Du\rvert)]^{\frac{2^{*}}{2}(\kappa+2)}+1\right]\,dx\right)^{2/2^{*}}$
$\displaystyle\quad\ \leq c(\kappa+1)^{2}\left(\lVert
h\rVert_{L^{d}(\mathcal{B})}^{2}+1\right)$ (6.29)
$\displaystyle\quad\quad\times\left(\int_{\mathcal{B}}(\eta^{2m}+\lvert
D\eta\rvert^{2m})\left[[G_{\varepsilon}(x,\lvert
Du\rvert)]^{m(\kappa+2+2\sigma+2\hat{\sigma})}+1\right]\,dx\right)^{1/m}$

for $c\equiv c(n,N,\nu,\gamma,T,c_{\textnormal{b}})$.

###  6.2 Moser’s iteration in finite steps

With $1\leq i\in\mathbb{N}$, we inductively define the exponents

$\kappa_{1}:=0,\quad\kappa_{i+1}:=\chi(\kappa_{i}+2)-2(1+\sigma+\hat{\sigma}),\quad
s_{i}:=m(\kappa_{i}+2+2\sigma+2\hat{\sigma})\,.$ (6.30)

Note that $s_{*}/n-s_{*}/d<\chi-1$ if and only if $s_{*}<2n/(n-2)$, and
therefore the second condition in (6.15) implies, again together with the
first one, that

$\sigma+\hat{\sigma}<\frac{s_{*}}{n}-\frac{s_{*}}{d}<\chi-1\,.$ (6.31)

This, in turn, implies that $\\{\kappa_{i}\\}$ and $\\{s_{i}\\}$ are
increasing sequences; moreover, it holds that

$\begin{cases}\displaystyle\kappa_{i+1}=2\chi^{i}-2(\sigma+\hat{\sigma})\sum_{j=0}^{i-1}\chi^{j}-2\quad\mbox{
for $i\geq 1$}\\\ \displaystyle
s_{i+1}=2m\chi^{i}-2m(\sigma+\hat{\sigma})\sum_{j=1}^{i-1}\chi^{j}\quad\mbox{for
$i\geq 2$ \ \ and \ \ $s_{2}=2m\chi=2^{*}$}.\end{cases}$

Note that this implies

$\begin{cases}\displaystyle
s_{i+1}=\frac{2^{*}}{2}(\kappa_{i}+2)=\frac{n(\kappa_{i}+2)}{n-2}\Longrightarrow
k_{i+1}\leq s_{i+1}\leq 2m\chi^{i}\\\ \displaystyle k_{i+1}\leq
2\chi^{i}\,.\end{cases}$ (6.32)

Again (6.31) implies, for $i\geq 1$

$\displaystyle
s_{i+1}=2m\chi^{i}\left[1-\frac{(\sigma+\hat{\sigma})(1-\chi^{1-i})}{\chi-1}\right]>2m\chi^{i}\left(1-\frac{\sigma+\hat{\sigma}}{\chi-1}\right)$
$\displaystyle\qquad\Longrightarrow\lim_{i\to\infty}s_{i+1}=\infty\,,$ (6.33)

so that from the first relation in (6.32) it also follows that
$\kappa_{i}\to\infty$. For $0<\varsigma\leq\tau_{1}<\tau_{2}\leq\varrho$, we
consider a sequence $\\{B_{\varrho_{i}}\\}$ of shrinking, concentric balls,
$B_{\tau_{1}}\Subset B_{\varrho_{i}+1}\Subset B_{\varrho_{i}}\Subset
B_{\tau_{2}}$, where $\varrho_{i}:=\tau_{1}+(\tau_{2}-\tau_{1})2^{-i+1}$. Note
that $\\{\varrho_{i}\\}$ is a decreasing sequence with $\varrho_{1}=\tau_{2}$
and $\varrho_{i}\to\tau_{1}$, therefore it is
$\bigcap_{i\in\mathbb{N}}B_{\varrho_{i}}=\overline{B_{\tau_{1}}}$ and
$B_{\varrho_{1}}=B_{\tau_{2}}$. Accordingly, we fix corresponding cut-off
functions $\\{\eta_{i}\\}\subset C^{1}_{{\rm c}}(B)$ with
$\mathds{1}_{B_{\varrho_{i+1}}}\leq\eta_{i}\leq\mathds{1}_{B_{\varrho_{i}}}$
and $\lvert D\eta_{i}\rvert\lesssim 1/(\varrho_{i}-\varrho_{i+1})\approx
2^{i}/(\tau_{2}-\tau_{1})$. Choosing $\eta\equiv\eta_{i}$ in (6.29),
elementary manipulations also based on (6.30) and (6.32) give that

$\displaystyle\left(\int_{B_{\varrho_{i+1}}}\left[[G_{\varepsilon}(x,\lvert
Du\rvert)]^{s_{i+1}}+1\right]\,dx\right)^{1/\chi}$
$\displaystyle\qquad\leq\frac{\left(c\lVert
h\rVert_{L^{d}(\mathcal{B})}+c\right)^{2m}2^{2mi}(\kappa_{i}+1)^{2m}}{(\tau_{2}-\tau_{1})^{2m}}\int_{B_{\varrho_{i}}}\left[[G_{\varepsilon}(x,\lvert
Du\rvert)]^{s_{i}}+1\right]\,dx$ (6.34)

holds whenever $\kappa_{i}\leq\bar{\kappa}$ and $i\geq 1$, with $c\equiv
c(n,N,\nu,\gamma,T,c_{\textnormal{b}})$. Finally, we set $c_{h}:=(c\lVert
h\rVert_{L^{d}(\mathcal{B})}+c)^{2^{*}}$ and

$\displaystyle\mathcal{G}_{i}:=\left(\int_{B_{\varrho_{i}}}\left[[G_{\varepsilon}(x,\lvert
Du\rvert)]^{s_{i}}+1\right]\,dx\right)^{1/s_{i}}\,,$

so that (6.2) (recall that $2^{*}=2m\chi$ by (6.14)) becomes

$\mathcal{G}_{i+1}\leq\left[\frac{c_{h}2^{2^{*}i}(\kappa_{i}+1)^{2^{*}}}{(\tau_{2}-\tau_{1})^{2^{*}}}\right]^{\frac{1}{s_{i+1}}}\mathcal{G}_{i}^{\frac{\chi
s_{i}}{s_{i+1}}}\,.$

Iterating the above inequality yields that

$\displaystyle\mathcal{G}_{i+1}\leq\prod_{j=0}^{i-1}\left[\frac{c_{h}2^{2^{*}(i-j)}(\kappa_{i-j}+1)^{2^{*}}}{(\tau_{2}-\tau_{1})^{2^{*}}}\right]^{\frac{\chi^{j}}{s_{i+1}}}\mathcal{G}_{1}^{\frac{\chi^{i}s_{1}}{s_{i+1}}}\quad\mbox{holds
provided $\kappa_{i}\leq\bar{\kappa}$}\,.$ (6.35)

Now, from (6.2) we deduce that

$\frac{\chi^{i+1}}{s_{i+1}}\leq\frac{\chi(\chi-1)}{2m(\chi-1-\sigma-\hat{\sigma})}\,.$
(6.36)

The function $t\mapsto t/\chi^{t}$ is decreasing on $[1/\log\chi,\infty)$ and
using this fact one sees that

$\sum_{j=1}^{\infty}\frac{j}{\chi^{j}}\lesssim\frac{1}{(\log\chi)^{2}}\leq\frac{c}{(\chi-1)^{2}}\,.$
(6.37)

We then write

$\displaystyle\prod_{j=0}^{i-1}\left[\frac{c_{h}2^{2^{*}(i-j)}(\kappa_{i-j}+1)^{2^{*}}}{(\tau_{2}-\tau_{1})^{2^{*}}}\right]^{\frac{\chi^{j}}{s_{i+1}}}$
$\displaystyle=\exp\left\\{\log\left(\frac{c_{h}}{(\tau_{2}-\tau_{1})^{2^{*}}}\right)\frac{1}{s_{i+1}}\sum_{j=0}^{i-1}\chi^{j}+\frac{2^{*}\log
2}{s_{i+1}}\sum_{j=0}^{i-1}(i-j)\chi^{j}\right.$ $\displaystyle\left.\hskip
56.9055pt+\frac{2^{*}}{s_{i+1}}\sum_{j=0}^{i-1}\chi^{j}\log(\kappa_{i-j}+1)\right\\},$

and note that, for every integer $i\geq 1$, we have

$\frac{1}{s_{i+1}}\sum_{j=0}^{i-1}\chi^{j}\stackrel{{\scriptstyle(\ref{recalls2})}}{{\leq}}c\,,\quad\frac{1}{s_{i+1}}\sum_{j=0}^{i-1}(i-j)\chi^{j}\leq
c\sum_{j=0}^{i-1}\frac{i-j}{\chi^{i-j}}\stackrel{{\scriptstyle(\ref{recalls3})}}{{\leq}}c$

and

$\frac{2^{*}\log
2}{s_{i+1}}\sum_{j=0}^{i-1}\chi^{j}\log(\kappa_{i-j}+1)\stackrel{{\scriptstyle(\ref{recallss}),(\ref{recalls2})}}{{\leq}}c\sum_{j=0}^{i-1}\frac{i-j}{\chi^{i-j}}\stackrel{{\scriptstyle(\ref{recalls3})}}{{\leq}}c\,,$

where $c\equiv c(n,d,\sigma,\tilde{\sigma})$ in all cases. Using the content
of the last three displays yields

$\prod_{j=0}^{i-1}\left[\frac{c_{h}2^{2^{*}(i-j)}(\kappa_{i-j}+1)^{2^{*}}}{(\tau_{2}-\tau_{1})^{2^{*}}}\right]^{\frac{\chi^{j}}{s_{i+1}}}\leq\frac{c}{(\tau_{2}-\tau_{1})^{\beta}}<\infty\,,$
(6.38)

where $c\equiv c(\texttt{data},\lVert h\rVert_{L^{d}(\mathcal{B})})\geq 1$ and
$\beta\equiv\beta(n,d,\sigma,\hat{\sigma})\geq 1$. Note that such constants
blow-up when $\chi\to 1+\sigma+\hat{\sigma}$; in particular, this happens when
$d\to n$. Using (6.38) in (6.35), and keeping (6.36) in mind, yields

$\displaystyle\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{s_{i+1}}(B_{\tau_{1}})}$
$\displaystyle\leq\mathcal{G}_{i+1}\leq\frac{c}{(\tau_{2}-\tau_{1})^{\beta}}\mathcal{G}_{1}^{\frac{\chi^{i}s_{1}}{s_{i+1}}}$
$\displaystyle\leq\frac{c}{(\tau_{2}-\tau_{1})^{\beta}}\left[\lVert
G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{s_{1}}(B_{\tau_{2}})}^{\frac{\chi^{i}s_{1}}{s_{i+1}}}+1\right]\,,$
(6.39)

with $c\equiv c(\texttt{data},\lVert h\rVert_{L^{d}(\mathcal{B})})$ and
$\beta\equiv\beta(n,d,\sigma,\hat{\sigma})$, for every $i\in\mathbb{N}$ such
that $\kappa_{i}\leq\bar{\kappa}$.

###  6.3 Sobolev regularity

By (6.15) we have $1\leq s_{*}<s_{1}=2m(1+\sigma+\hat{\sigma})$ so that, for
every integer index $i\geq 1$, we consider the interpolation inequality

$\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{s_{1}}(B_{\tau_{2}})}\leq\lVert
G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{s_{i+1}}(B_{\tau_{2}})}^{\lambda_{i+1}}\lVert
G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{s_{*}}(B_{\tau_{2}})}^{1-\lambda_{i+1}}\,,$ (6.40)

with $\lambda_{i+1}$ being defined by

$\frac{1}{s_{1}}=\frac{1-\lambda_{i+1}}{s_{*}}+\frac{\lambda_{i+1}}{s_{i+1}}\Rightarrow\lambda_{i+1}=\frac{s_{i+1}(s_{1}-s_{*})}{s_{1}(s_{i+1}-s_{*})}\,.$
(6.41)

Let us show that, thanks to (6.15), there exist
$\bar{\vartheta}\equiv\bar{\vartheta}(n,d,\sigma,\hat{\sigma})<1$ and
$i_{1}\in\mathbb{N}$ such that

$\displaystyle
i>i_{1}\Longrightarrow\frac{\lambda_{i+1}\chi^{i}s_{1}}{s_{i+1}}\leq\bar{\vartheta}<1\,.$
(6.42)

Indeed, for this it is sufficient to observe that

$\displaystyle\lim_{i\to\infty}\,\frac{\lambda_{i+1}\chi^{i}s_{1}}{s_{i+1}}$
$\displaystyle\stackrel{{\scriptstyle(\ref{il-
la})}}{{=}}\lim_{i\to\infty}\,\frac{\chi^{i}(s_{1}-s_{*})}{s_{i+1}-s_{*}}$
$\displaystyle=l_{*}:=\frac{\chi-1}{2m}\frac{2m(1+\sigma+\hat{\sigma})-s_{*}}{\chi-1-\sigma-\hat{\sigma}}\stackrel{{\scriptstyle(\ref{sigmasss})}}{{<}}1\,.$

Note that the last inequality is actually equivalent to (6.15) and that (6.15)
is in fact implied by (4.15) for $s_{*}=1$. Now, we consider the number $p>1$
for the statement of Proposition 1, and determine another index $k\equiv
k(p)>i_{1}$ such that $s_{k+1}\geq p$; accordingly, we consider the number
$\kappa_{k}$ related to $s_{k}$ via (6.32). We now choose the number
$\bar{\kappa}\equiv\bar{\kappa}(p)$ in (6.28) as $\bar{\kappa}:=\kappa_{k}$,
and accordingly we determine $R_{*}\equiv R_{*}(p)$ via (6.28). It follows
that (6.2) holds in the case $i\equiv k$ and therefore we can plug (6.40) in
it, thereby obtaining

$\displaystyle\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{s_{k+1}}(B_{\tau_{1}})}$
$\displaystyle\qquad\leq\frac{c}{(\tau_{2}-\tau_{1})^{\beta}}\lVert
G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{s_{k+1}}(B_{\tau_{2}})}^{\frac{\lambda_{k+1}\chi^{k}s_{1}}{s_{k+1}}}\lVert
G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{s_{*}}(B_{\tau_{2}})}^{\frac{(1-\lambda_{k+1})\chi^{k}s_{1}}{s_{k+1}}}$
$\displaystyle\qquad\quad+\frac{c}{(\tau_{2}-\tau_{1})^{\beta}}\,.$ (6.43)

On the other hand, as $k>i_{1}$, then (6.42) holds with $i\equiv k$; we can
therefore apply Young inequality in (6.3); this yields

$\displaystyle\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{s_{k+1}}(B_{\tau_{1}})}$
$\displaystyle\leq\frac{1}{2}\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{s_{k+1}}(B_{\tau_{2}})}$
$\displaystyle\qquad+\frac{c}{(\tau_{2}-\tau_{1})^{\beta_{k}}}\left[\lVert
G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{s_{*}}(B_{\varrho})}^{\theta_{k}}+1\right],$

where

$\beta_{k}:=\frac{s_{k+1}\beta}{s_{k+1}-\lambda_{k+1}\chi^{k}s_{1}}\quad\mbox{and}\quad\theta_{k}:=\frac{(1-\lambda_{k+1})\chi^{k+1}s_{1}}{s_{k+1}-\lambda_{k+1}\chi^{k}s_{1}}\,.$
(6.44)

Lemma 2 with the choice $\mathcal{Z}(t)\equiv\lVert
G_{\varepsilon}(\cdot,\lvert Du\rvert)\rVert_{L^{s_{k+1}}(B_{t})}$, finally
yields

$\displaystyle\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{s_{k+1}}(B_{\varsigma})}\leq\frac{c}{(\varrho-\varsigma)^{\beta_{k}}}\left[\lVert
G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{s_{*}}(B_{\varrho})}^{\theta_{k}}+1\right]\,,$

with $c\equiv c(\texttt{data},\lVert h\rVert_{L^{d}(\mathcal{B})})$. This last
inequality holds provided the bound in (6.15) holds. All in all, recalling
(5.18)2, (6.26) and that $p\leq s_{k+1}$, completes the proof of Proposition 1
with $\beta_{p}:=\beta_{k(p)},\theta_{p}:=\theta_{k(p)}$. We remark that in
(6.16) the exponents $\theta_{p},\beta_{p}$ can be replaced by exponents
$\tilde{\beta},\tilde{\theta}\equiv\tilde{\beta},\tilde{\theta}(n,d,\sigma,\hat{\sigma})$
that are independent of $p$. For this, observe that

$\begin{cases}\beta_{k},\beta_{p}\stackrel{{\scriptstyle(\ref{nuoviesponenti})}}{{\to}}\frac{\beta}{1-l_{*}}=:\tilde{\beta}\,,\quad\lambda_{k}\stackrel{{\scriptstyle(\ref{il-
la})}}{{\to}}1-\frac{s_{*}}{s_{1}}\\\
\theta_{k},\theta_{p}\stackrel{{\scriptstyle(\ref{nuoviesponenti})}}{{\to}}\frac{\chi
l_{*}s_{*}}{(s_{1}-s_{*})(1-l_{*})}:=\tilde{\theta}\,,\end{cases}$ (6.45)

when $k,p\to\infty$. The only dependence on $p$ in (6.16) comes through the
threshold radius $R_{*}$; it is $R_{*}(p)\to 0$ as $p\to\infty$ unless
$f\equiv 0$.

###### Remark 8

In the proof of Proposition 1, assume that
$g_{2,\varepsilon}/g_{2,\varepsilon}\leq c_{\rm b}$ as in (5.30), and that
$\sigma=\hat{\sigma}$ (compare with the assumptions of Theorem 4.4). It is
then easy to see that $\sigma+\hat{\sigma}$ can be replaced by $\hat{\sigma}$,
everywhere, starting from (6.29). This finally reflects in the same
replacement in (6.15); in particular, we can assume
$\hat{\sigma}<s_{*}/n-s_{*}/d$.

###  6.4 A Lipschitz bound in the homogeneous case $f\equiv 0$.

The above reasoning, eventually culminating in Proposition 1, immediately
leads to a priori Lipschitz estimates when $f\equiv 0$. The result, when
combined with the approximation of Section 7 below, extends those in M2 ; M3 ;
MP  to the case of nonautonomous functionals with superlinear growth as in
(1.1). For this, the key observation is that it is not any longer necessary to
consider balls with small radii $R_{*}$ as in (6.28), as the last term in
(6.27) does not appear. Therefore we can take everywhere, and in particular in
(6.16), $R_{*}=1$, independently of the value of $p$. It follows we can let
$p\to\infty$ in (6.16), and recalling (6.45) we conclude with

###### Proposition 2

Let $u\in W^{1,\gamma}(\mathcal{B};\mathbb{R}^{N})$ be a solution to
$\eqref{a0}$, under assumptions set in (4.20) for $n\geq 2$ and with $f\equiv
0$. Moreover, replace (4.15) by the weaker $\sigma+\hat{\sigma}<1/n-1/d$. If
$B_{\varsigma}\Subset B_{\varrho}$ are concentric balls contained in
$\mathcal{B}$, then

$\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\varsigma})}\leq\frac{c}{(\varrho-\varsigma)^{\tilde{\beta}}}\left[\|F_{\varepsilon}(\cdot,Du)\|_{L^{1}(B_{\varrho})}^{\tilde{\theta}}+1\right]\,,$

holds with $c\equiv c(\texttt{data},\lVert h\rVert_{L^{d}(\mathcal{B})})\geq
1$,
$\tilde{\beta},\tilde{\theta}\equiv\tilde{\beta},\tilde{\theta}(n,d,\sigma,\hat{\sigma})>0$.

Notice that here we are using Proposition 1 with the choice $s_{*}=1$. Note
also that Proposition 1 refers to the case $n>2$. The remaining two
dimensional case can be obtained via minor modifications to the proof of
Proposition 1, by choosing $2^{*}/2$ large enough (see (2.2)) in order to get
$\chi>1$ in (6.14). Anyway, the two dimensional case $n=2$ will be treated in
Section 6.7 directly for the general case $f\not=0$. In that situation the
proof cannot be readapted from the one of Proposition 1 as for Proposition 2.

###  6.5 Caccioppoli inequality on level sets

This is in the following:

######  Lemma 8 (Caccioppoli inequality)

Let $u\in W^{1,\gamma}(\mathcal{B};\mathbb{R}^{N})$ be a solution to
$\eqref{a0}$, under assumptions set in (4.20) for $n\geq 2$, and let
$B_{r}(x_{0})\Subset\mathcal{B}$ be a ball. Then

$\displaystyle\int_{B_{r/2}(x_{0})}\lvert D(G_{\varepsilon}(x,\lvert
Du\rvert)-\kappa)_{+}\rvert^{2}\ \,dx$
$\displaystyle\qquad\leq\frac{c}{r^{2}}\lVert\bar{G}_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{r}(x_{0}))}^{\vartheta\sigma}\int_{B_{r}(x_{0})}(G_{\varepsilon}(x,\lvert
Du\rvert)-\kappa)_{+}^{2}\ \,dx$
$\displaystyle\quad\qquad+c\int_{B_{r}(x_{0})}[h(x)]^{2}[\bar{G}_{\varepsilon}(x,\lvert
Du\rvert)]^{2(1+\sigma+\hat{\sigma})}\,dx$ $\displaystyle\quad\qquad+c\lVert
Du\rVert_{L^{\infty}(B_{r}(x_{0}))}^{2}\int_{B_{r}(x_{0})}\lvert f\rvert^{2}\
\,dx$ (6.46)

holds whenever $\kappa\geq 0$, with $c\equiv c(\texttt{data})$ and
$\vartheta\equiv\vartheta(\gamma)$ is as in (4.15).

