{\frac{1}{c_\varphi c_\frac{1}{2}2^{\mu_2}}\varphi\left({\delta_*}\right),\frac{\varepsilon_\#}{c_\varphi}\right\}, \end{equation}

where the constants $c_\varphi$, $c_\frac {1}{2}$ are specified later and, correspondingly, with (3.60) we can find a radius $\bar {\varrho }$ such that

(3.62)\begin{equation} \bar{\varrho}\leq \frac{\varphi(\delta_*)}{3\cdot 2^{\mu_2+1} \varphi(m_{x_0}+1)} \end{equation}

and

(3.63)\begin{equation} \mathop {\int\hskip -1,05em -\,} \limits_{B_{\bar{\varrho}}(x_0)}|{\bf V}_{|(D{\bf u})_{x_0,\bar{\varrho}}|}(D{\bf u}-(D{\bf u})_{x_0,\bar{\varrho}})|^{2}\,\mathrm{d}x\leq\sigma\quad \mbox{ and }\quad |(D{\bf u})_{x_0,\bar{\varrho}}|\leq m_{x_0}+1. \end{equation}

Now, recalling that

\begin{equation*} \Phi(x_0,\bar{\varrho})\leq c_\varphi \mathop {\int\hskip -1,05em -\,} \limits_{B_{\bar{\varrho}}(x_0)}|{\bf V}_{|(D{\bf u})_{x_0,\bar{\varrho}}|}(D{\bf u}-(D{\bf u})_{x_0,\bar{\varrho}})|^{2}\,\mathrm{d}x \end{equation*}

and observing that, as a consequence of the shift-change formula (2.7) with $\displaystyle \eta =\frac {1}{2}$, conditions (3.63) imply

\begin{equation*} \begin{split} &\mathop {\int\hskip -1,05em -\,} \limits_{B_{\bar{\varrho}}(x_0)}\varphi(|D{\bf u}|)\,\mathrm{d}x \\ & \quad \leq 2^{\mu_2-1}\left(c_\frac{1}{2} c_\varphi \mathop {\int\hskip -1,05em -\,} \limits_{B_{\bar{\varrho}}(x_0)}|{\bf V}_{|(D{\bf u})_{x_0,\bar{\varrho}}|}(D{\bf u}-(D{\bf u})_{x_0,\bar{\varrho}})|^{2}\,\mathrm{d}x+\frac{3}{2}\varphi(|(D{\bf u})_{x_0,\bar{\varrho}}|)\right)\\ & \quad \leq 2^{\mu_2-1}\left(c_{\frac{1}{2}}c_\varphi\sigma + \frac{3}{2}\varphi(m_{x_0}+1)\right) \leq \frac{1}{2}\varphi(\delta_*) + \frac{1}{2\bar{\varrho}}\varphi(\delta_*)\leq \varphi\left(\frac{\delta_*}{\bar{\varrho}}\right), \end{split} \end{equation*}

with the choice of $\sigma$ (3.61), corresponding to the radius $\bar {\varrho }\in (0,\tilde {\varrho }]$ as in (3.62) there holds

\begin{equation*} \Phi(x_0,\bar{\varrho})<\varepsilon_\#\quad \mbox{ and }\quad \Theta(x_0,\bar{\varrho})<\delta_*. \end{equation*}

By the absolute continuity of the integral, we can find an open neighbourhood $U_{x_0}$ of $x_0$ such that

\begin{equation*} \Phi(x,\bar{\varrho})<\varepsilon_\#\quad \mbox{ and }\quad \Theta(x,\bar{\varrho})<\delta_* \end{equation*}

for every $x\in U_{x_0}$. We can apply lemma 3.15 at each point of $U_{x_0}$, proving that ${\bf u}\in C^{0,\alpha }(U_{x_0},\mathbb {R}^{N})$ for every $\alpha \in (0,1)$. Thus, $x_0\in \Omega _0$ and the proof is concluded.