###### Proof

For $s\in\\{1,\ldots,n\\}$, we take
$\varphi_{s}:=\eta^{2}(G_{\varepsilon}(x,\lvert Du\rvert)-\kappa)_{+}D_{s}u$
in (6.17), where $\eta\in C^{1}_{{\rm c}}(B_{r}(x_{0}))$ satisfies
$\mathds{1}_{B_{r/2}(x_{0})}\leq\eta\leq\mathds{1}_{B_{r}(x_{0})}$ and $\lvert
D\eta\rvert\lesssim 1/r$. Admissibility of $\varphi_{s}$ follows by (6.5) and
(6.10). Note that all the integrals stemming from (6.17) extend over
$\mathcal{B}^{\kappa}:=\mathcal{B}\cap\\{G_{\varepsilon}(\cdot,\lvert
Du\rvert)>\kappa\\}$; in particular, by the very definition of
$G_{\varepsilon}$, we can always restrict to the case it is $|Du|>T$ and
$g_{1,\varepsilon}(\cdot,|Du|)>0$ (recall (4.6)). We start expanding the terms
resulting from (6.17)

$\displaystyle\sum_{s=1}^{n}$
$\displaystyle\int_{\mathcal{B}^{\kappa}}\partial_{z}a_{\varepsilon}(x,Du)DD_{s}u\cdot
D\varphi_{s}\,dx$
$\displaystyle=\sum_{s=1}^{n}\int_{\mathcal{B}^{\kappa}}\eta^{2}(G_{\varepsilon}(x,\lvert
Du\rvert)-\kappa)_{+}\partial_{z}a_{\varepsilon}(x,Du)DD_{s}u\cdot
DD_{s}u\,dx$
$\displaystyle\quad+\sum_{s=1}^{n}\int_{\mathcal{B}^{\kappa}}\eta^{2}\partial_{z}a_{\varepsilon}(x,Du)DD_{s}u\cdot\left[D_{s}u\otimes
D(G_{\varepsilon}(x,\lvert Du\rvert)-\kappa)_{+}\right]\,dx$
$\displaystyle\quad+2\sum_{s=1}^{n}\int_{\mathcal{B}^{\kappa}}\eta(G_{\varepsilon}(x,\lvert
Du\rvert)-\kappa)_{+}\partial_{z}a_{\varepsilon}(x,\lvert
Du\rvert)DD_{s}u\cdot D_{s}u\otimes D\eta\,dx$
$\displaystyle=:\mbox{(IV)}_{z}+\mbox{(V)}_{z}+\mbox{(VI)}_{z}\,.$ (6.47)

Moreover, it is

$\displaystyle\sum_{s=1}^{n}\int_{\mathcal{B}^{\kappa}}\partial_{x_{s}}a_{\varepsilon}(x,Du)\cdot
D\varphi_{s}\,dx$
$\displaystyle=\sum_{s=1}^{n}\int_{\mathcal{B}^{\kappa}}\eta^{2}(G_{\varepsilon}(x,\lvert
Du\rvert)-\kappa)_{+}\partial_{x_{s}}a_{\varepsilon}(x,Du)\cdot DD_{s}u\,dx$
$\displaystyle+\sum_{s=1}^{n}\int_{\mathcal{B}^{\kappa}}\eta^{2}\partial_{x_{s}}a_{\varepsilon}(x,Du)\cdot\left(D_{s}u\otimes
D(G_{\varepsilon}(x,\lvert Du\rvert)-\kappa)_{+}\right)\,dx$
$\displaystyle+2\sum_{s=1}^{n}\int_{\mathcal{B}^{\kappa}}\eta(G_{\varepsilon}(x,\lvert
Du\rvert)-\kappa)_{+}\partial_{x_{s}}a_{\varepsilon}(x,Du)\cdot\left(D_{s}u\otimes
D\eta\right)\,dx$
$\displaystyle=:\mbox{(IV)}_{x}+\mbox{(V)}_{x}+\mbox{(VI)}_{x}\,.$ (6.48)

By (6.9) we have

$\mbox{(IV)}_{z}\geq\int_{\mathcal{B}^{\kappa}}\eta^{2}(G_{\varepsilon}(x,\lvert
Du\rvert)-\kappa)_{+}g_{1,\varepsilon}(x,\lvert Du\rvert)\lvert
D^{2}u\rvert^{2}\,dx=:\mathcal{S}_{3}$ (6.49)

and for later use we also define

$\mathcal{S}_{4}:=\int_{\mathcal{B}^{\kappa}}\eta^{2}\lvert
D(G_{\varepsilon}(x,\lvert Du\rvert)-k)_{+}\rvert^{2}\,dx\,.$ (6.50)

We then consider two different cases.

Case 1: $1<\gamma<2$ in (4.4), and therefore it is $\vartheta=2$. We proceed
estimating the terms $\mbox{(V)}_{z}$ and $\mbox{(VI)}_{z}$. The estimate for
the term (V)z is similar to the one for (II)z in (6.20), see in particular
those for the terms $\mbox{(II)}_{z,3}$ and $\mbox{(II)}_{z,4}$; indeed, again
using (5.15), (5.29), (5.31), (6.3) and (6.13), we have

$\displaystyle\mbox{(V)}_{z}$
$\displaystyle\geq\mathcal{S}_{4}-c\int_{\mathcal{B}^{\kappa}}\eta^{2}h(x)[\bar{G}_{\varepsilon}(x,s)]^{1+\sigma+\hat{\sigma}}\lvert
D(G_{\varepsilon}(x,\lvert Du\rvert)-k)_{+}\rvert\,dx$
$\displaystyle\geq\frac{1}{2}\mathcal{S}_{4}-c\int_{B_{r}(x_{0})}[h(x)]^{2}[\bar{G}_{\varepsilon}(x,\lvert
Du\rvert)]^{2(1+\sigma+\hat{\sigma})}\,dx\,,$

with $c\equiv c(n,N,\nu,\gamma,c_{\textnormal{b}})$. Using (6.7)-(6.12), we
then have

$\displaystyle\mbox{(VI)}_{z}$
$\displaystyle=2\int_{\mathcal{B}^{\kappa}}\eta(G_{\varepsilon}(x,\lvert
Du\rvert)-\kappa)_{+}\frac{\bar{a}_{\varepsilon}(x,\lvert
Du\rvert)}{g_{1,\varepsilon}(x,\lvert Du\rvert)}DG_{\varepsilon}(x,\lvert
Du\rvert)\cdot D\eta\,dx$
$\displaystyle\qquad+2\sum_{\alpha=1}^{N}\int_{\mathcal{B}^{\kappa}}\eta\mathds{1}_{\mathcal{D}}(|Du|)(G_{\varepsilon}(x,\lvert
Du\rvert)-\kappa)_{+}\frac{\bar{a}_{\varepsilon}^{\prime}(x,\lvert
Du\rvert)\lvert Du\rvert}{g_{1,\varepsilon}(x,\lvert Du\rvert)\lvert
Du\rvert^{2}}$ $\displaystyle\hskip 99.58464pt\times(DG_{\varepsilon}(x,\lvert
Du\rvert)\cdot Du^{\alpha})(Du^{\alpha}\cdot D\eta)\,dx$
$\displaystyle\qquad-2\int_{\mathcal{B}^{\kappa}}\eta(G_{\varepsilon}(x,\lvert
Du\rvert)-\kappa)_{+}\frac{\bar{a}_{\varepsilon}(x,\lvert
Du\rvert)}{g_{1,\varepsilon}(x,\lvert Du\rvert)}$ $\displaystyle\hskip
99.58464pt\times\left(\int_{T}^{\lvert
Du\rvert}\partial_{x}g_{1,\varepsilon}(x,t)t\,dt\cdot D\eta\right)\,dx$
$\displaystyle\qquad-2\sum_{\alpha=1}^{N}\int_{\mathcal{B}^{\kappa}}\eta\mathds{1}_{\mathcal{D}}(|Du|)(G_{\varepsilon}(x,\lvert
Du\rvert)-\kappa)_{+}\frac{\bar{a}_{\varepsilon}^{\prime}(x,\lvert
Du\rvert)\lvert Du\rvert}{g_{1,\varepsilon}(x,\lvert Du\rvert)\lvert
Du\rvert^{2}}$ $\displaystyle\hskip 71.13188pt\times\left(\int_{T}^{\lvert
Du\rvert}\partial_{x}g_{1,\varepsilon}(x,t)t\,dt\cdot
Du^{\alpha}\right)(Du^{\alpha}\cdot D\eta)\,dx\,.$

Using (5.29), (5.31), (6.3)-(6.4), and Young inequality, we have

$\displaystyle|\mbox{(VI)}_{z}|$
$\displaystyle\leq\frac{1}{4}\mathcal{S}_{4}+\frac{c}{r^{2}}\lVert\bar{G}_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{r}(x_{0}))}^{2\sigma}\int_{B_{r}(x_{0})}(G_{\varepsilon}(x,\lvert
Du\rvert)-\kappa)_{+}^{2}\,dx$
$\displaystyle\quad+c\int_{B_{r}(x_{0})}[h(x)]^{2}[\bar{G}_{\varepsilon}(x,\lvert
Du\rvert)]^{2(1+\hat{\sigma})}\,dx\,,$ (6.51)

where $c\equiv c(n,N,\nu,\gamma,c_{\textnormal{b}})$. Gathering the content of
displays from (6.47) to (6.51), and using them in (6.17), some further
elementary estimations we have

$\displaystyle\mathcal{S}_{3}+\mathcal{S}_{4}$ $\displaystyle\leq
c|\mbox{(IV)}_{x}|+c|\mbox{(V)}_{x}|+c|\mbox{(VI)}_{x}|$
$\displaystyle\quad+\frac{c}{r^{2}}\lVert\bar{G}_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{r}(x_{0}))}^{\vartheta\sigma}\int_{B_{r}(x_{0})}(G_{\varepsilon}(x,\lvert
Du\rvert)-\kappa)_{+}^{2}\,dx$
$\displaystyle\quad+c\int_{B_{r}(x_{0})}[h(x)]^{2}[\bar{G}_{\varepsilon}(x,\lvert
Du\rvert)]^{2(1+\sigma+\hat{\sigma})}\,dx$
$\displaystyle\quad+c\sum_{s=1}^{n}\int_{\mathcal{B}^{\kappa}}\lvert f\cdot
D_{s}\varphi_{s}\rvert\ \,dx$ (6.52)

where $\vartheta=2$ and $c\equiv c(n,N,\nu,\gamma,c_{\textnormal{b}})$.

Case 2: $\gamma\geq 2$ in (4.4), and therefore it is $\vartheta=1$. In this
case we use that the function $t\mapsto\bar{a}_{\varepsilon}(\cdot,t)$ is non-
decreasing, so that $\bar{a}_{\varepsilon}^{\prime}(\cdot)$ is non-negative
(when it exists). We note that

$\displaystyle\mbox{(V)}_{z}+\mbox{(VI)}_{z}$
$\displaystyle\quad\stackrel{{\scriptstyle(\ref{a8})}}{{=}}\sum_{s=1}^{n}\int_{\mathcal{B}^{\kappa}}\eta^{2}g_{1,\varepsilon}(x,\lvert
Du\rvert)\lvert Du\rvert\partial_{z}a_{\varepsilon}(x,Du)DD_{s}u\cdot
D_{s}u\otimes D\lvert Du\rvert\,dx$
$\displaystyle\qquad+2\sum_{s=1}^{n}\int_{\mathcal{B}^{\kappa}}\eta(G_{\varepsilon}(x,\lvert
Du\rvert)-k)_{+}\partial_{z}a_{\varepsilon}(x,Du)DD_{s}u\cdot D_{s}u\otimes
D\eta\,dx$
$\displaystyle\qquad+\sum_{s=1}^{n}\int_{\mathcal{B}^{\kappa}}\eta^{2}\partial_{z}a_{\varepsilon}(x,Du)DD_{s}u\cdot
D_{s}u\otimes\int_{T}^{\lvert
Du\rvert}\partial_{x}g_{1,\varepsilon}(x,t)t\,dt\,dx$ $\displaystyle\quad\ \
=:\mbox{(V)}_{z,1}+\mbox{(VI)}_{z}+\mbox{(V)}_{z,2}\,.$ (6.53)

In turn, we have

$\displaystyle\mbox{(V)}_{z,1}\stackrel{{\scriptstyle(\ref{diso2})}}{{=}}\int_{\mathcal{B}^{\kappa}}\eta^{2}g_{1,\varepsilon}(x,\lvert
Du\rvert)\bar{a}_{\varepsilon}(x,|Du|)|D|Du||^{2}\lvert Du\rvert^{2}\,dx$
$\displaystyle\
+\sum_{\alpha=1}^{N}\int_{\mathcal{B}^{\kappa}}\eta^{2}g_{1,\varepsilon}(x,\lvert
Du\rvert)\mathds{1}_{\mathcal{D}}(|Du|)\bar{a}_{\varepsilon}^{\prime}(x,|Du|)|Du|(D|Du|\cdot
Du^{\alpha})^{2}\,dx$
$\displaystyle\stackrel{{\scriptstyle(\ref{a1}),\bar{a}_{\varepsilon}^{\prime}(\cdot)\geq
0}}{{\geq}}\int_{\mathcal{B}^{\kappa}}\eta^{2}[g_{1,\varepsilon}(x,\lvert
Du\rvert)\lvert Du\rvert]^{2}|D|Du||^{2}\,dx\,.$ (6.54)

For $\mbox{(VI)}_{z}$, keeping in mind the identity in the last display, we
again use (6.7) and $\bar{a}_{\varepsilon}^{\prime}(\cdot)\geq 0$ to estimate
via Young inequality as follows:

$\displaystyle 4|\mbox{(VI)}_{z}|\leq
8\int_{\mathcal{B}^{\kappa}}\eta(G_{\varepsilon}(x,\lvert
Du\rvert)-k)_{+}\bar{a}_{\varepsilon}(x,\lvert Du\rvert)|Du||D|Du|\cdot
D\eta|\,dx$ $\displaystyle\qquad\qquad\
+8\sum_{\alpha=1}^{N}\int_{\mathcal{B}^{\kappa}}\eta(G_{\varepsilon}(x,\lvert
Du\rvert)-k)_{+}\mathds{1}_{\mathcal{D}}(|Du|)\bar{a}_{\varepsilon}^{\prime}(x,|Du|)$
$\displaystyle\hskip 113.81102pt\times|(D|Du|\cdot
Du^{\alpha})(Du^{\alpha}\cdot D\eta)|\,dx$
$\displaystyle\qquad\leq\mbox{(V)}_{z,1}+c\int_{\mathcal{B}^{\kappa}}\frac{(G_{\varepsilon}(x,\lvert
Du\rvert)-k)_{+}^{2}}{g_{1,\varepsilon}(x,\lvert
Du\rvert)}\bar{a}_{\varepsilon}(x,|Du|)|D\eta|^{2}\,dx$
$\displaystyle\quad\qquad+c\sum_{\alpha=1}^{N}\int_{\mathcal{B}^{\kappa}}\frac{(G_{\varepsilon}(x,\lvert
Du\rvert)-k)_{+}^{2}}{g_{1,\varepsilon}(x,\lvert
Du\rvert)}\mathds{1}_{\mathcal{D}}(|Du|)\bar{a}_{\varepsilon}^{\prime}(x,|Du|)\frac{(Du^{\alpha}\cdot
D\eta)^{2}}{|Du|}\,dx$
$\displaystyle\qquad\leq\mbox{(V)}_{z,1}+\frac{c}{r^{2}}\lVert\bar{G}_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{r}(x_{0}))}^{\sigma}\int_{B_{r}(x_{0})}(G_{\varepsilon}(x,\lvert
Du\rvert)-\kappa)_{+}^{2}\,dx\,.$

As for $\mbox{(V)}_{z,2}$, by letting

$\mathcal{I}:=\int_{T}^{\lvert
Du\rvert}\partial_{x}g_{1,\varepsilon}(x,t)t\,dt\Longrightarrow|\mathcal{I}|\stackrel{{\scriptstyle(\ref{z10})}}{{\leq}}ch(x)[\bar{G}_{\varepsilon}(x,\lvert
Du\rvert)]^{1+\hat{\sigma}}$ (6.55)

we have, again using that $\bar{a}_{\varepsilon}^{\prime}(\cdot)\geq 0$

$\displaystyle 4|\mbox{(V)}_{z,2}|$
$\displaystyle\stackrel{{\scriptstyle(\ref{diso2})}}{{\leq}}$ $\displaystyle
4\int_{\mathcal{B}^{\kappa}}\eta^{2}\bar{a}_{\varepsilon}(x,\lvert
Du\rvert)|Du|\left|D|Du|\cdot\mathcal{I}\right|\,dx$ (6.56)
$\displaystyle\quad+4\sum_{\alpha=1}^{N}\int_{\mathcal{B}^{\kappa}}\eta^{2}\mathds{1}_{\mathcal{D}}(|Du|)\bar{a}_{\varepsilon}^{\prime}(x,|Du|)$
$\displaystyle\hskip 56.9055pt\times|(D|Du|\cdot
Du^{\alpha})(Du^{\alpha}\cdot\mathcal{I})|\,dx$
$\displaystyle\stackrel{{\scriptstyle(\ref{VVVV})}}{{\leq}}$
$\displaystyle\mbox{(V)}_{z,1}+c\int_{B_{r}(x_{0})}\frac{\eta^{2}[\bar{a}_{\varepsilon}(x,|Du|)+\bar{a}_{\varepsilon}^{\prime}(x,|Du|)|Du|]}{g_{1,\varepsilon}(x,\lvert
Du\rvert)}|\mathcal{I}|^{2}\,dx$
$\displaystyle\stackrel{{\scriptstyle\eqref{a2}_{2}}}{{\leq}}$
$\displaystyle\mbox{(V)}_{z,1}+c\int_{B_{r}(x_{0})}\frac{g_{2,\varepsilon}(x,\lvert
Du\rvert)}{g_{1,\varepsilon}(x,\lvert
Du\rvert)}\left|\mathcal{I}\right|^{2}\,dx$
$\displaystyle\stackrel{{\scriptstyle(\ref{z9}),(\ref{letting})}}{{\leq}}$
$\displaystyle\mbox{(V)}_{z,1}+c\int_{B_{r}(x_{0})}[h(x)]^{2}[\bar{G}_{\varepsilon}(x,\lvert
Du\rvert)]^{\sigma+2(1+\hat{\sigma})}\,dx.$

On the other hand, we have

$\displaystyle\mathcal{S}_{4}$
$\displaystyle\stackrel{{\scriptstyle(\ref{a8})}}{{\leq}}$ $\displaystyle
c\int_{\mathcal{B}^{\kappa}}\eta^{2}\left([g_{1,\varepsilon}(x,\lvert
Du\rvert)\lvert Du\rvert]^{2}|D|Du||^{2}+|\mathcal{I}|^{2}\right)\,dx$ (6.57)
$\displaystyle\stackrel{{\scriptstyle(\ref{letting})}}{{\leq}}$ $\displaystyle
c\int_{\mathcal{B}^{\kappa}}\eta^{2}\left([g_{1,\varepsilon}(x,\lvert
Du\rvert)\lvert Du\rvert]^{2}|D|Du||^{2}\right.$ $\displaystyle\hskip
56.9055pt\left.+|h(x)|^{2}[\bar{G}_{\varepsilon}(x,\lvert
Du\rvert)]^{2(1+\hat{\sigma})}\right)\,dx$
$\displaystyle\stackrel{{\scriptstyle(\ref{VVVV})}}{{\leq}}$ $\displaystyle
c\mbox{(V)}_{z,1}+c\int_{B_{r}(x_{0})}[h(x)]^{2}[\bar{G}_{\varepsilon}(x,\lvert
Du\rvert)]^{2(1+\hat{\sigma})}\,dx\,.$

Assembling the content of displays from (6.53) to (6.57) and using it in
(6.17), we again conclude with (6.52), but this time with $\vartheta=1$. We
proceed estimating the $x$-terms coming from (6.52) (these have been defined
in (6.48)), using $\eqref{growthd}_{3,4}$, (5.37) as follows:

$\displaystyle\lvert\mbox{(IV)}_{x}\rvert+\lvert\mbox{(V)}_{x}\rvert$
$\displaystyle\leq\bar{\varepsilon}\mathcal{S}_{3}+\bar{\varepsilon}\mathcal{S}_{4}+\frac{c}{\bar{\varepsilon}}\int_{B_{r}(x_{0})}\eta^{2}[h(x)]^{2}[\bar{G}_{\varepsilon}(x,\lvert
Du\rvert)]^{2(1+\sigma)}\,dx\,,$ $\displaystyle\lvert\mbox{(VI)}_{x}\rvert$
$\displaystyle\stackrel{{\scriptstyle\eqref{z11}_{1}}}{{\leq}}\frac{c}{r^{2}}\int_{B_{r}(x_{0})}(G_{\varepsilon}(x,\lvert
Du\rvert)-\kappa)_{+}^{2}\,dx$ $\displaystyle\qquad\ \
+c\int_{B_{r}(x_{0})}[h(x)]^{2}[\bar{G}_{\varepsilon}(x,\lvert
Du\rvert)]^{2(1+\sigma)}\,dx\,,$

with $c\equiv c(\texttt{data})$ and arbitrary $\bar{\varepsilon}\in(0,1)$.
Finally, the estimate of the terms involving $f$ can be done by using Young
inequality and (5.27) as follows:

$\displaystyle\sum_{s=1}^{n}\int_{\mathcal{B}^{\kappa}}\lvert f\cdot
D_{s}\varphi_{s}\rvert\,dx\leq\bar{\varepsilon}\mathcal{S}_{3}+\bar{\varepsilon}\mathcal{S}_{4}+\frac{c}{\bar{\varepsilon}}\int_{\mathcal{B}^{\kappa}}|D\eta|^{2}(G_{\varepsilon}(x,\lvert
Du\rvert)-\kappa)_{+}^{2}\,dx$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+\frac{c}{\bar{\varepsilon}}\int_{\mathcal{B}^{\kappa}}\eta^{2}\left[\frac{(G_{\varepsilon}(x,\lvert
Du\rvert)-\kappa)_{+}}{g_{1,\varepsilon}(x,\lvert Du\rvert)}+\lvert
Du\rvert^{2}\right]\lvert f\rvert^{2}\,dx\,$
$\displaystyle\quad\leq\bar{\varepsilon}\mathcal{S}_{3}+\bar{\varepsilon}\mathcal{S}_{4}+\frac{c}{\bar{\varepsilon}r^{2}}\int_{\mathcal{B}^{\kappa}}(G_{\varepsilon}(x,\lvert
Du\rvert)-\kappa)_{+}^{2}\,dx$
$\displaystyle\qquad\qquad+\frac{c}{\bar{\varepsilon}}\lVert
Du\rVert_{L^{\infty}(B_{r_{0}}(x_{0}))}^{2}\int_{\mathcal{B}^{\kappa}}\eta^{2}\lvert
f\rvert^{2}\,dx\,,$ (6.58)

for $c\equiv c(\texttt{data})$ and arbitrary $\bar{\varepsilon}\in(0,1)$.
Collecting the estimates in the last three displays to (6.52), recalling that
recalling that $\bar{G}_{\varepsilon}(\cdot)\geq 1$, and selecting
$\bar{\varepsilon}>0$ sufficiently small in order to reabsorb terms, we
complete the proof of Lemma 8.

###  6.6 Nonlinear iterations

In this section we finally derive pointwise gradient bounds. This goes via
Lemma 9 and Proposition 3 below.

###### Lemma 9

Let $u\in W^{1,\gamma}(\mathcal{B};\mathbb{R}^{N})$ be a solution to
$\eqref{a0}$, under assumptions set in (4.20) for $n>2$. If
$B_{2r_{0}}(x_{0})\Subset\mathcal{B}$ is a ball such that $x_{0}$ is a
Lebesgue point of $|Du|$, then

$\displaystyle G_{\varepsilon}(x_{0},\lvert Du(x_{0})\rvert)$
$\displaystyle\quad\leq\kappa+c\lVert\bar{G}_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{r}(x_{0}))}^{n\vartheta\sigma/4}\left(\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{B_{r_{0}}(x_{0})}(G_{\varepsilon}(x,\lvert
Du\rvert)-\kappa)_{+}^{2}\,dx\right)^{1/2}$ $\displaystyle\quad\ \ \ \
+c\lVert\bar{G}_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{r}(x_{0}))}^{(n-2)\vartheta\sigma/4}$
$\displaystyle\quad\qquad\
\times\left[\mathbf{P}_{1}^{\mathfrak{h}}(x_{0},2r_{0})+\lVert
Du\rVert_{L^{\infty}(B_{r_{0}}(x_{0}))}\mathbf{P}_{1}^{f}(x_{0},2r_{0})\right]\,,$
(6.59)

holds for all $\kappa\geq 0$, with $c\equiv c(\texttt{data})$, where
$\vartheta\equiv\vartheta(\gamma)$ is as in (4.15), and

$\displaystyle\mathfrak{h}(x):=h(x)[\bar{G}_{\varepsilon}(x,\lvert
Du\rvert)]^{1+\sigma+\hat{\sigma}}\,.$ (6.60)

###### Proof

Note that we can assume that $|Du(x_{0})|>T$, otherwise (9) is trivial by the
very definition of $G_{\varepsilon}$; this obviously implies
$\|Du\|_{L^{\infty}(B_{r}(x_{0}))}>T$. Let us first note that $x_{0}$ is also
a Lebesgue point of $x\mapsto G_{\varepsilon}(x,\lvert Du(x)\rvert)$ and it is

$\lim_{r\to
0}\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{B_{r}(x_{0})}G_{\varepsilon}(x,\lvert
Du(x)\rvert)\,dx=G_{\varepsilon}(x_{0},\lvert Du(x_{0})\rvert)\,,$ (6.61)

i.e., the right-hand side denotes the precise representative of
$G_{\varepsilon}(\cdot,\lvert Du(\cdot)\rvert)$ at the point $x_{0}$. Indeed,
note that

$\displaystyle\limsup_{r\to
0}\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{B_{r}(x_{0})}|G_{\varepsilon}(x,\lvert
Du(x)\rvert)-G_{\varepsilon}(x_{0},\lvert Du(x_{0})\rvert)|\,dx$
$\displaystyle\qquad\leq\limsup_{r\to
0}\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{B_{r}(x_{0})}|G_{\varepsilon}(x,\lvert
Du(x)\rvert)-G_{\varepsilon}(x_{0},\lvert Du(x)\rvert)|\,dx$
$\displaystyle\qquad\qquad+\limsup_{r\to
0}\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{B_{r}(x_{0})}|G_{\varepsilon}(x_{0},\lvert
Du(x)\rvert)-G_{\varepsilon}(x_{0},\lvert Du(x_{0})\rvert)|\,dx$
$\displaystyle\qquad\qquad\qquad=:\limsup_{r\to
0}\mathcal{C}_{1}(r)+\limsup_{r\to 0}\mathcal{C}_{2}(r)\,.$ (6.62)

As $x_{0}$ is a Lebesgue point for $Du$, $t\mapsto G_{\varepsilon}(x_{0},t)$
is locally Lipschitz-regular, and $Du$ is locally bounded, we have

$\lim_{r\to 0}\,\mathcal{C}_{2}(r)\lesssim\lim_{r\to
0}\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{B_{r}(x_{0})}|\lvert
Du(x)\rvert-\lvert Du(x_{0})\rvert|\,dx=0\,.$ (6.63)

As for the term $\mathcal{C}_{1}(\cdot)$, we have, also using Fubini’s

$\displaystyle\mathcal{C}_{1}(r)\leq\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{B_{r}(x_{0})}\int_{T}^{\max\\{|Du(x)|,T\\}}|g_{1,\varepsilon}(x,s)-g_{1,\varepsilon}(x_{0},s)|s\,ds\,dx$
$\displaystyle\quad\leq\int_{T}^{\|Du\|_{L^{\infty}(B_{r}(x_{0}))}}\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{B_{r}(x_{0})}|g_{1,\varepsilon}(x,s)-g_{1,\varepsilon}(x_{0},s)|\,dx\,s\,ds$
$\displaystyle\quad\leq\|Du\|_{L^{\infty}(B_{r}(x_{0}))}^{2}\sup_{s\in[T,\|Du\|_{L^{\infty}(B_{r}(x_{0}))}]}\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{B_{r}(x_{0})}|g_{1,\varepsilon}(x,s)-g_{1,\varepsilon}(x_{0},s)|\,dx.$

Recall that $g_{1}(\cdot)$ is assumed to be continuous on
$\Omega\times(0,\infty)$. By the definition in (5.7), this implies that also
$g_{1,\varepsilon}$ is continuous and therefore it is uniformly continuous on
$\overline{B_{r_{0}}(x_{0})}\times[T,\|Du\|_{L^{\infty}(B_{r}(x_{0}))}]$. This
and the content of the last display it is sufficient to infer that
$\mathcal{C}_{1}(r)\to 0$ as $r\to 0$. This fact, together with (6.62)-(6.63),
yields (6.61). Thanks to (8) we can verify (1) with the choices
$v(\cdot)\equiv G_{\varepsilon}(\cdot,\lvert Du(\cdot)\rvert)$,
$f_{1}\equiv\mathfrak{h}$, $f_{2}\equiv f$,
$M_{1}\equiv\lVert\bar{G}_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{r}(x_{0}))}^{\vartheta\sigma/2}$, $M_{2}\equiv
1$ and $M_{3}\equiv\lVert Du\rVert_{L^{\infty}(B_{r_{0}}(x_{0}))}$. Applying
Lemma 1, inequality (1) yields (9).

###### Proposition 3

Let $u\in W^{1,\gamma}(\mathcal{B};\mathbb{R}^{N})$ be a solution to
$\eqref{a0}$, under assumptions set in (4.20) for $n>2$. There exists a
positive radius $R_{*}\equiv R_{*}(\texttt{data},\mathfrak{f}(\cdot))\leq 1$
such that if $\texttt{r}(\mathcal{B})\leq R_{*}$ and $B_{\varsigma}\Subset
B_{\varrho}$ are concentric balls contained in $\mathcal{B}$, then

$\displaystyle[E_{\varepsilon}(\lVert
Du\rVert_{L^{\infty}(B_{\varsigma})})]^{\gamma}+\lVert
G_{\varepsilon}(\cdot,\lvert Du\rvert)\rVert_{L^{\infty}(B_{\varsigma})}$
$\displaystyle\qquad\leq\frac{c}{(\varrho-\varsigma)^{\beta}}\left[\|F_{\varepsilon}(\cdot,Du)\|_{L^{1}(B_{\varrho})}^{\theta}+\lVert
f\rVert_{L(n,1)(B_{\varrho})}^{\theta}+1\right]$ (6.64)

holds with $c\equiv c(\texttt{data},\lVert h\rVert_{L^{d}(\mathcal{B})})\geq
1$ and $\beta,\theta\equiv\beta,\theta(n,d,\gamma,\sigma,\hat{\sigma})>0$.

###### Proof

We take numbers $\tau,p\equiv\tau,p(n,d,\sigma,\hat{\sigma})$ such that

$\displaystyle 0<\tau<d-n\quad\mbox{and}\quad
p:=\frac{(1+\sigma+\hat{\sigma})(n+\tau)d}{d-n-\tau}\,,$ (6.65)

where $d>n$ is the exponent from (4.15), and fix $R_{*}\equiv
R_{*}(\texttt{data},\mathfrak{f}(\cdot))>0$ as the radius from Proposition 1,
so that (6.16) holds such $p$. With $B_{\varsigma}\Subset B_{\varrho}$ being
the balls considered in the statement, with no loss of generality we can
assume that $\lVert Du\rVert_{L^{\infty}(B_{\varsigma})}\geq T$ otherwise the
assertion in (3) is trivial by choosing $c$ large enough. Let
$\varrho_{1}:=\varsigma+{(\varrho-\varsigma)/2}$ and consider concentric balls
$B_{\varsigma}\Subset B_{\tau_{1}}\Subset B_{\tau_{2}}\Subset
B_{\varrho_{1}}\Subset B_{\varrho}$, a point $x_{0}\in B_{\tau_{1}}$ which is
a Lebesgue point for $|Du|$, and $r_{0}:=(\tau_{2}-\tau_{1})/8$, so that
$B_{2r_{0}}(x_{0})\Subset B_{\tau_{2}}$. Needless to say, a.e. point in
$B_{\tau_{1}}$ qualifies. By (6.26) we find

$\displaystyle\lVert\bar{G}_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{r_{0}}(x_{0}))}\leq(T^{2}+1)^{\gamma/2}\left[\lVert
G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{2}})}+1\right].$ (6.66)

This and (5.19)1 imply

$\displaystyle\lVert Du\rVert_{L^{\infty}(B_{\tau_{2}})}\leq c\lVert
G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{2}})}^{1/\gamma}+c\,,$ (6.67)

for $c\equiv c(\nu,\gamma,T)$. We then apply (9) on $B_{r_{0}}(x_{0})$, and
also using (6.66)-(6.67) and Hölder inequality (by (6.65) it is $p\geq 2$), we
obtain

$\displaystyle G_{\varepsilon}(x_{0},\lvert Du(x_{0})\rvert)$
$\displaystyle\leq cr_{0}^{-n/p}\left[\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{2}})}^{n\vartheta\sigma/4}+1\right]\|G_{\varepsilon}(\cdot,\lvert
Du\rvert)\|_{L^{p}(B_{r_{0}}(x_{0}))}$ $\displaystyle\qquad\quad+c\left[\lVert
G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{2}})}^{(n-2)\vartheta\sigma/4}+1\right]$
$\displaystyle\qquad\qquad\quad\times\left[\mathbf{P}_{1}^{\mathfrak{h}}(x_{0},2r_{0})+\lVert
Du\rVert_{L^{\infty}(B_{\tau_{2}})}\mathbf{P}_{1}^{f}(x_{0},2r_{0})\right],$
(6.68)

with $c\equiv c(\texttt{data})$. With (6.16) (where we take $\xi\equiv r_{0}$
and $\varrho\equiv 2r_{0}$) we further bound

$\|G_{\varepsilon}(\cdot,\lvert Du\rvert)\|_{L^{p}(B_{r_{0}}(x_{0}))}\leq
cr_{0}^{-\beta_{p}}\left[\|F_{\varepsilon}(\cdot,Du)\|_{L^{1}(B_{\varrho})}^{\theta_{p}}+1\right],$

for $c\equiv c(\texttt{data},\lVert h\rVert_{L^{d}(\mathcal{B})})$. Using also
(6.67) in (6.6), recalling that $x_{0}\in B_{\tau_{1}}$ is a arbitrary
Lebesgue point for $|Du|$, we obtain

$\displaystyle\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{1}})}$
$\displaystyle\leq\frac{c}{(\tau_{2}-\tau_{1})^{n/p+\beta_{p}}}\lVert
G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{2}})}^{n\vartheta\sigma/4}\left[\lVert
F_{\varepsilon}(\cdot,Du)\rVert_{L^{1}(B_{\varrho})}^{\theta_{p}}+1\right]$
$\displaystyle\quad+c\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{2}})}^{(n-2)\vartheta\sigma/4}\|\mathbf{P}_{1}^{\mathfrak{h}}(\cdot,(\tau_{2}-\tau_{1})/4)\|_{L^{\infty}(B_{\tau_{1}})}$
$\displaystyle\quad+c\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{2}})}^{(n-2)\vartheta\sigma/4+1/\gamma}\|\mathbf{P}_{1}^{f}(\cdot,(\tau_{2}-\tau_{1})/4)\|_{L^{\infty}(B_{\tau_{1}})}$
$\displaystyle\quad+\frac{c}{(\tau_{2}-\tau_{1})^{n/p+\beta_{p}}}\left[\lVert
F_{\varepsilon}(\cdot,Du)\rVert_{L^{1}(B_{\varrho})}^{\theta_{p}}+1\right]$
$\displaystyle\quad+c\|\mathbf{P}_{1}^{\mathfrak{h}}(\cdot,(\tau_{2}-\tau_{1})/4)\|_{L^{\infty}(B_{\tau_{1}})}+c\|\mathbf{P}_{1}^{f}(\cdot,(\tau_{2}-\tau_{1})/4)\|_{L^{\infty}(B_{\tau_{1}})}\,,$
(6.69)

where $c\equiv c(\texttt{data},\lVert h\rVert_{L^{d}(\mathcal{B})})$. Now,
observe that (4.15) implies

$n\sigma\vartheta/4<1\quad\mbox{and}\quad(n-2)\sigma\vartheta/4<1-1/\gamma\,,$
(6.70)

therefore we can apply Young inequality in (6.6) to end up with

$\displaystyle\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{1}})}$
$\displaystyle\leq\frac{1}{2}\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{2}})}$
$\displaystyle\quad+\frac{c}{(\tau_{2}-\tau_{1})^{\beta_{*}}}\left[\lVert
F_{\varepsilon}(\cdot,Du)\rVert_{L^{1}(B_{\varrho})}^{\theta_{*}}+1\right]$
$\displaystyle\quad+c\|\mathbf{P}_{1}^{\mathfrak{h}}(\cdot,(\tau_{2}-\tau_{1})/4)\|_{L^{\infty}(B_{\tau_{1}})}^{\theta_{*}}$
$\displaystyle\quad+c\|\mathbf{P}_{1}^{f}(\cdot,(\tau_{2}-\tau_{1})/4)\|_{L^{\infty}(B_{\tau_{1}})}^{\theta_{*}}+c\,,$
(6.71)

where $c\equiv c(\texttt{data},\lVert h\rVert_{L^{d}(\mathcal{B})})$ where it
is
$\beta_{*},\theta_{*}\equiv\beta_{*},\theta_{*}(n,d,\gamma,\sigma,\hat{\sigma})>0$.
Inequality (6.6) allows to apply Lemma 2 with the obvious choice
$\mathcal{Z}(t):=\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{t})}$, and this leads to

$\displaystyle\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\varsigma})}$
$\displaystyle\,\leq\frac{c}{(\varrho-\varsigma)^{\beta_{*}}}\left[\lVert
F_{\varepsilon}(\cdot,Du)\rVert_{L^{1}(B_{\varrho})}^{\theta_{*}}+1\right]$
$\displaystyle\quad+c\|\mathbf{P}_{1}^{\mathfrak{h}}(\cdot,(\varrho-\varsigma)/4)\|_{L^{\infty}(B_{\varrho_{1}})}^{\theta_{*}}$
$\displaystyle\quad+c\|\mathbf{P}_{1}^{f}(\cdot,(\varrho-\varsigma)/4)\|_{L^{\infty}(B_{\varrho_{1}})}^{\theta_{*}}+c$
(6.72)

for $c\equiv c(\texttt{data},\lVert h\rVert_{L^{d}(\mathcal{B})})$. By using
(3.1)1 we infer

$\|\mathbf{P}_{1}^{f}(\cdot,(\varrho-\varsigma)/4)\|_{L^{\infty}(B_{\varrho_{1}})}\leq
c\lVert f\rVert_{L(n,1)(B_{\varrho})}\,.$

Moreover, with $\varrho_{2}:=\varrho_{1}+(\varrho-\varsigma)/4$; also using
(6.65) and Hölder inequality yields (recall that $s_{*}=1$)

$\displaystyle\|\mathbf{P}_{1}^{\mathfrak{h}}(\cdot,(\varrho-\varsigma)/4)\|_{L^{\infty}(B_{\varrho_{1}})}$
$\displaystyle\stackrel{{\scriptstyle\eqref{potlore}_{1}}}{{\leq}}$
$\displaystyle c\lVert\mathfrak{h}\rVert_{L(n,1)(B_{\varrho_{2}})}$
$\displaystyle\stackrel{{\scriptstyle(\ref{normine})}}{{\leq}}$ $\displaystyle
c(\tau)\lVert\mathfrak{h}\rVert_{L^{n+\tau}(B_{\varrho_{2}})}$
$\displaystyle\stackrel{{\scriptstyle(\ref{fk0})}}{{\leq}}$ $\displaystyle
c\lVert h\rVert_{L^{d}(B_{\varrho})}\lVert\bar{G}_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{p}(B_{\varrho_{2}})}^{1+\sigma+\hat{\sigma}}$
$\displaystyle\stackrel{{\scriptstyle(\ref{a44})}}{{\leq}}$
$\displaystyle\frac{c\lVert h\rVert_{L^{d}(B_{\varrho})}\left[\lVert
F_{\varepsilon}(\cdot,Du)\rVert_{L^{1}(B_{\varrho})}^{\theta_{p}(1+\sigma+\hat{\sigma})}+1\right]}{(\varrho-\varsigma)^{\beta_{p}(1+\sigma+\hat{\sigma})}}$

where $c\equiv c(\texttt{data},\lVert h\rVert_{L^{d}(\mathcal{B})})$. We have
applied Proposition 1 with the choice $s_{*}=1$, so that (6.15) is verified by
the assumption (4.15)1. Inserting the above two estimates in (6.6) and
recalling also (5.11) and (5.19)1, we finally end up with (3), where
$\beta:=\max\\{\beta_{*},\beta_{p}\theta_{*}(1+\sigma+\hat{\sigma})\\}$ and
$\theta:=\max\\{\theta_{*},\theta_{p}\theta_{*}(1+\sigma+\hat{\sigma})\\}.$

###  6.7 The case $n=2$

Here we consider the missing two-dimensional case. We start with the following
lemma, which is a hybrid counterpart of Proposition 1, in the sense that the a
priori estimate involved still contains the $L^{\infty}$-norm of $Du$ in the
right-hand side. We recall that the number $m$ has been defined in (6.14).

###### Lemma 10

Let $u\in W^{1,\gamma}(\mathcal{B};\mathbb{R}^{N})$ be a solution to
$\eqref{a0}$ under assumptions set in (4.20) for $n=2$, where we replace
(4.15)2 by

$\sigma+\hat{\sigma}<\frac{s_{*}}{2}-\frac{s_{*}}{d}=\frac{s_{*}}{2m},\ \
\mbox{for some $s_{*}$ such that}\ 1\leq s_{*}<2m(1+\sigma+\hat{\sigma})\,.$
(6.74)

If $B_{\varsigma}\Subset B_{\varrho}$ are concentric balls contained in
$\mathcal{B}$, then, for every $p\geq 1$, there holds

$\displaystyle\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{p}(B_{\varsigma})}$
$\displaystyle\leq\frac{c}{(\varrho-\varsigma)^{\beta_{p}}}\left[\lVert
G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{s_{*}}(B_{\varrho})}^{\theta_{p}}+1\right]$
$\displaystyle\quad+c\lVert
Du\rVert_{L^{\infty}(B_{\varrho})}\|f\|_{L^{2}(B_{\varrho})}$ (6.75)

with $c\equiv c(\texttt{data},\lVert h\rVert_{L^{d}(\mathcal{B})},p)\geq 1$,
$\beta_{p},\theta_{p}\equiv\beta_{p},\theta_{p}(d,\sigma,\hat{\sigma},p)>0$.

###### Proof

We can confine ourselves to prove (6.75) for sufficiently large $p$, and we
consider

$\displaystyle
p>\max\left\\{2m(1+\sigma+\hat{\sigma}),\frac{2ms_{*}}{s_{*}-2m(\sigma+\hat{\sigma})}\right\\}=\frac{2ms_{*}}{s_{*}-2m(\sigma+\hat{\sigma})}\,.$
(6.76)

The last equality comes from the second inequality in (6.74). Note that such a
choice is possible thanks to (6.74), making the denominator of the last
quantity different than zero. In the following lines all the balls will be
concentric to $B_{\varrho}$. We look back at the proof of Proposition 1, take
$\kappa=0$ to obtain the test function
$\varphi_{s}=\eta^{2}G_{\varepsilon}(x,\lvert Du\rvert)D_{s}u$ for
$s\in\\{1,\ldots,n\\}$, and perform exactly the same calculations made there
up to (6.23). For the terms $\mbox{(I)}_{f}$-$\mbox{(III)}_{f}$ involving the
right-hand side $f$, we note that, as $\kappa=0$, the test maps $\varphi_{s}$
used in the proof of Propositions 1 and Lemma 8 do coincide (actually, we take
$\kappa=0$ in both Propositions 1 and Lemma 8). Therefore we can use estimate
(6.58), where $c\equiv c(\texttt{data})$ and $\bar{\varepsilon}\in(0,1)$; here
$\mathcal{S}_{3}$ and $\mathcal{S}_{4}$ have been defined in (6.49) and
(6.50), respectively. All together, choosing $\bar{\varepsilon}>0$ small
enough and re-absorbing terms in a standard way, we obtain

$\displaystyle\mathcal{S}_{3}+\mathcal{S}_{4}$ $\displaystyle\leq c(\lVert
h\rVert_{L^{d}(\mathcal{B})}^{2}+1)$
$\displaystyle\qquad\times\left(\int_{\mathcal{B}}(\eta^{2m}+\lvert
D\eta\rvert^{2m})\left[[G_{\varepsilon}(x,\lvert
Du\rvert)]^{2m(1+\sigma+\hat{\sigma})}+1\right]\,dx\right)^{1/m}$
$\displaystyle\quad+c\lVert
Du\rVert_{L^{\infty}(B_{\varrho})}^{2}\int_{\mathcal{B}}\eta^{2}\lvert
f\rvert^{2}\,dx\,,$ (6.77)

for $c\equiv c(\texttt{data})$. As it is $|D\eta|\lesssim
1/(\tau_{2}-\tau_{1})$, elementary manipulations on (6.7) yield

$\displaystyle\lVert D(\eta G_{\varepsilon}(\cdot,\lvert
Du\rvert))\rVert_{L^{2}(B_{\tau_{2}})}^{2}$ $\displaystyle\leq\frac{c(\lVert
h\rVert_{L^{d}(\mathcal{B})}+1)^{2}}{(\tau_{2}-\tau_{1})^{2}}\lVert
G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{2m(1+\sigma+\hat{\sigma})}(B_{\tau_{2}})}^{2(1+\sigma+\hat{\sigma})}$
$\displaystyle\quad+\frac{c(\lVert
h\rVert_{L^{d}(\mathcal{B})}+1)^{2}}{(\tau_{2}-\tau_{1})^{2}}+c\lVert
Du\rVert_{L^{\infty}(B_{\varrho})}^{2}\int_{B_{\tau_{2}}}\lvert
f\rvert^{2}\,dx$

so that Sobolev embedding gives

$\displaystyle\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{p}(B_{\tau_{1}})}$ $\displaystyle\leq
c_{p}\tau_{2}^{2/p}\lVert D(\eta G_{\varepsilon}(\cdot,\lvert
Du\rvert))\rVert_{L^{2}(B_{\tau_{2}})}$
$\displaystyle\leq\frac{c}{(\tau_{2}-\tau_{1})}\lVert
G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{2m(1+\sigma+\hat{\sigma})}(B_{\tau_{2}})}^{1+\sigma+\hat{\sigma}}$
$\displaystyle\quad+\frac{c}{(\tau_{2}-\tau_{1})}+c\lVert
Du\rVert_{L^{\infty}(B_{\varrho})}\|f\|_{L^{2}(B_{\tau_{2}})}\,,$ (6.78)

with $c\equiv c(\texttt{data},\|h\|_{L^{d}(\mathcal{B})})$. With
$\lambda_{p}\in(0,1)$ being defined through

$\displaystyle\frac{1}{2m(1+\sigma+\hat{\sigma})}=\frac{1-\lambda_{p}}{s_{*}}+\frac{\lambda_{p}}{p}\Rightarrow\lambda_{p}=\frac{p[2m(1+\sigma+\hat{\sigma})-s_{*}]}{2m(p-s_{*})(1+\sigma+\hat{\sigma})}\,,$

using the interpolation inequality

$\displaystyle\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{2m(1+\sigma+\hat{\sigma})}(B_{\tau_{2}})}\leq\lVert
G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{p}(B_{\tau_{2}})}^{\lambda_{p}}\lVert
G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{s_{*}}(B_{\tau_{2}})}^{1-\lambda_{p}}$

in (6.7), we get

$\displaystyle\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{p}(B_{\tau_{1}})}$
$\displaystyle\qquad\leq\frac{c}{(\tau_{2}-\tau_{1})}\lVert
G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{p}(B_{\tau_{2}})}^{\lambda_{p}(1+\sigma+\hat{\sigma})}\lVert
G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{s_{*}}(B_{\tau_{2}})}^{(1-\lambda_{p})(1+\sigma+\hat{\sigma})}$
$\displaystyle\qquad\quad+\frac{c}{(\tau_{2}-\tau_{1})}+c\lVert
Du\rVert_{L^{\infty}(B_{\varrho})}\|f\|_{L^{2}(B_{\tau_{2}})}\,.$

Using (6.74) and (6.76), we see that $\lambda_{p}(1+\sigma+\hat{\sigma})<1$,
thus Young inequality gives

$\displaystyle\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{p}(B_{\tau_{1}})}\leq\frac{1}{2}\lVert
G_{\varepsilon}(\cdot,\lvert Du\rvert)\rVert_{L^{p}(B_{\tau_{2}})}$
$\displaystyle\qquad+\frac{c}{(\tau_{2}-\tau_{1})^{\beta_{p}}}\left[\lVert
G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{s_{*}}(B_{\tau_{2}})}^{\theta_{p}}+1\right]+c\lVert
Du\rVert_{L^{\infty}(B_{\varrho})}\|f\|_{L^{2}(B_{\tau_{2}})}$

with $c\equiv c(\texttt{data},\lVert h\rVert_{L^{d}(\mathcal{B})})$ and

$\beta_{p}:=\frac{2m(p-s_{*})}{s_{*}p-2m[s_{*}+p(\sigma+\hat{\sigma})]}\ \
\mbox{and}\ \
\theta_{p}:=\frac{s_{*}p-2ms_{*}(1+\sigma+\hat{\sigma})}{s_{*}p-2m[s_{*}+p(\sigma+\hat{\sigma})]}\,.$
(6.79)

Lemma 2 with the choice $\mathcal{Z}(t)\equiv\lVert
G_{\varepsilon}(\cdot,\lvert Du\rvert)\rVert_{L^{p}(B_{t})}$ now gives (6.75).

We finally come to the a priori gradient bound in the two dimensional case.

###### Proposition 4

Let $u\in W^{1,\gamma}(\mathcal{B};\mathbb{R}^{N})$ be a solution to
$\eqref{a0}$ under assumptions set in (4.20) for $n=2$. If
$B_{\varsigma}\Subset B_{\varrho}$ are concentric balls contained in
$\mathcal{B}$, then

$\displaystyle[E_{\varepsilon}(\lVert
Du\rVert_{L^{\infty}(B_{\varsigma})})]^{\gamma}+\lVert
G_{\varepsilon}(\cdot,\lvert Du\rvert)\rVert_{L^{\infty}(B_{\varsigma})}$
$\displaystyle\qquad\leq\frac{c}{(\varrho-\varsigma)^{\beta}}\left[\|F_{\varepsilon}(\cdot,Du)\|_{L^{1}(B_{\varrho})}^{\theta}+\lVert
f\rVert_{L^{2}(\log L)^{\mathfrak{a}}(B_{\varrho})}^{\theta}+1\right]\,,$
(6.80)

holds with $c\equiv c(\texttt{data},\lVert h\rVert_{L^{d}(\mathcal{B})})\geq
1$, $\beta,\theta\equiv\beta,\theta(d,\gamma,\sigma,\hat{\sigma})>0$.

###### Proof

We proceed as for the proof of Proposition 3, keeping the notation used there,
including (6.65). By Lemma 8 we use Lemma 1 with the choice made at the end of
the proof of Lemma 9, and this for $n=2$ gives

$\displaystyle G_{\varepsilon}(x_{0},\lvert Du(x_{0})\rvert)$
$\displaystyle\quad\leq cr_{0}^{-2/p}\left[\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{2}})}^{(1+\delta)\vartheta\sigma/2}+1\right]\|G_{\varepsilon}(\cdot,\lvert
Du\rvert)\|_{L^{p}(B_{r_{0}}(x_{0}))}$ $\displaystyle\qquad\quad+c\left[\lVert
G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{2}})}^{\delta\vartheta\sigma/2}+1\right]$
$\displaystyle\qquad\qquad\quad\times\left[\mathbf{P}_{1}^{\mathfrak{h}}(\cdot,2r_{0})+\lVert
Du\rVert_{L^{\infty}(B_{\tau_{2}})}\mathbf{P}_{1}^{f}(\cdot,2r_{0})\right],$
(6.81)

for every $\delta\in(0,1/2)$, where $c\equiv c(\texttt{data},\lVert
h\rVert_{L^{d}(\mathcal{B})},\delta)$ and $\vartheta$ is as in (4.15). Then
(6.75) gives

$\displaystyle\|G_{\varepsilon}(\cdot,\lvert
Du\rvert)\|_{L^{p}(B_{r_{0}}(x_{0}))}$ $\displaystyle\quad\leq
cr_{0}^{-\beta_{p}}\left[\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{1}(B_{2r_{0}})}^{\theta_{p}}+1\right]+c\lVert
Du\rVert_{L^{\infty}(B_{\tau_{2}})}\|f\|_{L^{2}(B_{2r_{0}})}$

where $p$ is as in (6.65) with $n=2$, $c\equiv c(\texttt{data},\lVert
h\rVert_{L^{d}(\mathcal{B})},p)$ and $\beta_{p},\theta_{p}$ are as in (6.79).
Note that here we are using Lemma 10 with the choice $s_{*}=1$, which is
allowed as the assumption gives that $\sigma+\hat{\sigma}<1/2-1/d$, which is
(4.15) for $n=2$. Combining the last two estimates, that hold for a.e.
$x_{0}\in B_{\tau_{1}}$, we have

$\displaystyle\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{1}})}$ $\displaystyle\ \
\leq\frac{c}{(\tau_{2}-\tau_{1})^{2/p+\beta_{p}}}\left[\lVert
G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{2}})}^{(1+\delta)\vartheta\sigma/2}+1\right]\left[\lVert
G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{1}(B_{\varrho})}^{\theta_{p}}+1\right]$ $\displaystyle\ \
\qquad+\frac{c}{(\tau_{2}-\tau_{1})^{2/p}}\left[\lVert
G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{2}})}^{(1+\delta)\vartheta\sigma/2}+1\right]\lVert
Du\rVert_{L^{\infty}(B_{\tau_{2}})}\|f\|_{L^{2}(B_{\varrho})}$ $\displaystyle\
\ \qquad+c\left[\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{2}})}^{\delta\vartheta\sigma/2}+1\right]$
$\displaystyle\hskip 42.67912pt\times\lVert
Du\rVert_{L^{\infty}(B_{\tau_{2}})}\|\mathbf{P}_{1}^{f}(\cdot,(\tau_{2}-\tau_{1})/4)\|_{L^{\infty}(B_{\tau_{1}})}$
$\displaystyle\ \ \qquad+c\left[\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{2}})}^{\delta\vartheta\sigma/2}+1\right]\|\mathbf{P}_{1}^{\mathfrak{h}}(\cdot,(\tau_{2}-\tau_{1})/4)\|_{L^{\infty}(B_{\tau_{1}})}$
$\displaystyle\ \ =:T_{1}+T_{2}+T_{3}+T_{4}\,,$ (6.82)

for $c\equiv c(\texttt{data},\lVert h\rVert_{L^{d}(\mathcal{B})},\delta)$. To
estimate the $T$-terms we take $\delta$ such that

$(1+\delta)\vartheta\sigma/2+1/\gamma<1\quad\mbox{and}\quad\delta\vartheta\sigma/2+(1+\sigma+\hat{\sigma})/\gamma<1$
(6.83)

hold, which is in turn possible by (4.15)2; this fixes $\delta$ as a function
of $\sigma,\tilde{\sigma},\gamma$. We then have

$\displaystyle T_{1}$
$\displaystyle\stackrel{{\scriptstyle\eqref{sceglidelta}}}{{\leq}}$
$\displaystyle\frac{1}{8}\,\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{2}})}+\frac{c\lVert
G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{1}(B_{\varrho})}^{\theta}+c}{(\tau_{2}-\tau_{1})^{\beta}}\,,$
$\displaystyle T_{2}$
$\displaystyle\stackrel{{\scriptstyle\eqref{sos0.1}}}{{\leq}}$
$\displaystyle\frac{c}{(\tau_{2}-\tau_{1})^{2/p}}[\lVert
G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{2}})}+1]^{(1+\delta)\vartheta\sigma/2+1/\gamma}\|f\|_{L^{2}(B_{\varrho})}$
$\displaystyle\stackrel{{\scriptstyle\eqref{sceglidelta}}}{{\leq}}$
$\displaystyle\frac{1}{8}\,\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{2}})}+\frac{c\|f\|_{L^{2}(\log
L)^{\mathfrak{a}}(B_{\varrho})}^{\theta}+c}{(\tau_{2}-\tau_{1})^{\beta}}\,,$
$\displaystyle T_{3}$
$\displaystyle\stackrel{{\scriptstyle\eqref{potlore}_{2},\eqref{sos0.1}}}{{\leq}}$
$\displaystyle c[\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{2}})}+1]^{\delta\vartheta\sigma/2+1/\gamma}\|f\|_{L^{2}(\log
L)^{\mathfrak{a}}(B_{\varrho})}$
$\displaystyle\stackrel{{\scriptstyle\eqref{sceglidelta}}}{{\leq}}$
$\displaystyle\frac{1}{8}\,\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{2}})}+c\|f\|_{L^{2}(\log
L)^{\mathfrak{a}}(B_{\varrho})}^{\theta}+1\,,$

with $c\equiv c(\texttt{data},\lVert h\rVert_{L^{d}(\mathcal{B})})$. Here, as
in the following lines, $\theta,\beta$ denote positive exponents depending on
$d,\gamma,\sigma,\hat{\sigma}$; they might change from line to line according
to the same convention used to denote a generic constant $c$. To estimate the
last term $T_{4}$ we again use Lemma 10 with $s_{*}=1$. Therefore, we set
$\tau_{3}:=(\tau_{1}+\tau_{2})/2$ so that
$\tau_{2}-\tau_{3}=(\tau_{2}-\tau_{1})/2$, and, recalling (6.65) for $n=2$, we
have

$\displaystyle\|\mathbf{P}_{1}^{\mathfrak{h}}(\cdot,(\tau_{2}-\tau_{1})/4)\|_{L^{\infty}(B_{\tau_{1}})}$
$\displaystyle\stackrel{{\scriptstyle\eqref{potlore}_{2}}}{{\leq}}$
$\displaystyle c\lVert\mathfrak{h}\rVert_{L^{2}(\log
L)^{\mathfrak{a}}(B_{\tau_{3}})}$
$\displaystyle\stackrel{{\scriptstyle(\ref{normine})}}{{\leq}}$ $\displaystyle
c(\tau)\lVert\mathfrak{h}\rVert_{L^{2+\tau}(B_{\tau_{3}})}$
$\displaystyle\stackrel{{\scriptstyle(\ref{fk0})}}{{\leq}}$ $\displaystyle
c\lVert h\rVert_{L^{d}(B_{\tau_{3}})}\lVert\bar{G}_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{p}(B_{\tau_{3}})}^{1+\sigma+\hat{\sigma}}$
$\displaystyle\stackrel{{\scriptstyle(\ref{2.2})}}{{\leq}}$
$\displaystyle\frac{c}{(\tau_{2}-\tau_{1})^{\beta_{*}}}\left[\lVert
G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{1}(B_{\tau_{2}})}^{\theta_{*}}+1\right]$ $\displaystyle\ \
+c\left[\|Du\|_{L^{\infty}(B_{\tau_{2}})}\|f\|_{L^{2}(B_{\tau_{2}})}\right]^{1+\sigma+\hat{\sigma}}$

for $c\equiv c(\texttt{data},\lVert h\rVert_{L^{d}(\mathcal{B})})$. Using the
above inequality and (5.19)1, we end up with

$\displaystyle T_{4}$
$\displaystyle\leq\frac{c}{(\tau_{2}-\tau_{1})^{\beta_{*}}}\left[\lVert
G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{2}})}^{\delta\vartheta\sigma/2}+1\right]\left[\lVert
G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{1}(B_{\varrho})}^{\theta_{*}}+1\right]$
$\displaystyle\quad+c\left[\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{2}})}^{\delta\vartheta\sigma/2+(1+\sigma+\hat{\sigma})/\gamma}+1\right]\|f\|_{L^{2}(\log
L)^{\mathfrak{a}}(B_{\varrho})}^{1+\sigma+\hat{\sigma}}$
$\displaystyle\leq\frac{1}{8}\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{2}})}+\frac{c\lVert
G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{1}(B_{\varrho})}^{\theta}+c}{(\tau_{2}-\tau_{1})^{\beta}}+c\|f\|_{L^{2}(\log
L)^{\mathfrak{a}}(B_{\varrho})}^{\theta}$

where $c\equiv c(\texttt{data},\lVert h\rVert_{L^{d}(\mathcal{B})})$, and we
have used the second condition in (6.83). Plugging the estimates for the
$T$-terms in (6.82) yields

$\displaystyle\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{1}})}$
$\displaystyle\leq\frac{1}{2}\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{2}})}$ $\displaystyle\quad+\frac{c\lVert
G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{1}(B_{\varrho})}^{\theta}+c\|f\|_{L^{2}(\log
L)^{\mathfrak{a}}(B_{\varrho})}^{\theta}+c}{(\tau_{2}-\tau_{1})^{\beta}}$

for new exponents $\theta,\beta$ as in the statement. Lemma 2 allows now to
conclude with

$\displaystyle\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\varsigma})}\leq\frac{c\lVert
G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{1}(B_{\varrho})}^{\theta}+c\|f\|_{L^{2}(\log
L)^{\mathfrak{a}}(B_{\varrho})}^{\theta}+c}{(\varrho-\varsigma)^{\beta}}\,,$

where $c\equiv c(\texttt{data},\lVert h\rVert_{L^{d}(\mathcal{B})})$. Finally,
(4) follows from this last estimate, (5.18) and (5.19)1. The proof is
complete.

##  7 Proofs of Theorems 4.1 and 4.3

We start with the proof of Theorem 4.3, where, in particular, we assume $f\in
L^{n}(\Omega;\mathbb{R}^{N})$ and $n>2$. We fix $p$ as in statement and,
without loss of generality, we assume that $p>1+\sigma$ ($\sigma$ being as in
(4.15)). Then, for every integer $j\geq 1$, we define $f_{j}\in
L^{\infty}(\Omega;\mathbb{R}^{N})$ as $f_{j}(x):=f(x)$ if $\lvert
f(x)\rvert\leq j$, and $f_{j}(x):=j|f(x)|^{-1}f(x)$ otherwise. It clearly
follows that

$|f_{j}|\leq|f|\quad\mbox{for every $j\geq 1$}\,,\qquad f_{j}\to f\ \mbox{in
$L^{n}(\Omega;\mathbb{R}^{N})$}\,.$ (7.1)

Next, we determine $R_{*}\equiv R_{*}(\texttt{data},f(\cdot),p)\leq 1$
according to Proposition 1. Pay attention here; with some abuse of notation,
the $f$ used here is not the same from Proposition 1, but rather corresponds
to $\mathfrak{f}$ from (6.1) in the context of Proposition 1 (and thanks to
(7.1), $f_{j}$ corresponds to $f$ in Proposition 1). Accordingly, we fix a
ball $\mathcal{B}\Subset\Omega$ such that $\texttt{r}(\mathcal{B})\leq R_{*}$.
We consider a decreasing sequence of positive numbers $\\{\varepsilon_{j}\\}$
such that $\varepsilon_{j}\leq\min\\{1,T\\}/4$ for every $j\in\mathbb{N}$,
and, accordingly, we consider the families of functions
$\\{F_{j}\\}\equiv\\{F_{\varepsilon_{j}}\\}$,
$\\{G_{j}\\}\equiv\\{G_{\varepsilon_{j}}\\}$ constructed in (5.4). Note now
that any minimizer $u$ of the functional $\mathcal{F}$ in (1.1) belongs to
$W^{1,\gamma}_{{\rm loc}}(\Omega;\mathbb{R}^{N})$ by (5.20)1. This allows to
define $u_{j}\in u+W^{1,\gamma}_{0}(\mathcal{B};\mathbb{R}^{N})$ as the
solution to

$\displaystyle u_{j}\mapsto\min_{w\in
u+W^{1,\gamma}_{0}(\mathcal{B};\mathbb{R}^{N})}\,\mathcal{F}_{j}(w;\mathcal{B})$
$\displaystyle\qquad:=\min_{w\in
u+W^{1,\gamma}_{0}(\mathcal{B};\mathbb{R}^{N})}\,\int_{\mathcal{B}}\left[F_{j}(x,Dw)-f_{j}\cdot
w\right]\,dx\,.$

Directs Methods of the Calculus of Variations apply here and ensure existence
(see for instance (BM, , Section 4.4)). As for (BM, , (4.55)), and recalling
(5.20)2 and (5.22), we find

$\displaystyle\|F_{j}(\cdot,Du_{j})\|_{L^{1}(\mathcal{B})}+\|Du_{j}\|_{L^{\gamma}(\mathcal{B})}^{\gamma}$
$\displaystyle\qquad\leq
c\left[\|F(\cdot,Du)+1\|_{L^{1}(\mathcal{B})}+[\texttt{r}(\mathcal{B})]^{n}\lVert
f\rVert_{L^{n}(\mathcal{B})}^{\gamma/(\gamma-1)}\right]$ (7.2)

for every $j\geq 1$, where $c\equiv c(n,N,\nu,\gamma,T)$. This implies we can
assume that, up to a not relabelled subsequence, $Du_{j}\rightharpoonup
D\tilde{u}$ weakly in $L^{\gamma}(\mathcal{B};\mathbb{R}^{N\times n})$ and
$u_{j}\to\tilde{u}$ strongly in
$L^{\frac{n}{n-1}}(\mathcal{B};\mathbb{R}^{N})$, for some $\tilde{u}\in
u+W^{1,\gamma}_{0}(\mathcal{B};\mathbb{R}^{N})$. Note that Proposition 1
applies to $u_{j}$ with the choice $s_{*}=1$, as (6.15) is satisfied assuming
that $\sigma+\hat{\sigma}<1/n-1/d$, which is the case here by (4.15) in set.
The application of Proposition 1, and (6.16), now give that

$\displaystyle\|G_{j}(\cdot,\lvert Du_{j}\rvert)\|_{L^{p}(s\mathcal{B})}$
$\displaystyle\qquad\leq\frac{c}{(1-s)^{\beta_{p}}[\texttt{r}(\mathcal{B})]^{\beta_{p}}}\left[\|F(\cdot,Du)\|_{L^{1}(\mathcal{B})}^{\theta_{p}}+\lVert
f\rVert_{L^{n}(\mathcal{B})}^{\theta_{p}}+1\right]$ (7.3)

holds for every $s\in(0,1)$, for new exponents
$\beta_{p},\theta_{p}\equiv\beta_{p},\theta_{p}(n,\gamma,d,\sigma,\hat{\sigma},p)>0$,
where we have also used (7) to bound the right-side coming from (6.16); $c$
depends on data and $\lVert h\rVert_{L^{d}(\mathcal{B})}$. We fix
$j_{0}\in\mathbb{N}$, and apply (5.33) with $j\geq j_{0}$ in (7.3), to get

$\displaystyle\|G_{j_{0}}(\cdot,\lvert Du_{j}\rvert)\|_{L^{p}(s\mathcal{B})}$
$\displaystyle\qquad\leq\frac{c}{(1-s)^{\beta_{p}}[\texttt{r}(\mathcal{B})]^{\beta_{p}}}\left[\|F(\cdot,Du)\|_{L^{1}(\mathcal{B})}^{\theta_{p}}+\lVert
f\rVert_{L^{n}(\mathcal{B})}^{\theta_{p}}+1\right]\,.$ (7.4)

Letting $j\to\infty$ in the above inequality, and using weak lower
semicontinuity (recall that $G_{j_{0}}(\cdot)$ is convex and non-decreasing,
by (5.17), $\varepsilon\equiv\varepsilon_{j_{0}}$, therefore $z\mapsto
G_{j_{0}}(\cdot,\lvert z\rvert)$ is convex), yields

$\displaystyle\|G_{j_{0}}(\cdot,\lvert
D\tilde{u}\rvert)\|_{L^{p}(s\mathcal{B})}$
$\displaystyle\qquad\leq\frac{c}{(1-s)^{\beta_{p}}[\texttt{r}(\mathcal{B})]^{\beta_{p}}}\left[\|F(\cdot,Du)\|_{L^{1}(\mathcal{B})}^{\theta_{p}}+\lVert
f\rVert_{L^{n}(\mathcal{B})}^{\theta_{p}}+1\right]\,.$ (7.5)

This holds for every $j_{0}\in\mathbb{N}$ and therefore, by finally letting
$j_{0}\to\infty$ (by Fatou’s lemma) and recalling (5.33), we conclude with

$\displaystyle\|G(\cdot,\lvert D\tilde{u}\rvert)\|_{L^{p}(s\mathcal{B})}$
$\displaystyle\qquad\leq\frac{c}{(1-s)^{\beta_{p}}[\texttt{r}(\mathcal{B})]^{\beta_{p}}}\left[\|F(\cdot,Du)\|_{L^{1}(\mathcal{B})}^{\theta_{p}}+\lVert
f\rVert_{L^{n}(\mathcal{B})}^{\theta_{p}}+1\right]\,,$ (7.6)

for every $s\in(0,1)$, where $c\equiv c(\texttt{data},\lVert
h\rVert_{L^{d}(\mathcal{B})})$ and
$\beta_{p},\theta_{p}\equiv\beta_{p},\theta_{p}(n,\gamma,d,\sigma,\linebreak\hat{\sigma},p)>0$.
Next, we trivially write

$\displaystyle\mathcal{F}_{j_{0}}(u_{j};s\mathcal{B})$
$\displaystyle\leq\mathcal{F}_{j}(u_{j};s\mathcal{B})+\|F_{j}(\cdot,Du_{j})-F_{j_{0}}(\cdot,Du_{j})\|_{L^{1}(s\mathcal{B})}$
$\displaystyle\quad+\|(f_{j_{0}}-f_{j})\cdot u_{j}\|_{L^{1}(s\mathcal{B})}$
(7.7)

whenever $s\in(0,1)$. Properties (7.1)-(7), Hölder and Sobolev-Poincaré
inequalities give

$\mathcal{F}_{j}(u_{j};s\mathcal{B})\leq\mathcal{F}_{j}(u_{j};\mathcal{B})+\|{f_{j}\cdot
u_{j}\|_{L^{1}(\mathcal{B}\setminus
s\mathcal{B})}}\leq\mathcal{F}_{j}(u;\mathcal{B})+c\lVert
f\rVert_{L^{n}(\mathcal{B}\setminus s\mathcal{B})},$ (7.8)

where $c$ is independent of $s,j$ and we have used minimality of $u_{j}$.
Using (5.42) with $\varepsilon_{1}\equiv\varepsilon_{j}$,
$\varepsilon_{2}\equiv\varepsilon_{j_{0}}$, $w\equiv u_{j}$, $B\equiv
s\mathcal{B}$ (recall it is $p>1+\sigma$) and (7.4), we have

$\|F_{j}(\cdot,Du_{j})-F_{j_{0}}(\cdot,Du_{j})\|_{L^{1}(s\mathcal{B})}\leq
c(1-s)^{-\beta_{p}}\texttt{o}(j_{0})$ (7.9)

where $c$ is independent of $s,j,j_{0}$. Again Sobolev-Poincaré inequality
(7.1) and (7) give $\|(f_{j_{0}}-f_{j})\cdot u_{j}\|_{L^{1}(s\mathcal{B})}\leq
c\|f_{j_{0}}-f_{j}\|_{L^{n}(s\mathcal{B})}$, with $c$ independent of
$s,j,j_{0}$. Using this last inequality and (7.8)-(7.9) in (7.7), and finally
letting $j\to\infty$, lower semicontinuity yields

$\displaystyle\mathcal{F}_{j_{0}}(\tilde{u};s\mathcal{B})$
$\displaystyle\leq\mathcal{F}(u;\mathcal{B})+c\lVert
f\rVert_{L^{n}(\mathcal{B}\setminus s\mathcal{B})}$
$\displaystyle\quad+c\lVert
f_{j_{0}}-f\rVert_{L^{n}(\mathcal{B})}+c(1-s)^{-\beta_{p}}\texttt{o}(j_{0})\,.$
(7.10)

Note that we have used (5.22) and Lebesgue’s dominated convergence theorem to
get $\mathcal{F}_{j}(u;\mathcal{B})\to\mathcal{F}(u;\mathcal{B})$. In turn,
again note that (7.6) ensures that $G(\cdot,\lvert D\tilde{u}\rvert)\in
L^{p}(s\mathcal{B})$ and therefore allows to apply (5.43); this yields

$\|F(\cdot,D\tilde{u})-F_{j_{0}}(\cdot,D\tilde{u})\|_{L^{1}(s\mathcal{B})}\leq
c(1-s)^{-\beta_{p}}\texttt{o}(j_{0}),$

so that
$\mathcal{F}_{j_{0}}(\tilde{u};s\mathcal{B})\to\mathcal{F}(\tilde{u};s\mathcal{B})$
as $j_{0}\to\infty$, where we also use (7.1). In view of this, letting first
$j_{0}\to\infty$ and then $s\to 1$ in (7.10), yields
$\mathcal{F}(\tilde{u};\mathcal{B})\leq\mathcal{F}(u;\mathcal{B})$. This and
the minimality of $u$ finally give
$\mathcal{F}(u;\mathcal{B})=\mathcal{F}(\tilde{u};B)$, therefore, by standard
convexity arguments, see for instance (BM, , Section 4.4), we end up with

$\displaystyle\mbox{either}\ \max\\{\lvert Du(x)\rvert,\lvert
D\tilde{u}(x)\rvert\\}\leq T\ \ \mbox{or}\ \ Du(x)=D\tilde{u}(x)$ (7.11)

for a.e. $x\in\mathcal{B}$. Using this information in (7.6) yields (4.24) and
concludes the proof of Theorem 4.3. Observe that, in order to justify the
content of Remark 4 it is sufficient to note that making the a priori estimate
(7.3) only requires the bound $\sigma+\hat{\sigma}<1/n-1/d$.

We now come to the proof of Theorem 4.1. We can again use the same
approximation employed for Theorem 4.3. Using this time estimate (3) for the
case $n>2$ and estimate (4) when $n=2$, together with (7), we find

$\displaystyle[E_{\varepsilon_{j}}(\lVert
Du_{j}\rVert_{L^{\infty}(s\mathcal{B})})]^{\gamma}+\lVert G_{j}(\cdot,\lvert
Du_{j}\rvert)\rVert_{L^{\infty}(s\mathcal{B})}$
$\displaystyle\qquad\quad\leq\frac{c}{(1-s)^{\beta}[\texttt{r}(\mathcal{B})]^{\beta}}\left[\|F(\cdot,Du)\|_{L^{1}(\mathcal{B})}^{\theta}+\lVert
f\rVert_{\mathfrak{X}(\mathcal{B})}^{\theta}+1\right]$ (7.12)

for every $s\in(0,1)$, where $c,\beta,\theta$ have the same dependencies as in
(3). It follows that for every $s\in(0,1)$ there exists $M_{s}$ such that
$\|Du_{j}\|_{L^{\infty}(s\mathcal{B})}\leq M_{s}$, for every $j\in\mathbb{N}$.
Using a standard diagonalization argument we infer that, up to a not
relabelled subsequences, we have that $u_{j}\rightharpoonup^{*}\tilde{u}$ in
$W^{1,\infty}_{{\rm loc}}(\mathcal{B};\mathbb{R}^{N})$ for some $\tilde{u}\in
u+W^{1,\gamma}_{0}(\mathcal{B};\mathbb{R}^{N})$ being such that
$\|D\tilde{u}\|_{L^{\infty}(s\mathcal{B})}\leq M_{s}$. Moreover, we can repeat
verbatim the argument of Theorem 4.3 leading to (7.11). Now, denoting $j_{s}$
the first integer such that $1/\varepsilon_{j_{s}}>M_{s}$, from the very
definition of $G_{j}\equiv G_{\varepsilon_{j}}$ in (5.11), it follows that
$\lVert G_{j}(\cdot,\lvert
Du_{j}\rvert)\rVert_{L^{\infty}(s\mathcal{B})}=\mathfrak{g}_{1}\lVert
G(\cdot,\lvert Du_{j}\rvert)\rVert_{L^{\infty}(s\mathcal{B})}$, for every
$j\geq j_{s}$. Therefore, letting $j\to\infty$ in (7), and using (7.11),
yields (4.22) and the proof of Theorem 4.1 is complete too. Notice that here
we are using standard lower semicontinuity theorems for supremal functionals
with respect to the weak∗ convergence in $W^{1,\infty}_{{\rm
loc}}(\mathcal{B};\mathbb{R}^{N})$ (see for instance BJW ).

###### Remark 9

Let us discuss the case $f\equiv 0$. We start by the case $n>2$; the first
relation in (6.70) is already implied by $\sigma+\hat{\sigma}<1/n-1/d$.
Moreover, note that the second condition in (6.70) appears only when
$f\not\equiv 0$. This is the only point in the proof of Theorem 4.1 where the
full bound in (4.15)1 is required; otherwise $\sigma+\hat{\sigma}<1/n-1/d$ is
sufficient. Alternatively, one can apply directly Proposition 2 instead of
Proposition 1 in the proof of Theorem 4.1. When $n=2$ and $f\equiv 0$, (6.83)
turns into $(1+\delta)\vartheta\sigma/2<1$ and $\delta\vartheta\sigma/2<1$,
that are implied by $\sigma+\hat{\sigma}<1/2-1/d$ by taking $\delta$ small
enough (see estimate of $T_{4}$ in Proposition 4). Finally, note that in the
case $\gamma\geq 2$, the minimum in (4.15) is attained by $1/n-1/d$. All in
all, we have justified the content of Remark 3.

##  8 Uniform ellipticity and proof of Theorem 4.2

Here we work under assumptions $\texttt{set}_{\textnormal{u}}$ in (4.20); we
keep the full notation introduced in Sections 5 and 6. In particular, we keep
on considering a solution $u\in W^{1,\gamma}(\mathcal{B};\mathbb{R}^{N})$ to
(6.1) in a ball $\mathcal{B}\Subset\Omega$ such that
$\texttt{r}(\mathcal{B})\leq 1$. With the current choice of
$g_{1},g_{2},g_{3}$ as in (4.19), we apply the constructions laid down in
Section 5, thereby obtaining, in particular, the new functions
$\tilde{a}_{\varepsilon}$, $g_{1,\varepsilon}$, $g_{2,\varepsilon}$,
$g_{3,\varepsilon}$ in (5.3) and (5.7)-(5.9). The last three functions are now
independent of $x$, as well as $G_{\varepsilon}$ defined in (5.11). Note that
(4.19)2 from $\texttt{set}_{\textnormal{u}}$ implies the validity of (4.18)
with $c_{\textnormal{b}}\equiv c_{\textnormal{u}}$ and $\sigma=0$. Therefore
we can use all the properties from Section 5, and displayed through Lemmas
3-6, implied by assumptions (4.1)-(4.4) and (4.18). In particular, we can use
Lemma 6 with $\sigma=0$. In addition to such properties, we have

###### Lemma 11

Under assumptions $\texttt{set}_{\textnormal{u}}$ in (4.20)

  * •

The following inequalities hold for every $t\in[T,\infty)$:

$\begin{cases}\displaystyle\ g_{1,\varepsilon}(t)\leq
g_{2,\varepsilon}(t)\leq\tilde{c}_{\textnormal{u}}g_{1,\varepsilon}(t)\\\
g_{3,\varepsilon}(t)\leq\tilde{c}_{\textnormal{u}}g_{1,\varepsilon}(t)t\end{cases}$
(8.1)

where
$\tilde{c}_{\textnormal{u}}\equiv\tilde{c}_{\textnormal{u}}(n,N,\nu,\gamma,c_{\textnormal{u}})\geq
1$, and

$\displaystyle\frac{1}{2\tilde{c}_{\textnormal{u}}}g_{1,\varepsilon}(t)t^{2}\leq
G_{\varepsilon}(t)+\frac{\mathfrak{g}_{1}}{2}g_{1}(T)T^{2}\leq\frac{\tilde{c}_{\textnormal{u}}}{2}g_{1,\varepsilon}(t)t^{2}\,.$
(8.2)

  * •

Moreover, for every $x\in\Omega$ and $t\geq 0$, it holds that

$G_{\varepsilon}(t)\leq\tilde{F}_{\varepsilon}(x,t)\leq
cG_{\varepsilon}(t)+cg_{1}(T)(T^{2}+\mu_{\varepsilon}^{2})\,,\quad c\equiv
c(\texttt{data}_{\textnormal{u}})\,.$ (8.3)

###### Proof

Properties (8.1) are immediate and follow from (4.19), also recalling (4.6),
therefore we concentrate on (8.2). As for the right-hand side of (8.2),
integration by parts yields

$\int_{T}^{t}\left[\tilde{a}_{\varepsilon}(x,s)+\tilde{a}_{\varepsilon}^{\prime}(x,s)s\right]s\,ds=-\int_{T}^{t}\tilde{a}_{\varepsilon}(x,s)s\,ds+\tilde{a}_{\varepsilon}(x,t)t^{2}-\tilde{a}_{\varepsilon}(x,T)T^{2}$
(8.4)

and therefore, recalling the definition in (5.7), we have

$\displaystyle G_{\varepsilon}(t)$
$\displaystyle\stackrel{{\scriptstyle\eqref{a2}_{2}}}{{\leq}}$
$\displaystyle\int_{T}^{t}\left[\tilde{a}_{\varepsilon}(x,s)+\tilde{a}_{\varepsilon}^{\prime}(x,s)s\right]s\,ds$
$\displaystyle\stackrel{{\scriptstyle(\ref{a1}),(\ref{uni3})}}{{\leq}}$
$\displaystyle-
G_{\varepsilon}(t)+\tilde{c}_{\textnormal{u}}g_{1,\varepsilon}(t)t^{2}-\mathfrak{g}_{1}g_{1}(T)T^{2}$

that is, the right-hand side of (8.2). As for the left-hand side, we similarly
have

$\displaystyle G_{\varepsilon}(t)$
$\displaystyle\stackrel{{\scriptstyle(\ref{a2})_{2},(\ref{uni3})}}{{\geq}}$
$\displaystyle\frac{1}{\tilde{c}_{\textnormal{u}}}\int_{T}^{t}\left[\tilde{a}_{\varepsilon}(x,s)+\tilde{a}_{\varepsilon}^{\prime}(x,s)s\right]s\,ds$
$\displaystyle\stackrel{{\scriptstyle(\ref{a1}),(\ref{intpart})}}{{\geq}}$
$\displaystyle-\frac{1}{\tilde{c_{\textnormal{u}}}}\int_{T}^{t}g_{2,\varepsilon}(s)s\,ds+\frac{1}{\tilde{c_{\textnormal{u}}}}\left[\tilde{a}_{\varepsilon}(x,t)t^{2}-\tilde{a}_{\varepsilon}(x,T)T^{2}\right]$
$\displaystyle\stackrel{{\scriptstyle\eqref{g1e},(\ref{uni3})}}{{\geq}}$
$\displaystyle-
G_{\varepsilon}(t)+\frac{1}{\tilde{c_{\textnormal{u}}}}g_{1,\varepsilon}(t)t^{2}-\mathfrak{g}_{1}g_{1}(T)T^{2}\,.$

We turn to (8.3). The left-hand side inequality is nothing but (5.18)3 (that
holds whenever $|z|\geq 0$). For the right-hand side inequality, when $t\leq
T$ we have, similarly to (5.28)

$\tilde{F}_{\varepsilon}(x,t)\stackrel{{\scriptstyle\eqref{uni0}_{2},(\ref{slipffff})}}{{\leq}}c\left[\frac{g_{1}(T)}{(T^{2}+\mu^{2})^{\frac{\gamma-2}{2}}}+\varepsilon\right](T^{2}+\mu_{\varepsilon}^{2})^{\gamma/2}\stackrel{{\scriptstyle(\ref{00bis})}}{{\leq}}cg_{1}(T)(T^{2}+\mu^{2}).$

When $t>T$, we note that (5.15) and (8.1) imply
$\bar{a}_{\varepsilon}(x,s)\leq cg_{1,\varepsilon}(s)$, that implies
$\tilde{F}_{\varepsilon}(x,t)-\tilde{F}_{\varepsilon}(x,T)\leq
cG_{\varepsilon}(t)$, from which (8.3) follows again using the content of the
last display.

###### Proposition 5

Let $u\in W^{1,\gamma}(\mathcal{B};\mathbb{R}^{N})$ be a solution to
$\eqref{a0}$, under assumptions $\texttt{set}_{\textnormal{u}}$ in (4.20) for
$n\geq 2$. There exists a positive radius $R_{*}\equiv
R_{*}(\texttt{data}_{\textnormal{u}},h(\cdot))\leq 1$ such that if
$\texttt{r}(\mathcal{B})\leq R_{*}$ and $B_{\varsigma}\Subset B_{\varrho}$ are
concentric balls contained in $\mathcal{B}$, then

$\displaystyle\|F_{\varepsilon}(\cdot,Du)\|_{L^{\infty}(B_{\xi})}+\|G_{\varepsilon}(\lvert
Du\rvert)\|_{L^{\infty}(B_{\xi})}$ $\displaystyle\hskip
56.9055pt\leq\frac{c}{(\varrho-\xi)^{n}}\|F_{\varepsilon}(\cdot,Du)\|_{L^{1}(B_{\varrho})}+c\left[\|f\|_{\mathfrak{X}(B_{\varrho})}^{\gamma/(\gamma-1)}+1\right]$
(8.5)

holds with $c\equiv c(\texttt{data}_{\textnormal{u}})$, where
$\mathfrak{X}(\cdot)$ has been defined in (1.9).

###### Proof

We start taking a ball $B_{r}(x_{0})\Subset\mathcal{B}$ (therefore it is
$r<1$), and prove that

$\displaystyle\int_{B_{r/2}(x_{0})}\lvert D(G_{\varepsilon}(\lvert
Du\rvert)-\kappa)_{+}\rvert^{2}\,dx\leq\frac{c}{r^{2}}\int_{B_{r}(x_{0})}(G_{\varepsilon}(\lvert
Du\rvert)-\kappa)_{+}^{2}\,dx$ $\displaystyle\hskip 71.13188pt+c\left[\lVert
G_{\varepsilon}(\lvert
Du\rvert)\rVert_{L^{\infty}(B_{r}(x_{0}))}+1\right]^{2}\int_{B_{r}(x_{0})}|h|^{2}\,dx$
$\displaystyle\hskip 85.35826pt+c\lVert
Du\rVert_{L^{\infty}(B_{r}(x_{0}))}^{2}\int_{B_{r}(x_{0})}\lvert
f\rvert^{2}\,dx$ (8.6)

holds whenever $\kappa\geq 0$, with $c\equiv
c(\texttt{data}_{\textnormal{u}})$. This is an analogue of (8) and to get it
we modify the proof of Lemma 8, keeping the notation used there. Proceeding as
for the bounds for $\mbox{(IV)}_{z}$-$\mbox{(VI)}_{z}$ in Lemma 8, we have

$\displaystyle\mathcal{S}_{3}+\mathcal{S}_{4}$ $\displaystyle\leq
c|\mbox{(IV)}_{x}|+c|\mbox{(V)}_{x}|+c|\mbox{(VI)}_{x}|+\frac{c}{r^{2}}\int_{B_{r}(x_{0})}(G_{\varepsilon}(\lvert
Du\rvert)-\kappa)_{+}^{2}\,dx$
$\displaystyle\qquad+c\sum_{s=1}^{n}\int_{\mathcal{B}^{\kappa}}\lvert f\cdot
D_{s}\varphi_{s}\rvert\,dx\,,$ (8.7)

with $c\equiv c(\texttt{data}_{\textnormal{u}})$. This estimate can be
obtained by adapting those in (6.49)-(6.51), and also those for the terms in
(6.20). One must take into account that now $g_{1,\varepsilon}$ is independent
of $x$ (therefore the terms coming from the use of (5.31) and featuring
$h(\cdot)$ disappear), and the fact that we can formally take $\sigma\equiv 0$
as the ratio $g_{2,\varepsilon}/g_{1,\varepsilon}$ is bounded by a constant by
(8.1)1. In turn, the last term in (8.7) involving the right-hand side $f$ can
be treated exactly as in (6.58). As for the remaining $x$-terms appearing in
the first line of (8.7), with the help of $\eqref{growthd}_{3}$, (8.1) and
(8.2), we estimate

$\displaystyle c|\mbox{(IV)}_{x}|+c|\mbox{(V)}_{x}|+c|\mbox{(VI)}_{x}|$
$\displaystyle\leq\bar{\varepsilon}\mathcal{S}_{3}+\bar{\varepsilon}\mathcal{S}_{4}+\frac{c}{r^{2}}\int_{B_{r}(x_{0})}(G_{\varepsilon}(\lvert
Du\rvert)-\kappa)_{+}^{2}\,dx$
$\displaystyle\quad+\frac{c}{\bar{\varepsilon}}\int_{\mathcal{B}}\eta^{2}|h|^{2}[G_{\varepsilon}(\lvert
Du\rvert)+1]^{2}\,dx\,,$

for $c\equiv c(\texttt{data}_{\textnormal{u}})$ and
$\bar{\varepsilon}\in(0,1)$. Merging the content of the above two displays,
choosing
$\bar{\varepsilon}\equiv\bar{\varepsilon}(\texttt{data}_{\textnormal{u}})$
small enough, and reabsorbing terms, we end up with (8.6), where $c\equiv
c(\texttt{data}_{\textnormal{u}})$. As a consequence, we proceed as for the
proofs of Lemma 9 and Proposition 3. An application of Lemma 1 gives that, if
$B_{2r_{0}}(x_{0})\Subset\mathcal{B}$ is any ball, then

$\displaystyle G_{\varepsilon}(\lvert Du(x_{0})\rvert)$ $\displaystyle\leq
cr_{0}^{-n/2}\left[\lVert G_{\varepsilon}(\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{2}})}+1\right]^{1/2}\|G_{\varepsilon}(\lvert
Du\rvert)\|_{L^{1}(B_{\tau_{2}})}^{1/2}$ $\displaystyle\quad+c\left[\lVert
G_{\varepsilon}(\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{2}})}+1\right]\mathbf{P}_{1}^{h}(x_{0},2r_{0})$
$\displaystyle\quad+c\left[\lVert G_{\varepsilon}(\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{2}})}+1\right]^{1/\gamma}\mathbf{P}_{1}^{f}(x_{0},2r_{0})$
(8.8)

holds provided $x_{0}$ is a Lebesgue point of $|Du|$, where $c\equiv
c(\texttt{data}_{\textnormal{u}})$, and we have also used (5.19)1. Next, (3.1)
gives that

$\|\mathbf{P}_{1}^{h}(\cdot,2r_{0})\|_{L^{\infty}(B_{\tau_{1}})}\lesssim\|h\|_{\mathfrak{X}(B_{\varrho})}\quad\mbox{and}\quad\|\mathbf{P}_{1}^{f}(\cdot,2r_{0})\|_{L^{\infty}(B_{\tau_{1}})}\lesssim\|f\|_{\mathfrak{X}(B_{\varrho})}\,.$

Using these informations in (8.8) yields

$\displaystyle\|G_{\varepsilon}(\lvert
Du\rvert)\|_{L^{\infty}(B_{\tau_{1}})}+1$ $\displaystyle\qquad\leq
cr_{0}^{-n/2}\left[\lVert G_{\varepsilon}(\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{2}})}+1\right]^{1/2}\|G_{\varepsilon}(\lvert
Du\rvert)\|_{L^{1}(B_{\tau_{2}})}^{1/2}+1$
$\displaystyle\qquad\quad+c_{*}\left[\lVert G_{\varepsilon}(\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{2}})}+1\right]\|h\|_{\mathfrak{X}(B_{\varrho})}$
$\displaystyle\qquad\quad+c\left[\lVert G_{\varepsilon}(\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{2}})}+1\right]^{1/\gamma}\|f\|_{\mathfrak{X}(B_{\varrho})}\,,$
(8.9)

where $c,c_{*}\equiv c,c_{*}(\texttt{data}_{\textnormal{u}})$. By absolute
continuity, we now determine the radius $R_{*}\equiv
R_{*}(\texttt{data}_{\textnormal{u}},h(\cdot))$ mentioned in the statement in
such a way that

$\displaystyle\texttt{r}(\mathcal{B})\leq R_{*}\Longrightarrow c_{*}\lVert
h\rVert_{\mathfrak{X}(B_{\varrho})}\leq c_{*}\lVert
h\rVert_{\mathfrak{X}(\mathcal{B})}\leq 1/6\,.$ (8.10)

Using this and Young inequality in (8), and yet recalling that
$r_{0}:=(\tau_{2}-\tau_{1})/8$, gives

$\displaystyle\|G_{\varepsilon}(\lvert Du\rvert)\|_{L^{\infty}(B_{\tau_{1}})}$
$\displaystyle\leq\frac{1}{2}\|G_{\varepsilon}(\lvert
Du\rvert)\|_{L^{\infty}(B_{\tau_{2}})}$
$\displaystyle\quad+\frac{c}{(\tau_{2}-\tau_{1})^{n}}\|G_{\varepsilon}(\lvert
Du\rvert)\|_{L^{1}(B_{\varrho})}+c\left[\|f\|_{\mathfrak{X}(B_{\varrho})}^{\gamma/(\gamma-1)}+1\right]\,.$

Inequality (8.5) now follows using Lemma 2 with
$\mathcal{Z}(t):=\|G_{\varepsilon}(\lvert Du\rvert)\|_{L^{\infty}(B_{t})}$ and
(8.3).

With Proposition 5 available, we can now complete the proof of Theorem 4.2.
Arguing as in the proof of Theorem 4.1 in Section 7 (as usual with
$F_{j}\equiv F_{\varepsilon_{j}}$ and so on), and arrive up to (7), the analog
of which in the present context is

$\|F_{j}(\cdot,Du_{j})\|_{L^{\infty}(s\mathcal{B})}\leq\frac{c}{(1-s)^{n}[\texttt{r}(\mathcal{B})]^{n}}\|F(\cdot,Du)\|_{L^{1}(\mathcal{B})}+c\left[\|f\|_{\mathfrak{X}(\mathcal{B})}^{\gamma/(\gamma-1)}+1\right]$

where $c$ depends on $\texttt{data}_{\textnormal{u}}$ (again recall the
equivalence in (8.3)). The rest of the proof again follows with minor
modifications to the proof for Theorem 4.1.

##  9 Proof of Theorem 4.4

We revisit the proof of Theorem 4.1 starting from the part concerning the a
priori estimates of Section 6 in the case $n>2$. In turn, this uses as a
preliminary result Proposition 1 for $s_{*}=1$, that now works only assuming
$\hat{\sigma}<1/n-1/d$, as allowed by (4.25). This relies on the condition
$g_{2}/g_{1}\leq c_{\textnormal{b}}$ in (4.25), that via (5.29) also implies
$g_{2,\varepsilon}/g_{1,\varepsilon}\leq c$. Then Remark 8 can be used and we
can replace $\sigma+\hat{\sigma}$ by $\hat{\sigma}$ in (6.15). The same
arguments of Remark 8 also apply in the proof of Lemma 8, and yield the
following simplified form of (8):

$\displaystyle\int_{B_{r/2}(x_{0})}\lvert D(G_{\varepsilon}(x,\lvert
Du\rvert)-\kappa)_{+}\rvert^{2}\
\,dx\leq\frac{c}{r^{2}}\int_{B_{r}(x_{0})}(G_{\varepsilon}(x,\lvert
Du\rvert)-\kappa)_{+}^{2}\ \,dx$ $\displaystyle\hskip
125.19212pt+c\int_{B_{r}(x_{0})}[h(x)]^{2}[\bar{G}_{\varepsilon}(x,\lvert
Du\rvert)]^{2(1+\hat{\sigma})}\,dx$ $\displaystyle\hskip 125.19212pt+c\lVert
Du\rVert_{L^{\infty}(B_{r}(x_{0}))}^{2}\int_{B_{r}(x_{0})}\lvert f\rvert^{2}\
\,dx\,.$

This last inequality still holds when $n=2$. Using it as in Lemma 9 yields

$\displaystyle G_{\varepsilon}(x_{0},\lvert Du(x_{0})\rvert)$
$\displaystyle\leq\kappa+c\left(\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{B_{r_{0}}(x_{0})}(G_{\varepsilon}(x,\lvert
Du\rvert)-\kappa)_{+}^{2}\,dx\right)^{1/2}$ $\displaystyle\ \
+c\left[\mathbf{P}_{1}^{\mathfrak{h}}(x_{0},2r_{0})+\lVert
Du\rVert_{L^{\infty}(B_{r_{0}}(x_{0}))}\mathbf{P}_{1}^{f}(x_{0},2r_{0})\right]\,,$
(9.1)

for every $\kappa\geq 0$, which is in fact the analog of (9) in the present
setting. Here it is $\mathfrak{h}(x):=h(x)[\bar{G}_{\varepsilon}(x,\lvert
Du\rvert)]^{1+\hat{\sigma}}$. From this last inequality we again arrive at (3)
exactly as in the proof of Proposition 3 (needless to say, for different
exponents $\theta$ and $\beta$). Here the key observation is that we do not
have to verify the two conditions in (6.70), that in fact are not occurring
(actually they occur taking formally $\sigma=0$). It is only the second, and
more restrictive one, that requires the stronger bound in (4.15), that can
therefore be relaxed in (4.25). Starting from this fact, the rest of the proof
is the same as the one of Theorem 4.1 in the case $n>2$, again replacing
$\sigma+\hat{\sigma}$ by $\hat{\sigma}$. In the case $n=2$ the argument is
similar. First, we use Lemma 10, again with $s_{*}=1$, and, as for the case
$n>2$, this works replacing $\sigma+\hat{\sigma}$ by $\hat{\sigma}$ in (6.74).
Next, we apply this time (9), thereby getting (6.81) (with formally
$\sigma=0$) and then (6.82), with $\lVert G_{\varepsilon}(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(B_{\tau_{2}})}$ replaced by $1$. From this point
on, the proof goes as in the previous case; the only remark is that now the
first condition in (6.83) does not appear as a consequence of the new version
of (6.82), and this new bound is exactly the one appearing in (4.25) upon
considering only $\hat{\sigma}$.

##  10 Proof of Theorems 1.1-1.7

###  10.1 Theorems 1.1 and 1.2

We just verify that the assumptions of Theorem 1.1 imply those of Theorem 4.1
for suitable choices of objects and parameters. For this we take $\gamma=p$,
$g_{i}(x,t)\equiv g_{i}(t)$, $i\in\\{1,2,3\\}$, where
$g_{1}(t):=\nu(t^{2}+\mu^{2})^{(p-2)/2}$,
$g_{2}(t):=\Lambda(t^{2}+\mu^{2})^{(q-2)/2}+\Lambda(t^{2}+\mu^{2})^{(p-2)/2}$
and $g_{3}(t):=(t^{2}+\mu^{2})^{(q-2)/2}t+(t^{2}+\mu^{2})^{(p-2)/2}t$, for all
$t\in(0,\infty)$. It follows that $G(x,t)\equiv
G(t)=\nu[(t^{2}+\mu^{2})^{p/2}-(T^{2}+\mu^{2})^{p/2}]/p$ (for $t\geq T$ and it
is zero otherwise) and
$\bar{G}(x,t)\equiv\bar{G}(t)=G(t)+(T^{2}+1)^{p/2}\geq\nu(t^{2}+\mu^{2})^{p/2}+(p-\nu)/p$.
Here, any number $T\in(0,\infty)$ is fine. By (1.13)-(1.15) we see that $F$
satisfies (4.1)-(4.4). Note that $\eqref{7}$ holds with $\hat{\sigma}=0$ since
$g_{1}$ is $x$-independent. As for (4.17)-(4.18), we verify them choosing
$\sigma=q/p-1$, for a suitably large constant $c_{\textnormal{b}}\equiv
c_{\textnormal{b}}(n,N,\nu,p,q,\Lambda)\geq 1$. With such a choice of
$\sigma$, and $\hat{\sigma}=0$ as above, the condition in (4.15) is implied by
the assumed one in (1.16). All in all, assumptions (4.14)-(4.18) are verified
too and and we can apply Theorem 4.1 to get Theorem 1.1. Finally, the last
assertion concerning the improved bound in (1.17) follows from the content of
Remark 3 and this completes the proof of Theorem 1.1. Theorem 1.2 is a special
case of Theorem 1.1, upon taking $h(\cdot)\approx|D\mathfrak{c}(\cdot)|$. For
this, just observe that the convexity of $z\to F(z)$, and the growth
assumptions (1.13), imply also that
$\tilde{F}^{\prime}(t)\lesssim(t^{2}+\mu^{2})^{(q-2)/2}t+(t^{2}+\mu^{2})^{(p-2)/2}t$,
thereby allowing to verify also (1.15) for the integrand
$(x,t)\mapsto\mathfrak{c}(x)\tilde{F}(t)$.

###  10.2 Theorem 1.3

As all our estimates are local in nature, we can assume that $H(\cdot,Du)\in
L^{1}(\Omega)$, where the integrand $H(\cdot)$ has been defined in (1.10);
moreover, by localizing Morrey’s embedding theorem, we can also assume that
$[a]_{0,1-n/d;\Omega}+\lVert a\rVert_{L^{\infty}(\Omega)}$ is finite.
Similarly to (4.21), we denote

$\texttt{data}_{\textnormal{h}}:=(n,\nu,N,p,q,d,[a]_{0,1-n/d;\Omega},\lVert
a\rVert_{L^{\infty}},\lVert H(\cdot,Du)\rVert_{L^{1}},\lVert
f\rVert_{L^{n}})\,.$ (10.1)

Here $[a]_{0,1-n/d;\Omega}$ denotes the usual $(1-n/d)$-Hölder seminorm of
$a(\cdot)$ on $\Omega$, and all the norms extend over $\Omega$. The proof now
proceeds in three steps.

#### Step 1: Quantification of ellipticity

We refer to the framework of Theorem 4.4. We define, for $x\in\Omega$ and
$t>0$, $g_{1}(x,t):=\min\\{p-1,1\\}[t^{p-2}+a(x)t^{q-2}]$,
$g_{2}(x,t):=2qNn[t^{p-2}+a(x)t^{q-2}]$, $g_{3}(x,t)\equiv g_{3}(t):=t^{q-1}$
that implies $G(x,t):=\min\\{p-1,1\\}[(t^{p}/p+a(x)t^{q}/q)-(1/p+a(x)/q)]$
(here we take $T=1$ and $\mu=0$), $\bar{G}(x,t)=G(x,t)+2^{p/2}$ and
$g_{2}/g_{1}\leq c_{\rm b}$ for $t\geq 1$, where $c_{\rm b}\equiv c_{\rm
b}(\texttt{data}_{\textnormal{h}})$. We also set $h(\cdot)=\lvert
Da(\cdot)\rvert$, $\sigma=\hat{\sigma}:=q/p-1$, $\gamma=p$. We now observe
that, replacing $\leq$ by $<$ in the first bound from (1.19), we can
immediately conclude with the local Lipschitz continuity of minima invoking
Theorem 4.4, whose assumptions, and in particular (4.25), are satisfied but in
the limiting/equality case. In order to get the delicate equality in (1.19),
we have to readapt some points from the proof of Theorem 4.4 using some
additional results available when the peculiar structure in (1.10) is
considered.

#### Step 2: Uniform higher integrability

We modify and extend arguments from CoM0 ; CoM ; demicz .

###### Lemma 12

Let $u\in W^{1,1}(\Omega;\mathbb{R}^{N})$ be a minimizer of the functional in
(1.10), under assumptions (1.2) and $0\leq a(\cdot)\in W^{1,d}(\Omega)$ with
(1.17). There exists $s_{**}\equiv s_{**}(\texttt{data}_{\textnormal{h}})>1$
such that $H(\cdot,Du)\in L^{s_{**}}_{{\rm loc}}(\Omega)$.

###### Proof

Consider concentric balls $B_{\varrho}\subset B_{t}\Subset B_{s}\subset B_{R}$
contained in $\Omega$, then take a standard cut-off function $\eta\in
C^{1}_{{\rm c}}(B_{R})$ so that
$\mathds{1}_{B_{t}}\leq\eta\leq\mathds{1}_{B_{s}}$ and $\lvert
D\eta\rvert\lesssim 1/(s-t)$. We take $w:=u-\eta(u-(u)_{B_{R}})$ as competitor
to minimality, that yields
$\|H(\cdot,Du)\|_{L^{1}(B_{s})}\leq\|H(\cdot,Dw)\|_{L^{1}(B_{s})}+\|\eta
f\cdot(u-(u)_{B_{R}})\|_{L^{1}(B_{s})}$. The first term in the right-hand side
of this last inequality can be handled as in (CoM0, , Section 9), while by
Sobolev and Young inequalities we have

$\displaystyle\|\eta
f\cdot(u-(u)_{B_{R}})\|_{L^{1}(B_{s})}\leq\|f\|_{L^{n}(B_{s})}\|\eta(u-(u)_{B_{R}})\|_{L^{n/(n-1)}(B_{s})}$
$\displaystyle\quad\leq cR^{n-n/p}\lVert f\rVert_{L^{n}(\Omega)}\lVert
Du\rVert_{L^{p}(B_{s})}+\frac{cR^{n-n/p}}{s-t}\lVert
f\rVert_{L^{n}(\Omega)}\|u-(u)_{B_{R}}\|_{L^{p}(B_{s})}$
$\displaystyle\quad\leq\frac{1}{4}\|H(\cdot,Du)\|_{L^{1}(B_{s})}+c\int_{B_{R}}\tilde{H}\left(x,\frac{|u-(u)_{B_{R}}|}{s-t}\right)\,dx+cR^{n}\,,$
(10.2)

where $c\equiv c(n,p,q,\lVert f\rVert_{L^{n}(B)})$. This last estimate
together with the arguments developed in (CoM0, , Section 9), leads to the
reverse Hölder inequality

$\left(\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{B_{R/2}}[H(\cdot,Du)]^{1+\delta}\,dx\right)^{1/(1+\delta)}\leq
c\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{B_{R}}[H(\cdot,Du)]\,dx+c\,,$

for $c\equiv c(\texttt{data}_{\textnormal{h}})\geq 1$,
$\delta\equiv\delta(\texttt{data}_{\textnormal{h}})>0$, which in turn allows
us to conclude via a covering argument.

Let $B\Subset\Omega$ be a ball with $\texttt{r}(B)\leq 1$. As is CoM0 ; CoM ,
we say that the $p$-phase occurs when

$a_{\textnormal{i}}(B):=\inf_{x\in B}a(x)\leq
4[\texttt{r}(B)]^{1-n/d}[a]_{0,1-n/d;B}$ (10.3)

holds, otherwise we say that the $(p,q)$-phase occurs. Accordingly, we define
the integrand $\tilde{H}^{-}_{B}(|z|):=\lvert
z\rvert^{p}/p+a_{\textnormal{i}}(B)\lvert z\rvert^{q}/q.$ Finally, let
$\tilde{H}_{\varepsilon}(\cdot)$, $\tilde{H}^{-}_{B,\varepsilon}(\cdot)$ be
the integrands defined by applying the construction in Section 5.1,
(5.3)-(5.4), to $\tilde{H}(\cdot)$ and $\tilde{H}^{-}_{B}(\cdot)$,
respectively, with $\gamma=p$ and $g_{1},g_{2},g_{3}$ defined in _Step 1_ and
$T=1$. Such a construction provides us with approximating integrands that are
still of double phase type, i.e.,

$\tilde{H}_{\varepsilon}(x,t)=\int_{0}^{t}\tilde{h}_{\varepsilon}(x,s)s\,ds+\varepsilon\int_{0}^{t}(s^{2}+\varepsilon^{2})^{\frac{p-2}{2}}s\,ds\,,$
(10.4)

where

$\displaystyle\tilde{h}_{\varepsilon}(x,t):=\begin{cases}\
\frac{\varepsilon^{p-2}}{(2\varepsilon^{2})^{\frac{p-2}{2}}}(t^{2}+\varepsilon^{2})^{\frac{p-2}{2}}\\\
\qquad+a(x)\frac{\varepsilon^{q-2}}{(2\varepsilon^{2})^{\frac{p-2}{2}}}(t^{2}+\varepsilon^{2})^{\frac{p-2}{2}}\
\ &\mbox{if}\ t\in[0,\varepsilon)\\\ \ t^{p-2}+a(x)t^{q-2}\ \ &\mbox{if}\
t\in[\varepsilon,T_{\varepsilon})\\\ \
\frac{T_{\varepsilon}^{p-2}}{(T_{\varepsilon}^{2}+\varepsilon^{2})^{\frac{p-2}{2}}}(t^{2}+\varepsilon^{2})^{\frac{p-2}{2}}\\\
\qquad+a(x)\frac{T_{\varepsilon}^{q-2}}{(T_{\varepsilon}^{2}+\varepsilon^{2})^{\frac{p-2}{2}}}(t^{2}+\varepsilon^{2})^{\frac{p-2}{2}}\
\ &\mbox{if}\ t\in[T_{\varepsilon},\infty)\,,\end{cases}$ (10.5)

with the same representation holding for
$\tilde{H}^{-}_{B,\varepsilon}(\cdot)$, replacing $a(\cdot)$ by
$a_{\textnormal{i}}(B)$ in (10.5) above; here it is
$T_{\varepsilon}=1+1/\varepsilon$. Next, we develop an intrinsic Sobolev-
Poincaré inequality involving $H_{\varepsilon}(\cdot)$; the main point is that
the implied constants and exponents are independent of $\varepsilon$.

###### Lemma 13

Assume that $0\leq a(\cdot)\in W^{1,d}(\Omega)$ with (1.17). Let
$B\Subset\Omega$ be a ball with $\texttt{r}(B)\leq 1$ and $w\in
W^{1,p}_{0}(B;\mathbb{R}^{N})$ and $v\in W^{1,p}(B;\mathbb{R}^{N})$ be such
that $H_{\varepsilon}(\cdot,Dw),H_{\varepsilon}(\cdot,Dv)\in L^{1}(B)$. Then,
the following Sobolev and Sobolev-Poincaré inequalities hold:

$\displaystyle\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{B}\tilde{H}_{\varepsilon}\left(x,\frac{|w|}{\texttt{r}(B)}\right)\,dx\leq
c\left(\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{B}[\tilde{H}_{\varepsilon}(x,|Dw|)+1]^{\tau}\,dx\right)^{1/\tau}$
(10.6)

$c\equiv c(n,N,p,q,d,[a]_{0,1-n/d;B},\lVert Dw\rVert_{L^{p}(B)})$ and

$\displaystyle\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{B}\tilde{H}_{\varepsilon}\left(x,\frac{|v-(v)_{B}|}{\texttt{r}(B)}\right)\,dx\leq
c\left(\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{B}[\tilde{H}_{\varepsilon}(x,|Dv|)+1]^{\tau}\,dx\right)^{1/\tau}\,,$
(10.7)

for $c\equiv c(n,N,p,q,d,[a]_{0,1-n/d;B},\lVert Dv\rVert_{L^{p}(B)})$. In both
cases it is $\tau\equiv\tau(n,p,q)\linebreak\in(0,1)$.

###### Proof

We prove (10.6), the proof of (10.7) being totally similar. We start
considering the $p$-phase; this is when (10.3) occurs. In this case, recalling
$\eqref{hhdopo}$-$\eqref{hhdopo2}$, we bound

$\displaystyle\int_{B}\tilde{H}_{\varepsilon}\left(x,\frac{|w|}{\texttt{r}(B)}\right)\,dx$
$\displaystyle\qquad\leq
c\varepsilon^{p}[\texttt{r}(B)]^{n}+c\int_{B\cap\\{\varepsilon\leq\lvert
w\rvert/\texttt{r}(B)<T_{\varepsilon}\\}}\left(\int_{\varepsilon}^{\lvert
w\rvert/\texttt{r}(B)}g_{1}(x,s)s\,ds\right)\,dx$
$\displaystyle\qquad\qquad+c\int_{B\cap\\{\lvert w\rvert/\texttt{r}(B)\geq
T_{\varepsilon}\\}}\left(\int_{\varepsilon}^{\lvert
w\rvert/\texttt{r}(B)}g_{1,\varepsilon}(x,s)s\,ds\right)\,dx$
$\displaystyle\qquad\qquad+c\varepsilon[\texttt{r}(B)]^{n}\left(\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{B}\lvert
Dw\rvert^{p_{*}}\,dx\right)^{p/p_{*}}\
\big{(}=:\mbox{(I)}+\mbox{(II)}+\mbox{(III)}++\mbox{(IV)}\big{)}\,,$

where $c\equiv c(n,p,[a]_{0,1-n/d;B})$. Now, note that the assumed bound in
(1.17) implies $p_{*}\leq q_{*}<p$, where $b_{*}:=\max\,\\{1,nb/(n+b)\\}$, for
$b\in\\{p,q\\}$. Therefore using again Sobolev-Poincaré and Hölder’s
inequalities, we estimate

$\displaystyle\mbox{(II)}+\mbox{(III)}\leq
c\int_{B}[(|w|/\texttt{r}(B))^{p}+a(x)(|w|/\texttt{r}(B))^{q}]\,dx$
$\displaystyle\qquad\qquad\quad+c\int_{B\cap\\{\lvert
w\rvert/\texttt{r}(B)\geq
T_{\varepsilon}\\}}[T_{\varepsilon}^{p}+a(x)T_{\varepsilon}^{q}+a(x)T_{\varepsilon}^{q-p}(|w|/\texttt{r}(B))^{p}]\,dx$
$\displaystyle\stackrel{{\scriptstyle(\ref{pphase})}}{{\leq}}c[\texttt{r}(B)]^{n}\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{B}[(|w|/\texttt{r}(B))^{p}+[\texttt{r}(B)]^{1-n/d}(|w|/\texttt{r}(B))^{q}]\,dx$
$\displaystyle\ \leq
c[\texttt{r}(B)]^{n}\left(\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{B}\lvert
Dw\rvert^{p_{*}}\,dx\right)^{p/p_{*}}$
$\displaystyle\qquad\qquad+c[\texttt{r}(B)]^{n+1-n/d}\left(\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{B}\lvert
Dw\rvert^{p}\,dx\right)^{q/p-1}\left(\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{B}\lvert
Dw\rvert^{p(q_{*}/p)}\,dx\right)^{p/q_{*}}$ $\displaystyle\
\stackrel{{\scriptstyle(\ref{lox2bis})}}{{\leq}}c[\texttt{r}(B)]^{n}\left(\lVert
Dw\rVert_{L^{p}(B)}^{q-p}+1\right)\left(\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{B}\lvert
Dw\rvert^{p\tilde{\tau}}\,dx\right)^{1/\tilde{\tau}},$

with $c\equiv c(n,N,p,q,d,[a]_{0,1-n/d;B})$ and $\tilde{\tau}:=q_{*}/p<1$.
Merging the content of the last two displays above easily yields

$\displaystyle\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{B}\tilde{H}_{\varepsilon}\left(x,\frac{|w|}{\texttt{r}(B)}\right)\,dx\leq
c\left(\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{B}[\tilde{H}_{\varepsilon}(x,|Dw|)+1]^{\tilde{\tau}}\,dx\right)^{1/\tilde{\tau}}$
(10.8)

for $c\equiv c(n,N,p,q,d,[a]_{0,1-n/d;B}\lVert Dw\rVert_{L^{p}(B)}^{q-p})$. We
then pass to the $(p,q)$-phase, that is, when the complementary condition to
(10.3) holds; here we follow the arguments from (CoM, , Section 4). It is then
easy to see that
$\tilde{H}_{\varepsilon}(x,z)\sim\tilde{H}^{-}_{B,\varepsilon}(z)$ for
$(x,z)\in B\times\mathbb{R}^{N\times n}$. Moreover, recalling that
$H^{-}_{B,\varepsilon}$ can be written as in (10.4)-(10.5) with $a(\cdot)$
replaced by $a_{\textnormal{i}}(B)$, it follows that
$\tilde{H}^{-}_{B,\varepsilon}$ satisfies both $\Delta_{2}$ and $\nabla_{2}$
conditions (with constants depending on $n,p,q$, but independent of
$\varepsilon$). We are therefore in position to argue as in the proof of (DiE,
, Theorem 7), thereby again arriving at (10.8), for a different exponent
$\tilde{\tau}\equiv\tau(n,p,q)\in(0,1)$. At this stage the proof of (10.6)
follows merging the inequalities found in the two cases, and taking the
largest of the two exponents $\tilde{\tau}$ found for the two phases.

Given the inequalities in (10.6)-(10.7), and the representation in
(10.4)-(10.5), we can proceed for instance as in (demicz, , Lemma 5) to get a
global gradient integrability result; this also involves estimates as in
(10.2) to treat the additional $f$-terms appearing here with respect to the
case considered in demicz . This involves a matching of local and up-to-the-
boundary versions of Gehring’s lemma (see KSt ).

###### Lemma 14

Let $\mathcal{B}\Subset\Omega$ be a ball with $\texttt{r}(\mathcal{B})\leq 1$.
Under assumptions (1.2), and $0\leq a(\cdot)\in W^{1,d}(\Omega)$ with (1.17),
let $u_{\varepsilon}\in W^{1,p}(\mathcal{B};\mathbb{R}^{N})$ solve

$\displaystyle
u_{\varepsilon}\mapsto\min_{w}\int_{\mathcal{B}}[\tilde{H}_{\varepsilon}(x,\lvert
Dw\rvert)-f\cdot w]\,dx\qquad w\in
u+W^{1,p}_{0}(\mathcal{B};\mathbb{R}^{N})\,,$ (10.9)

where $u\in W^{1,1}(\Omega;\mathbb{R}^{N})$ is a minimizer of the functional
in (1.10). Then there exists $s_{*}\in(1,s_{**})$, depending only on
$\texttt{data}_{\textnormal{h}}$, but not on $\varepsilon$, such that

$\displaystyle\|\tilde{H}_{\varepsilon}(\cdot,|Du_{\varepsilon}|)+1\|_{L^{s_{*}}(\mathcal{B})}$
$\displaystyle\leq
c\|\tilde{H}_{\varepsilon}(\cdot,|Du|)+1\|_{L^{s_{*}}(\mathcal{B})}$
$\displaystyle\leq c\|\tilde{H}(\cdot,|Du|)+1\|_{L^{s_{*}}(\mathcal{B})}$
(10.10)

holds for a constant $c\equiv c(\texttt{data}_{\textnormal{h}})$, where
$s_{**}$ is the exponent coming from Lemma 12.

We finally remark that the constants appearing in (10.10) should a priori
depend also on $\lVert Du_{\varepsilon}\rVert_{L^{p}(B)}$, as Lemma 13 is
involved in the derivation of Lemma 14 (see again demicz ), and it is used
with the choices $w\equiv u_{\varepsilon}$ and $v\equiv u_{\varepsilon}$. Such
a dependence on $\varepsilon$ does not actually occur; indeed, adapting
estimate (7) to the present case leads to bounds as $\lVert
Du_{\varepsilon}\rVert_{L^{p}(\mathcal{B})}^{p}\lesssim\|H(\cdot,Du)+1\|_{L^{1}(\mathcal{B})}+[\texttt{r}(\mathcal{B})]^{n}\lVert
f\rVert_{L^{n}(\mathcal{B})}^{p/(p-1)}$, thereby reducing the dependence of
the constants on $\texttt{data}_{\textnormal{h}}$ in (10.1). This fact is
indeed used to prove the intermediate last inequality in (10.10) with an
$\varepsilon$-independent constant $c$. The last inequality in (10.10) is
instead a direct consequence of (5.22). In turn, the last quantity in (10.10)
is finite by Lemma 12.

#### Step 3: Completion of the proof of Theorem 1.3

We proceed almost verbatim as in the proof of Theorem 4.4. We start with the
case $n>2$; the only difference here is that we apply Proposition 1 with
$s_{*}>1$ being equal to the higher integrability exponent found in Lemma 14.
With no loss of generality we may assume that
$s_{*}<\min\\{2m(1+\hat{\sigma}),2^{*}\\}=\min\\{2mq/p,2^{*}\\}$, as required
in (6.15) (applied with $\sigma=0$). Note that this is possible when
$\hat{\sigma}<s_{*}/n-s_{*}/d$ and therefore, in particular, when
$\hat{\sigma}\leq 1/n-1/d$, which is the case considered here. Indeed, with
the choice made in Step 1, this last condition translates in $q/p\leq
1+1/n-1/d$, which is (1.19) for $n>2$. This is the essential point where the
higher integrability estimates of Step 2 come into the play, allowing for
equality in (1.19). We then proceed exactly as for Theorem 4.4, as its
assumptions are verified but for the equality case in (1.19), as noticed in
Step 1. Proceeding as in the proof of Theorem 4.4 we arrive at (9), and all
the foregoing considerations remain the same, but, in order to get a suitable
a priori estimate, the term involving $\mathbf{P}_{1}^{\mathfrak{h}}$ must be
estimated slightly differently from Proposition 3. More precisely, using
(6.16) with the current choice of $s_{*}$ in (6.6), we finally come to the new
uniform bound

$\displaystyle\lVert G_{\varepsilon}(\cdot,\lvert
Du_{\varepsilon}\rvert)\rVert_{L^{\infty}(B_{\varsigma})}$
$\displaystyle\qquad\leq\frac{c}{(\varrho-\varsigma)^{\beta}}\left[\|\tilde{H}_{\varepsilon}(\cdot,|Du_{\varepsilon}|)\|_{L^{s_{*}}(B_{\varrho})}^{\theta}+\lVert
f\rVert_{L(n,1)(B_{\varrho})}^{\theta}+1\right]\,,$ (10.11)

which replaces (3) and holds whenever $B_{\varsigma}\Subset B_{\varrho}$ are
concentric balls contained in $\mathcal{B}$; the constant $c$ depends on
$\texttt{data}_{\textnormal{h}}$. Note that we have applied the argument of
Proposition 3 directly to $u_{\varepsilon}$ defined in (10.9). Using (10.10)
in (10.11) yields

$\displaystyle\|G_{\varepsilon}(\cdot,\lvert
Du_{\varepsilon}\rvert)\|_{L^{\infty}(s\mathcal{B})}$
$\displaystyle\qquad\leq\frac{c}{(1-s)^{\beta}[\texttt{r}(\mathcal{B})]^{\beta}}\left[\|H(\cdot,Du)\|_{L^{s_{*}}(\mathcal{B})}^{\theta}+\lVert
f\rVert_{L(n,1)(\mathcal{B})}^{\theta}+1\right]$

for every $s\in(0,1)$. Again, the right-hand side stays finite by Lemma 12.
This last estimate can be used as a replacement of (7) in an approximation
argument which is at this stage completely similar to the one used for the
proof of Theorem 4.1 and this completes the proof in the case $n>2$. It
remains to treat the case $n=2$. For this we again turn to the arguments of
Section 6.7, where we apply Lemma 10 with the choice of the number $s_{*}>1$
again determined in Lemma 14. Such an application is legal as we are assuming
that $\hat{\sigma}=q/p-1\leq 1/2-1/d$, while again we may assume that
$s_{*}<2m(1+\hat{\sigma})$. Then we proceed exactly as in the proof of Theorem
4.4 case $n=2$, again noting that only the second inequality in (6.83) is
needed (formally with $\sigma=0$), and this leads to require that $q/p<p$,
which is the second condition in (1.19) for the case $n=2$.

###  10.3 Proof of Theorems 1.4-1.5 and Theorem 1.7

We deduce Theorem 1.5, from Theorem 4.1; Theorem 1.4 then follows similarly.
In exactly the same way, we can then deduce Theorem 1.7 from Theorem 4.3. As
usual, we do it by making a suitable choice of the growth functions
$g_{1},g_{2},g_{3}$ and of the parameters
$\sigma,\hat{\sigma},c_{\textnormal{a}},c_{\textnormal{b}},\gamma,T,\mu$. This
requires some preparations; we split the proof in two steps. In the following
we denote
$\texttt{data}_{k}\equiv(n,N,k,\nu_{\mathfrak{m}},L,p_{\mathfrak{m}},p_{m},p_{M})$,
for every integer $k\geq 0$. In the following, with abuse of notation, we
shall indicate by $\partial_{zz}{\bf e}_{k}(\cdot)$ the Hessian matrix of
$z\to{\bf e}_{k}(\cdot,|z|)$, while ${\bf e}_{k}^{\prime}(x,t)$ keeps on
denoting the (partial) derivative of the function $t\to{\bf e}_{k}(\cdot,t)$.
By Sobolev-Morrey embedding theorem we can assume that
$\\{\mathfrak{c}_{k},p_{k}\\}$ are Hölder continuous functions with exponent
$\alpha=1-n/d$.

#### Step 1: Computations

By induction we have

$\displaystyle{\bf
e}_{k}^{\prime}(x,t)=\mathfrak{c}_{k}(x)p_{k}(x)t^{p_{0}(x)-1}\Pi_{k}(x,t){\bf
e}_{k}(x,t)\ \ \mbox{for}\ \ k\geq 0\,,$ (10.12)

for every $x\in\Omega$ and $t>0$, where

$\begin{cases}\Pi_{k}(x,t):=\mathfrak{c}_{k-1}(x)p_{k-1}(x)[{\bf
e}_{k-1}(x,t)]^{p_{k}(x)}\Pi_{k-1}(x,t)\ \ \mbox{for $k\geq 1$}\\\
\Pi_{0}(x,t):=1\,.\end{cases}$

Then, observing that

$\Pi_{k}^{\prime}(x,t)=\Pi_{k}(x,t)t^{p_{0}(x)-1}\sum_{j=0}^{k-1}p_{j+1}(x)\mathfrak{c}_{j}(x)p_{j}(x)\Pi_{j}(x,t)\
\ \mbox{for}\ \ k\geq 1\,,$

identity (10.12) gives

${\bf e}_{0}^{\prime\prime}(x,t)=t^{p_{0}(x)-1}{\bf
e}_{0}^{\prime}(x,t)\left[\mathfrak{c}_{0}(x)p_{0}(x)+\frac{p_{0}(x)-1}{t^{p_{0}(x)}}\right]$

and

$\displaystyle{\bf e}_{k}^{\prime\prime}(x,t)$
$\displaystyle\,=t^{p_{0}(x)-1}{\bf
e}_{k}^{\prime}(x,t)\mathfrak{c}_{k}(x)p_{k}(x)\Pi_{k}(x,t)$
$\displaystyle\qquad+t^{p_{0}(x)-1}{\bf
e}_{k}^{\prime}(x,t)\sum_{j=0}^{k-1}p_{j+1}(x)\mathfrak{c}_{j}(x)p_{j}(x)\Pi_{j}(x,t)$
$\displaystyle\qquad+t^{p_{0}(x)-1}{\bf
e}_{k}^{\prime}(x,t)\frac{p_{0}(x)-1}{t^{p_{0}(x)}}\qquad\mbox{for}\ \ k\geq
1\,.$ (10.13)

Finally, it is

$[{[\bf e}_{k}(x,t)]^{p_{k+1}(x)}]^{\prime}=[{\bf
e}_{k}(x,t)]^{p_{k+1}(x)}p_{k+1}(x)\mathfrak{c}_{k}(x)p_{k}(x)t^{p_{0}(x)-1}\Pi_{k}(x,t)\,.$

We now come to the $x$-derivatives; note that the following computations are
justified as in (1.21) we are considering a composition of bounded Sobolev
functions $\mathfrak{c}_{k},p_{k}$ with smooth functions, and therefore the
standard (vectorial) chain rule applies as in the traditional case, i.e., if
$\mathfrak{c}_{k},p_{k}$ were $C^{1}$-regular. We use properly defined,
auxiliary vector fields
$\mathcal{D}_{k},\mathcal{L}_{k}\colon\Omega\times(0,\infty)\to\mathbb{R}^{n}$
for $k\geq 0$, and the notation ${\bf e}_{-1}(x,t)\equiv t$. Then, we have,
for a.e. $x\in\Omega$ and $t>0$

$\displaystyle\begin{cases}\partial_{x}{\bf e}_{k}(x,t)={\bf e}_{k}(x,t)[{\bf
e}_{k-1}(x,t)]^{p_{k}(x)}\mathcal{D}_{k}(x,t)\,,\quad\mbox{for}\ k\geq 0\\\
\partial_{x}[{\bf e}_{k-1}(x,t)]^{p_{k}(x,t)}:=[{\bf
e}_{k-1}(x,t)]^{p_{k}(x,t)}[{\bf e}_{k-2}(x,t)]^{p_{k-1}(x,t)}\\\ \hskip
102.42992pt\times\left[\mathfrak{c}_{k-1}(x)Dp_{k}(x)+p_{k}(x)\mathcal{D}_{k-1}(x,t)\right],\
\mbox{for}\ k\geq 1\,,\end{cases}$ (10.14)

where, by induction, we have defined, for $k\geq 0$

$\displaystyle\begin{cases}\mathcal{D}_{0}(x,t):=D\mathfrak{c}_{0}(x)+\mathfrak{c}_{0}(x)\log
t\,Dp_{0}(x)\\\
\mathcal{D}_{k+1}(x,t):=D\mathfrak{c}_{k+1}(x)+\mathfrak{c}_{k+1}(x)\mathfrak{c}_{k}(x)[{\bf
e}_{k-1}(x,t)]^{p_{k}(x)}Dp_{k+1}(x)\\\
\qquad\qquad\qquad+\mathfrak{c}_{k+1}(x)p_{k+1}(x)[{\bf
e}_{k-1}(x,t)]^{p_{k}(x)}\mathcal{D}_{k}(x,t)\,.\end{cases}$ (10.15)

Using (10.14)-(10.15), for $k\geq 1$ we compute, again by induction

$\displaystyle\begin{cases}\
\partial_{x}\Pi_{0}(x,t)=0\,,\quad\mathcal{L}_{0}(x,t)=0_{\mathbb{R}^{n}}\\\ \
\partial_{x}\Pi_{k}(x,t)=\Pi_{k}(x,t)\mathcal{L}_{k}(x,t)\\\ \
\mathcal{L}_{k}(x,t):=D\log\left(\mathfrak{c}_{k-1}(x)p_{k-1}(x)\right)\\\ \
\quad+[{\bf
e}_{k-2}(x,t)]^{p_{k-1}(x)}\left[\mathfrak{c}_{k-1}(x)Dp_{k}(x)+p_{k}(x)\mathcal{D}_{k-1}(x,t)\right]+\mathcal{L}_{k-1}(x,t)\,.\end{cases}$
(10.16)

Finally, using (10.14)-(10.16) in (10.12), for every $k\geq 0$ we conclude
with

$\displaystyle\partial_{x}{\bf e}^{\prime}_{k}(x,t)={\bf
e}^{\prime}_{k}(x,t)\left[D\left(\log(\mathfrak{c}_{k}(x)p_{k}(x))\right)\right.$
$\displaystyle\left.\hskip 48.36967pt+\log
tDp_{0}(x)+\mathcal{L}_{k}(x,t)+[{\bf
e}_{k-1}(x,t)]^{p_{k}(x)}\mathcal{D}_{k}(x,t)\right].$ (10.17)

As for the tensor $\partial_{zz}{\bf e}_{k}(x,|z|)$, by a direct computation
we have that

$\displaystyle\begin{cases}\partial_{zz}{\bf
e}_{k}(x,|z|)\xi\cdot\xi\geq\min\left\\{{\bf
e}_{k}^{\prime\prime}(x,|z|),\frac{{\bf
e}_{k}^{\prime}(x,|z|)}{|z|}\right\\}|\xi|^{2}\\\ |\partial_{zz}{\bf
e}_{k}(x,|z|)|^{2}=[{\bf e}_{k}^{\prime\prime}(x,|z|)]^{2}+\left[\frac{{\bf
e}_{k}^{\prime}(x,|z|)}{|z|}\right]^{2}(Nn-1)\end{cases}$ (10.18)

hold for every choice of $z,\xi\in\mathbb{R}^{N\times n}$ with $z\not=0$ and
$x\in\Omega$.

#### Step 2: Determining $g_{1}$, $g_{2}$ and $g_{3}$.

For every fixed $k\geq 0$, the constants implied in the symbols $\lesssim$ and
$\approx$, will depend on $\texttt{data}_{k}$ and we shall use the auxiliary
functions

$\mathfrak{h}_{k}(x):=\sum_{i=0}^{k}\,[|D\mathfrak{c}_{i}(x)|+|Dp_{i}(x)|]+1\,,\qquad
k\geq 0\,.$

It follows that $\mathfrak{h}_{k}\in L^{d}(\Omega)$ for $d>n$, by assumptions.
By (10.15) it follows $|\mathcal{D}_{0}(x,t)|\lesssim\mathfrak{h}_{0}(x)[|\log
t|+1]$, so that induction gives that

$|\mathcal{D}_{k}(x,t)|\lesssim\mathfrak{h}_{k}(x)\left[(t^{p_{0}(x)}|\log
t|+1)\Pi_{k-1}(x,t)+1\right]\quad\mbox{holds for all $k\geq 1$}\,.$

In turn, this and (10.16) imply that

$|\mathcal{L}_{k}(x,t)|\lesssim\mathfrak{h}_{k}(x)\left[(t^{p_{0}(x)}|\log
t|+1)\Pi_{k-1}(x,t)+1\right]\quad\mbox{holds for all $k\geq 1$}\,.$

By (10.12) it is also ${\bf e}_{k}^{\prime}(x,t)\lesssim t^{p_{0}(x)-1}\leq
t^{p_{\mathfrak{m}}-1}$ for $t\approx 0$. From this and (10.3) it follows
there exit constants $m_{k}\equiv m_{k}(\texttt{data}_{k})\geq 1$ such that,
for a.e. $x\in\Omega$

$\begin{cases}\lvert\partial_{x}{\bf e}^{\prime}_{k}(x,t)\rvert\lesssim
m_{k}\mathfrak{h}_{k}(x)&\ \mbox{for all}\ \ t\in[0,{\rm e}]\\\
\lvert\partial_{x}{\bf
e}^{\prime}_{k}(x,t)\rvert\lesssim\mathfrak{h}_{k}(x){\bf
e}_{k}^{\prime}(x,t)t^{p_{0}(x)}\log t\Pi_{k}(x,t)&\ \mbox{for all}\ \
t\in[{\rm e},\infty)\,.\end{cases}$ (10.19)

Now, let $\\{c_{k},d_{k}\\}$ be constants larger than $1$, to be eventually
chosen large enough, again in dependence on $\texttt{data}_{k}$, and $\phi\in
C([0,\infty),[0,1])$ be a non-decreasing function such that $\phi(t)=0$ for
$t\in[0,{\rm e}/2]$ and $\phi(t)=1$ for $t\in[{\rm e},\infty)$. With
$p_{\mathfrak{m}}$ as in (1.20), and $x\in\Omega$ and $t>0$, we define

$\displaystyle\begin{cases}\ g_{1}(x,t)\equiv
g_{1,k}(x,t):=\phi(t)\min\\{p_{\mathfrak{m}}-1,1\\}{\bf
e}_{k}^{\prime}(x,t)/t\\\ \ g_{2}(x,t)\equiv
g_{2,k}(x,t):=\phi(t)c_{k}t^{p_{0}(x)}\Pi_{k}(x,t){\bf
e}_{k}^{\prime}(x,t)/t\\\ \ g_{3}(x,t)\equiv
g_{3,k}(x,t):=(1-\phi(t))c_{k}m_{k}\\\ \hskip
108.12047pt+\phi(t)c_{k}m_{k}{\bf e}_{k}^{\prime}(x,t)t^{p_{0}(x)}\log
t\Pi_{k}(x,t)\,.\end{cases}$

We are ready to check that $g_{1},g_{2},g_{3}$ satisfy the relations
prescribed in Section 4.1 and required to meet the assumptions of Theorem 4.1.
We take $c_{\textnormal{b}}\equiv c_{\textnormal{b}}(\texttt{data}_{k})\geq 1$
to be determined in due course of the proof, $T={\rm
e},\nu\approx\min\\{p_{\mathfrak{m}}-1,1\\},\gamma=p_{\mathfrak{m}},\mu=0$;
moreover, any choice of small numbers $\sigma,\hat{\sigma}>0$ will work. In
this way it is $\bar{G}(x,t)=\min\\{p_{\mathfrak{m}}-1,1\\}[{\bf
e}_{k}(x,t)-{\bf e}_{k}(x,{\rm e})+({\rm e}^{2}+1)^{p_{\mathfrak{m}}/2}]$ for
$t\geq{\rm e}$, so that $\bar{G}(x,t)\approx{\bf e}_{k}(x,t)$ for $t$ large.
By (10.12)-(10.3) and (10.18), it follows that (4.2)2,3 are satisfied provided
we take $c_{k}\equiv c_{k}(\texttt{data}_{k})$ large enough. Similarly, (4.2)4
follows using (10.19) and eventually increasing $c_{k}$. In the same way (4.4)
holds with the above choice of the parameters by (10.12). As for
(4.16)-(4.18), note that, given $k\in\mathbb{N}$ and $\varepsilon\in(0,1)$,
there exists $c\equiv c(\texttt{data}_{k},\varepsilon)$, such that
$\Pi_{k}(x,t)t^{p_{0}(x)+1}\log t\leq c[{\bf e}_{k}(x,t)]^{\varepsilon}$ for
$t\geq{\rm e}$. Using this, (4.16) follows, for all $\hat{\sigma}>0$, from
(10.19) by taking $h\equiv d_{k}\mathfrak{h}_{k}\in L^{d}(\Omega)$, provided
we take $d_{k}\equiv d_{k}(\texttt{data}_{k},\hat{\sigma})$ large enough. As
for (4.17), note that $(g_{3}^{2}/g_{1})(x,t)\lesssim[{\bf
e}_{k}^{\prime}(x,t)]^{1+\varepsilon}\lesssim[{\bf
e}_{k}(x,t)]^{1+2\varepsilon}$ for every $\varepsilon>0$ and $t\geq{\rm e}$;
therefore (4.17) follows for every $\sigma\in(0,1)$, again taking
$c_{\textnormal{b}}\equiv c_{\textnormal{b}}(\texttt{data}_{k},\sigma)$ large
enough. In the same way (4.18) follows, by eventually enlarging
$c_{\textnormal{b}}$. This means that the assumptions of Theorem 4.1 are
satisfied and the proof of Theorem 1.5 is complete.

###  10.4 Proof of Theorem 1.6

Conditions in (4.1) are verified by (1.4); indeed, an easy consequence of
(1.4) is that $t\mapsto\tilde{a}(t)/t^{i_{\textnormal{a}}}$ is non-decreasing.
It follows that $\tilde{a}(t)t\leq\tilde{a}(1)t^{i_{\textnormal{a}}+1}$ for
$t\in(0,1]$, and therefore $t\mapsto A(x,t)\in C^{1}_{{\rm loc}}[0,\infty)\cap
C^{2}_{{\rm loc}}(0,\infty)$ for every $x\in\Omega$ as it is
$i_{\textnormal{a}}>-1$. Moreover

$\displaystyle\begin{cases}\ \lvert\partial_{zz}A(x,\lvert z\rvert)\rvert\leq
L\sqrt{Nn}\max\\{1,s_{\textnormal{a}}+1\\}\tilde{a}(\lvert z\rvert)\\\ \
\nu\min\\{1,i_{\textnormal{a}}+1\\}\tilde{a}(\lvert
z\rvert)\lvert\xi\rvert^{2}\leq\partial_{zz}A(x,\lvert z\rvert)\xi\cdot\xi\\\
\ \lvert\partial_{xz}A(x,\lvert z\rvert)\rvert\leq\lvert
D\mathfrak{c}(x)\rvert\tilde{a}(\lvert z\rvert)\lvert z\rvert\,,\end{cases}$

hold for every choice of $x\in\Omega$ and $z,\xi\in\mathbb{R}^{N\times n}$,
$|z|\not=0$. Notice that here we again indicate by $\partial_{zz}A(\cdot)$ the
Hessian of $z\to A(\cdot,|z|)$. Having in mind to apply Theorem 4.2, this
leads to define
$g_{1}(t):=\nu\min\\{1,i_{\textnormal{a}}+1\\}\tilde{a}(t),g_{2}(t):=LNn\max\\{1,s_{\textnormal{a}}+1\\}\tilde{a}(t),g_{3}(t):=\tilde{a}(t)t$
and $h:=\lvert D\mathfrak{c}\rvert\in\mathfrak{X}(\Omega)$. As
$t\mapsto\tilde{a}(t)/t^{i_{\textnormal{a}}}$ is non-decreasing, (4.4) is also
verified with $\gamma=i_{\textnormal{a}}+2$, $\mu=0$. The definitions given
above make sure that conditions $\eqref{uni0}$ are satisfied with
$c_{\textnormal{u}}\approx\max\\{1,s_{\textnormal{a}}+1\\}/\min\\{1,i_{\textnormal{a}}+1\\}$.
This means that Theorem 4.2 applies and (1.24) follows from (4.23).

##  11 Obstacle problems and Theorem 1.8

###  11.1 A general result

We start with the following constrained analog of Theorem 4.1, which will be
used to ge the proof of Theorem 1.8:

###### Theorem 11.1

Let $u\in W^{1,1}_{{\rm loc}}(\Omega)$ be a constrained minimizer of the
functional $\mathcal{F}_{0}$ in (1.25), under assumptions set in (4.20) with
$g_{3},\partial_{zz}F$ being locally bounded on $\Omega\times[0,\infty)$ and
$\Omega\times\mathbb{R}^{N\times n}$, respectively, and $\gamma\geq 2$. If
$\psi\in W^{2,1}_{{\rm loc}}(\Omega)$ with $\lvert
D^{2}\psi\rvert\in\mathfrak{X}(\Omega)$, then $Du\in
L^{\infty}_{\textnormal{loc}}(\Omega;\mathbb{R}^{N\times n})$.

###### Proof

In set we have initially assumed that $g_{3}$ was locally bounded in
$\Omega\times(0,\infty)$, while here we are assuming it is locally bounded in
$\Omega\times[0,\infty)$, that means that for every $\Omega_{0}\Subset\Omega$
and $b\geq 0$ we have that $\sup_{\Omega_{0}\times[0,b]}\,g_{3}$ is finite.
This is is not really an additional assumption as it is automatically
satisfied in all the cases considered for instance in Theorem 1.8. Notice also
that $D^{2}\psi\in\mathfrak{X}(\Omega)\subset L(n,1)$ implies that $D\psi$ is
continuous in $\Omega$ St  and, in particular, $D\psi\in
L^{\infty}(\mathcal{B};\mathbb{R}^{n})$ for any ball
$\mathcal{B}\Subset\Omega$; accordingly, we let $T_{\psi}\equiv
T_{\psi}(\mathcal{B}):=\|D\psi\|_{L^{\infty}(\mathcal{B})}+T+1$. We now fix an
arbitrary ball $\mathcal{B}\Subset\Omega$ with $\texttt{r}(\mathcal{B})\leq
1$, we consider the family $\\{F_{\varepsilon}\\}$ constructed in Section 5.1.
This time we take $0<\varepsilon<\min\\{1/T_{\psi},T\\}/4$ and define
$u_{\varepsilon}$ as the solution to the auxiliary problem

$\displaystyle
u_{\varepsilon}\mapsto\min_{(u+W^{1,\gamma}_{0}(\mathcal{B}))\cap\mathcal{K}_{\psi}(\mathcal{B})}\int_{\mathcal{B}}F_{\varepsilon}(x,Dw)\,dx\,.$
(11.1)

The existence of $u_{\varepsilon}$ follows by standard theory, and the
variational inequality

$\displaystyle\int_{\mathcal{B}}\partial_{z}F_{\varepsilon}(x,Du_{\varepsilon})\cdot(Dw-
Du_{\varepsilon})\,dx\geq 0$ (11.2)

holds for all
$w\in(u+W^{1,\gamma}_{0}(\mathcal{B}))\cap\mathcal{K}_{\psi}(\mathcal{B})$.
Thanks to the $\gamma$-polynomial growth conditions in (5.14), we are now able
to perform the linearization procedure used in (fumin, , page 237) i.e., we
can rearrange (11.2) in the following way:

$\displaystyle\int_{\mathcal{B}}\left[\partial_{z}F_{\varepsilon}(x,Du_{\varepsilon})\cdot
D\varphi-\texttt{f}_{\varepsilon}\cdot\varphi\right]\,dx=0$ (11.3)

for all $\varphi\in W^{1,\gamma}_{0}(\mathcal{B})$, where

$\texttt{f}_{\varepsilon}(x):=-\theta_{\varepsilon}(x)\mathds{1}_{\\{x\in\mathcal{B}\colon
u_{\varepsilon}(x)=\psi(x)\\}}\,{\rm
div}\,\partial_{z}F_{\varepsilon}(x,D\psi)\,,$

for some measurable density $\theta_{\varepsilon}\colon\mathcal{B}\to[0,1]$.
Note that the definition of $\texttt{f}_{\varepsilon}$ makes sense in light of
the fact that $\psi\in W^{2,n}(\Omega)\cap W^{1,\infty}(\Omega)$ and of the
discussion made at the beginning of Section 6 to prove (6.8). This implies
that $\partial_{z}F_{\varepsilon}(\cdot,D\psi)\in
W^{1,n}(\mathcal{B};\mathbb{R}^{n})$ and the usual chain rule formula holds as
in Section 6, again thanks to the results in dele ; see Remark 10 below. We
then define the constant

$\displaystyle\mathfrak{g}_{\psi}(\mathcal{B})$
$\displaystyle:=\|g_{3}\|_{L^{\infty}(\mathcal{B}\times[0,T_{\psi}))}+\|\tilde{a}(\cdot,1)\|_{L^{\infty}(\mathcal{B})}$
$\displaystyle\qquad\quad+\sup_{x\in\mathcal{B},|z|\in[0,T_{\psi})}|\partial_{zz}F(x,z)|+T_{\psi}^{\gamma-2}+1\,.$
(11.4)

This quantity is always bounded as $\partial_{zz}F$ is in turn assumed to be
locally bounded and $\gamma\geq 2$, here we also use the fact that $g_{3}$ is
locally bounded as again described in the statement of the Theorem. This is
essentially the only place where such assumptions come into the play.
Elementary manipulations based on the first property in (4.4) and (5.13)3,
give

$\lvert\texttt{f}_{\varepsilon}(x)\rvert\leq
c(n,N,\nu,\gamma,c_{\textnormal{b}})\mathfrak{g}_{\psi}(\mathcal{B})[\lvert
D^{2}\psi(x)\rvert+h(x)]\,.$ (11.5)

Here we have also used that $\varepsilon\leq 1/T_{\psi}$ to exploit the
definitions in (5.3) and (5.9). In turn, (11.5) gives

$\displaystyle\texttt{f}_{\varepsilon}\in\mathfrak{X}(B)\quad\mbox{with}\ \
\lVert\texttt{f}_{\varepsilon}\rVert_{\mathfrak{X}(B)}\leq c\lVert
D^{2}\psi\rVert_{\mathfrak{X}(B)}+c\lVert h\rVert_{\mathfrak{X}(B)}\,,$ (11.6)

for all balls $B\subseteq\mathcal{B}$, where $c\equiv
c(n,N,\nu,\gamma,c_{\textnormal{b}},g_{\psi}(B))$. Since $u_{\varepsilon}$
verifies (11.3), the strict convexity of $F_{\varepsilon}(\cdot)$ prescribed
by $\eqref{z3}_{4}$ implies that $u_{\varepsilon}$ is the unique solution of
Dirichlet problem

$\displaystyle
u_{\varepsilon}\mapsto\min_{u+W^{1,\gamma}_{0}(\mathcal{B})}\int_{\mathcal{B}}\left[F_{\varepsilon}(x,Dw)-\texttt{f}_{\varepsilon}\cdot
w\right]\,dx\,.$ (11.7)

By (5.14) and (11.6), we see that problem (11.7) falls in the realm of those
covered by Proposition 6 in Section 12 below, therefore $u_{\varepsilon}\in
W^{1,\infty}_{{\rm loc}}(B)\cap W^{2,2}_{{\rm loc}}(B)$. This is exactly the
information in (6.5) allowing to justify the all the subsequent calculations
in Section 6 in view of an application to solutions to (11.7). Moreover,
thanks to (11.5), the coercivity estimate (7) applied to the case of (11.7),
becomes

$\displaystyle\|F_{\varepsilon}(\cdot,Du_{\varepsilon})\|_{L^{1}(\mathcal{B})}+\|Du_{\varepsilon}\|_{L^{\gamma}(\mathcal{B})}^{\gamma}$
$\displaystyle\qquad\leq c\|F(\cdot,Du)\|_{L^{1}(\mathcal{B})}+c\lVert
D^{2}\psi\rVert_{L^{n}(\mathcal{B})}^{\gamma/(\gamma-1)}+c\lVert
h\rVert_{L^{n}(\mathcal{B})}^{\gamma/(\gamma-1)}+c$ (11.8)

thanks to (11.5). Using Proposition 3 when $n>2$, and Proposition 4 when $n=2$
(with $u_{\varepsilon}\equiv u$ and $\texttt{f}_{\varepsilon}\equiv f$), we
arrive at

$\displaystyle[E_{\varepsilon}(\lVert
Du_{\varepsilon}\rVert_{L^{\infty}(s\mathcal{B})})]^{\gamma}+\lVert
G_{\varepsilon}(\cdot,\lvert
Du_{\varepsilon}\rvert)\rVert_{L^{\infty}(s\mathcal{B})}$ $\displaystyle\hskip
17.07164pt\leq\frac{c}{(1-s)^{\beta}[\texttt{r}(\mathcal{B})]^{\beta}}\left[\|F(\cdot,Du)\|_{L^{1}(\mathcal{B})}^{\theta}+\lVert
D^{2}\psi\rVert_{\mathfrak{X}(\mathcal{B})}^{\theta}+\lVert
h\rVert_{\mathfrak{X}(\mathcal{B})}^{\theta}+1\right]\,,$ (11.9)

where we have used (11.5) and (11.1). Here $c$ depends on
$\texttt{data},\lVert h\rVert_{L^{d}(\mathcal{B})}$ and
$\mathfrak{g}_{\psi}(\mathcal{B})$, but not on $\varepsilon$. Notice that,
accordingly to the content of Section 6 and in particular recalling (6.28),
estimate (11.1) does not hold for any ball, but it holds provided
$\texttt{r}(\mathcal{B})\leq R_{*}$ where, exactly as in (6.28), the threshold
radius z $R_{*}$ depends now also on $h(\cdot),D\psi(\cdot)$ and
$\mathfrak{g}_{\psi}(\mathcal{B})$. As for the dependence on this last
quantity, there is no vicious circle here since the set function
$\Omega_{0}\mapsto\mathfrak{g}_{\psi}(\Omega_{0})$ defined in (11.4) is
obviously non-decreasing with respect to general open subsets
$\Omega_{0}\Subset\Omega$ (specifically, fix $\Omega_{0}\Subset\Omega$,
determine $\mathfrak{g}_{\psi}(\Omega_{0})$ as in (11.4) and proceed for every
ball $\mathcal{B}\Subset\Omega_{0}$ with radius $\texttt{r}(\mathcal{B})$
whose smallness now depends on $\mathfrak{g}_{\psi}(\Omega_{0})$). Estimate
(11.1) can be now used to replace (7) in the approximation scheme of the proof
of Theorem 4.1, with $\varepsilon\equiv\varepsilon_{j}$. The rest of the proof
now follows exactly the proof of Theorem 4.1 and leads to the conclusion,
along with explicit a priori estimates obtainable by (11.1) after a suitable
passage to the limit. Note that also the argument of (7.11) can be repeated
verbatim, as the obstacle constraint involved here is still convex. Indeed,
note that here we are not passing to the limit in the linearized problems
(11.3), but rather directly in the obstacle problems (11.1); see for instance
(dejmaa, , Section 5.5) and fumin  for more details on such approximation
arguments.

###### Remark 10

The linearization procedure leading to (11.7) has been first introduced in
fuobs , and it is usually employed assuming that $\psi\in
W^{2,\infty}(\mathcal{B})$ fumin . A careful examination of the proof reveals
that $\psi\in W^{2,n}(\mathcal{B})\cap W^{1,\infty}(\mathcal{B})$ suffices,
that in turn implies $\partial_{z}F_{\varepsilon}(\cdot,D\psi)\in
W^{1,n}(\mathcal{B};\mathbb{R}^{n})$. For this, a first basic remark is that
the weak integration-by-parts formula
$\int_{\mathcal{B}}h\varphi_{x_{i}}\,dx=-\int_{\mathcal{B}}h_{x_{i}}\varphi\,dx$
holds for every $i\in\\{1,\ldots,n\\}$, whenever $h\in W^{1,n}(\mathcal{B})$
and $\varphi\in W^{1,\gamma}_{0}(\mathcal{B})$, for some $\gamma>1$. In turn,
this follows by a standard density argument and Sobolev embedding in the
limiting case.

###  11.2 Proof of Theorem 1.8 and additional results

The proof of Theorem 11.1 offers a route to get the obstacle version of all
the other results presented in the unconstrained case; in particular, Theorem
1.8 follows. The key point is again to employ the linearization procedure used
in fuobs ; fumin  to pass from a variational inequality as in (11.2) to an
equation as in (11.3), to which the estimates in the unconstrained case
immediately apply. The whole procedure then works provided the additional
assumptions $\gamma\geq 2$ and $g_{3},\partial_{zz}F\in L^{\infty}_{{\rm
loc}}$ are in force as described in the statement of Theorem 11.1. With this
path being settled, the reader can now easily obtain the constrained
extensions of all the results presented in this paper. A few remarks are in
order. First, notice that Theorems 1.1-1.2 and Theorems 1.4-1.5 in the
constrained version, are a direct consequence of Theorem 11.1 and this can be
checked exactly as in the unconstrained version. Next, again as for the case
of Theorem 11.1, Theorems 4.2 and 4.4 admit a constrained reformulation. In
turn, the former would imply a constrained version of Theorem 1.6. The latter
would instead imply a first constrained version of Theorem 1.3, where the
bounds in (1.19) appear in the $<$-version; see also Step 1 of the proof of
Theorem 1.3. As for the full $\leq$-version in (1.19), it is then necessary to
readapt to the obstacle case the arguments from Step 2 and 3 of the proof of
Theorem 1.3 in Section 10.2, along the lines of the proof of Theorem 11.1. In
this respect, the only worth mentioning difference is that the higher
integrability lemmas 12 and 13 can be easily obtained in the setting of
obstacle problems too, starting from the arguments indicated here. For this
see also chde . Finally, note that in the case of functionals with
$(p,q)$-growth, including the double phase one in (1.10), verifying the
assumptions $\gamma\geq 2$ and $\partial_{zz}F\in L^{\infty}_{{\rm loc}}$
boils down to assume that $p\geq 2$, as indeed done in Theorem 1.8. The
additional (micro)assumption on $g_{3}$ in the statement of Theorem 11.1 is
instead satisfied in every case.

###### Remark 11

In Theorem 11.1 we can trade the assumption $\gamma\geq 2$ with $\mu>0$ (non-
degenerate case). This eventually leads to constrained versions of Theorems
1.1-1.2 assuming that $p>1$ instead of $p\geq 2$, provided it is $\mu>0$.

##  12 Justification of a priori regularity

In this final section we justify the claim in (6.5), which is necessary to
carry out the rest of the estimates in Sections 6 and 8. Keeping (5.6) in
mind, we therefore consider a functional like (1.1), with the integrand $F$
being convex in the gradient variable, satisfying the structure condition
(4.1)1, where we assume that $\tilde{F}$ and $\tilde{F}^{\prime}$ are
continuous functions on $\Omega\times[0,\infty)$ such that
$\tilde{F}(x,0)\equiv 0$, that $\partial_{zz}F,\partial_{xz}F$ are measurable
and

$\displaystyle\begin{cases}\ t\mapsto\tilde{F}(x,t)\in
C^{1}_{\textnormal{loc}}[0,\infty)&\mbox{for all}\ \ x\in\Omega\\\ \
t\mapsto\tilde{F}^{\prime}(x,t)=:\tilde{a}(x,t)t\in\textnormal{Lip}_{{\rm
loc}}[0,\infty)&\mbox{for all}\ \ x\in\Omega\\\ \
x\mapsto\tilde{F}^{\prime}(x,t)\in W(1;\mathfrak{X})(\Omega)\quad&\mbox{for
every}\ \ t\geq 0\,.\end{cases}$ (12.1)

The last condition means that
$|\partial_{x}\tilde{F}^{\prime}(\cdot,t)|\in\mathfrak{X}(\Omega)$ for every
$t\geq 0$, where $\mathfrak{X}(\Omega)$ has been defined in (1.9). In
particular, this implies that $x\mapsto\tilde{F}^{\prime}(x,t)\in
W^{1,n}(\Omega)$. As in Section 6, we also assume that the set of non-
differentiability points of $t\mapsto\tilde{a}(x,t)t$ is independent of $x$
and that

$\displaystyle\begin{cases}\ \nu_{0}(\lvert
z\rvert^{2}+\mu^{2})^{\gamma/2}-\Lambda\mu^{\gamma}\leq
F(x,z)\leq\Lambda(\lvert z\rvert^{2}+\mu^{2})^{\gamma/2}\\\ \
\lvert\partial_{zz}F(x,z)\rvert\leq\Lambda(\lvert
z\rvert^{2}+\mu^{2})^{(\gamma-2)/2}\\\ \ \nu_{0}(\lvert
z\rvert^{2}+\mu^{2})^{(\gamma-2)/2}\lvert\xi\rvert^{2}\leq\partial_{zz}F(x,z)\xi\cdot\xi\\\
\ \lvert\partial_{xz}F(x,z)\rvert\leq\Lambda h(x)(\lvert
z\rvert^{2}+\mu^{2})^{(\gamma-1)/2}\\\ \
f,h\in\mathfrak{X}(\Omega)\end{cases}$ (12.2)

hold for every $x\in\Omega$, and for every $z\in\mathbb{R}^{N\times n}$,
provided $\partial_{zz}F(x,z)$ exists, and a.e. $x$ and
$z,\xi\in\mathbb{R}^{N\times n}$ in the case of (12.2)4. In $\eqref{ir0}$, it
is $\gamma>1$, $0<\nu_{0}\leq 1\leq\Lambda$, $0<\mu\leq 1$. This functional is
of the type considered in (6.2) by (5.14) and (5.23), so that the claim in
(6.5) is justified by the following:

###### Proposition 6

Let $u\in W^{1,\gamma}_{{\rm loc}}(\Omega;\mathbb{R}^{N})$ be a minimizer of
$\mathcal{F}$ in (1.1), under assumptions (12.1)-(12.2). Then $Du\in
L^{\infty}_{\textnormal{loc}}(\Omega;\mathbb{R}^{N\times n})$ and $u\in
W^{2,2}_{\textnormal{loc}}(\Omega;\mathbb{R}^{N})$.

###### Proof

The proof goes now in three different steps, where we essentially revisit and
readapt a few hidden facts in the literature. References DeMi  and KM  are
particularly relevant here.

#### Step 1: Regularized integrands.

We revisit the procedure we used in (DeMi, , Theorem 4, Step 1) and start
fixing a ball $\mathcal{B}\Subset\Omega$ such that
$\texttt{r}(\mathcal{B})\leq 1$. We first extend $\tilde{F}$ by even
reflection making it defined on $\Omega\times\mathbb{R}$, i.e.,
$\tilde{F}(x,t):=\tilde{F}(x,-t)$, and then we consider standard, radially
symmetric mollifiers $\phi_{1}\in C^{\infty}_{\rm{c}}(B_{1}),\phi_{2}\in
C^{\infty}_{\rm{c}}(-1,1),\|\phi_{1}\|_{L^{1}(\mathbb{R}^{n})}=\|\phi_{2}\|_{L^{1}(\mathbb{R})}=1,\phi_{1,\delta}(x):=\phi_{1}(x/\delta)/\delta^{n}$,
$\phi_{2,\delta}(t):=\phi(t/\delta)/\delta$,
$B_{3/4}(0)\subset\textnormal{supp}\,\phi_{1},(-3/4,3/4)\subset\textnormal{supp}\,\phi_{2}$.
With $\delta$ such that $0<\delta<\left(0,\,{\rm
dist}(\mathcal{B},\partial\Omega)/2\right)$, we define

$\begin{cases}\displaystyle\tilde{F}_{\delta}(x,t):=\int_{-1}^{1}\int_{B_{1}}\tilde{F}(x+\delta
y,t+\delta s)\phi_{1}(y)\phi_{2}(s)\,dy\,ds\\\
h_{\delta}(x):=(h*\phi_{1,\delta})(x)\end{cases}$ (12.3)

for all $x\in\mathcal{B}$ and $t\in\mathbb{R}$.

###### Lemma 15

If $F_{\delta}(x,z):=\tilde{F}_{\delta}(x,\langle z\rangle)$, where $\langle
z\rangle:=\sqrt{|z|^{2}+\delta^{2}}$, then

$\displaystyle\begin{cases}\ \frac{1}{c_{\textnormal{r}}}(\lvert
z\rvert^{2}+\mu^{2}_{\delta})^{\gamma/2}-c_{\textnormal{r}}\mu_{\delta}^{\gamma}\leq
F_{\delta}(x,z)\leq c_{\textnormal{r}}(\lvert
z\rvert^{2}+\mu^{2}_{\delta})^{\gamma/2}\\\ \
\lvert\partial_{zz}F_{\delta}(x,z)\rvert\leq c_{\textnormal{r}}(\lvert
z\rvert^{2}+\mu^{2}_{\delta})^{(\gamma-2)/2}\\\ \
\frac{1}{c_{\textnormal{r}}}(\lvert
z\rvert^{2}+\mu^{2}_{\delta})^{(\gamma-2)/2}\lvert\xi\rvert^{2}\leq\partial_{zz}F_{\delta}(x,z)\xi\cdot\xi\\\
\ \lvert\partial_{xz}F_{\delta}(x,z)\rvert\leq
c_{\textnormal{r}}h_{\delta}(x)(\lvert
z\rvert^{2}+\mu^{2}_{\delta})^{(\gamma-1)/2}\end{cases}$ (12.4)

hold for every $x\in\Omega^{\prime}$, $z,\xi\in\mathbb{R}^{N\times n}$, where
$c_{\textnormal{r}}\equiv c_{\textnormal{r}}(n,N,\nu_{0},\Lambda,\gamma)\geq
1$ is a positive constant, and, as usual, it is $\mu_{\delta}:=\mu+\delta$.
Moreover, we have

$\left\\{\begin{array}[]{c}F_{\delta}(x,z)\to F(x,z)\ \ \mbox{uniformly on
compact subsets of $\overline{\mathcal{B}}\times\mathbb{R}^{n}$ as $\delta\to
0$}\\\\[3.0pt] \lVert h_{\delta}\rVert_{\mathfrak{X}(B)}\leq c\lVert
h\rVert_{\mathfrak{X}(B+\delta B_{1}(0))}\end{array}\right.$ (12.5)

whenever $B\subset\mathcal{B}$ is a ball such that $B+\delta