# Lipschitz regularity for
degenerate elliptic integrals with $p,q$-growth

G. Cupini – P. Marcellini – E. Mascolo – A. Passarelli di Napoli  Giovanni
Cupini: Dipartimento di Matematica, Università di Bologna
Piazza di Porta S.Donato 5, 40126 - Bologna, Italy<EMAIL_ADDRESS>Paolo Marcellini and Elvira Mascolo: Dipartimento di Matematica “U. Dini”,
Università di Firenze
Viale Morgagni 67/A, 50134 - Firenze, Italy<EMAIL_ADDRESS><EMAIL_ADDRESS>Antonia Passarelli di Napoli: Dipartimento di
Matematica e Appl. “R. Caccioppoli”
Università di Napoli “Federico II”
Via Cintia, 80126 Napoli, Italy<EMAIL_ADDRESS>
###### Abstract.

We establish the local Lipschitz continuity and the higher differentiability
of vector-valued local minimizers of a class of energy integrals of the
Calculus of Variations. The main novelty is that we deal with possibly
degenerate energy densities with respect to the $x-$variable.

###### Key words and phrases:

Nonstandard growth conditions; $p,q$-growth; Degenerate ellipticity; Lipschitz
continuity.

###### 2010 Mathematics Subject Classification:

49N60, 35J50

Acknowledgements. The authors are members of GNAMPA (Gruppo Nazionale per
l’Analisi Matematica, la Probabilità e le loro Applicazioni) of INdAM
(Istituto Nazionale di Alta Matematica)

##  1\. Introduction

The paper deals with the regularity of minimizers of integral functionals of
the Calculus of Variations of the form

$F(u)=\int_{\Omega}f(x,Du)\,dx$ (1.1)

where $\Omega\subset\mathbb{R}^{n}$, $n\geq 2$, is a bounded open set,
$u:\Omega\rightarrow\mathbb{R}^{N}$, $N\geq 1$, is a Sobolev map. The main
feature of (1.1) is the possible degeneracy of the lagrangian $f(x,\xi)$ with
respect to the $x-$variable. We assume that the Carathéodory function
$f=f\left(x,\xi\right)$ is convex and of class $C^{2}$ with respect to
$\xi\in\mathbb{R}^{N\times n}$, with $f_{\xi\xi}\left(x,\xi\right)$, $f_{\xi
x}\left(x,\xi\right)$ also Carathéodory functions and $f(\cdot,0)\in
L^{1}(\Omega)$. We emphasize that the $N\times n$ matrix of the second
derivatives $f_{\xi\xi}\left(x,\xi\right)$ not necessarily is uniformly
elliptic and it may degenerate at some $x\in\Omega$.

In the vector-valued case $N>1$ minimizers of functionals with general
structure may lack regularity, see [17],[48],[42], and it is natural to assume
a modulus-gradient dependence for the energy density; i.e. that there exists
$g=g(x,t):\Omega\times[0,+\infty)\rightarrow[0,+\infty)$ such that

$f(x,\xi)=g(x,|\xi|).$ (1.2)

Without loss of generality we can assume $g(x,0)=0$; indeed the minimizers of
$F$ are minimizers of $u\mapsto\int_{\Omega}\left(f(x,Du)-f(x,0)\right)\,dx$
too. Moreover, by (1.2) and the convexity of $f$, $g\left(x,t\right)$ is a
non-negative, convex and increasing function of $t\in\left[0,+\infty\right)$.

As far as the growth and the ellipticity assumptions are concerned, we assume
that there exist exponents $p,q$, nonnegative measurable functions $a(x),k(x)$
and a constant $L>0$ such that

$\left\\{\begin{array}[]{l}a\left(x\right)\,(1+|\xi|^{2})^{\frac{p-2}{2}}|\lambda|^{2}\leq\langle
f_{\xi\xi}(x,\xi)\lambda,\lambda\rangle\leq
L\,(1+|\xi|^{2})^{\frac{q-2}{2}}|\lambda|^{2},\quad 2\leq p\leq q,\\\
\left|f_{\xi x}(x,\xi)\right|\leq
k\left(x\right)(1+|\xi|^{2})^{\frac{q-1}{2}}\end{array}\right.$ (1.3)

for a.e. $x\in\Omega$ and for every $\xi,\lambda\in\mathbb{R}^{N\times n}$. We
allow the coefficient $a\left(x\right)$ to be zero, so that (1.3)1 is a not
uniform ellipticity condition. As proved in Lemma 2.2, (1.3)1 implies the
following possibly degenerate $p,q-$growth conditions for $f$, for some
constant $c>0$,

$c\,a\left(x\right)(1+|\xi|^{2})^{\frac{p-2}{2}}|\xi|^{2}\leq f(x,\xi)\leq
L(1+|\xi|^{2})^{\frac{q}{2}},\;\;\;\;\text{a.e.
}x\in\Omega,\;\forall\;\xi\in\mathbb{R}^{N\times n}.$ (1.4)

Our main result concerns the local Lipschitz regularity and the higher
differentiability of the local minimizers of $F$.

######  Theorem 1.1.

Let the functional $F$ in (1.1) satisfy (1.2) and (1.3). Assume moreover that

$\frac{1}{a}\in L_{\mathrm{loc}}^{s}(\Omega),\qquad k\in
L_{\mathrm{loc}}^{r}(\Omega),$ (1.5)

with $r,s>n$ and

$\frac{q}{p}<\frac{s}{s+1}\left(1+\frac{1}{n}-\frac{1}{r}\right).$ (1.6)

If $u\in W_{\mathnormal{loc}}^{1,1}(\Omega)$ is a local minimizer of $F$, then
for every ball $B_{R_{0}}\Subset\Omega$ the following estimates

$\|Du\|_{L^{\infty}(B_{R_{0}/2})}\leq
C\mathcal{K}_{R_{0}}^{\vartheta}\left(\int_{B_{R_{0}}}\left(1+f(x,Du)\right)\,dx\right)^{\vartheta}$
(1.7)
$\int_{B_{R_{0}/2}}a(1+|Du|^{2})^{\frac{p-2}{2}}\left|D^{2}u\right|^{2}\,dx\leq
C\mathcal{K}_{R_{0}}^{\vartheta}\left(\int_{B_{R_{0}}}\left(1+f(x,Du)\right)\,dx\right)^{\vartheta},$
(1.8)

hold with the exponent $\vartheta$ depending on the data, the constant $C$
also depending on $R_{0}$ and where
$\mathcal{K}_{R_{0}}=1+\|a^{-1}\|_{L^{s}(B_{R_{0}})}\|k\|_{L^{r}(B_{R_{0}})}^{2}$.

It is well known that to get regularity under $p,q-$growth the exponents $q$
and $p$ cannot be too far apart; usually, the gap between $p$ and $q$ is
described by a condition relating $p,q$ and the dimension $n$. In our case we
take into account the possible degeneracy of $a(x)$ and the condition (1.3)2
on the mixed derivatives $f_{\xi x}$ in terms of a possibly unbounded
coefficient $k(x)$; then we deduce that the gap depends on $s$, the
summability exponent of $a^{-1}$ that “measures” how much $a$ is degenerate,
and the exponent $r$ that tell us how far $k(x)$ is from being bounded. If
$s=r=\infty$ then (1.6) reduces to $\frac{q}{p}<1+\frac{1}{n}$ that is what
one expects, see [12] and for instance [39]. Moreover, if $s=\infty$ and
$n<r\leq+\infty$, then (1.6) reduces to
$\frac{q}{p}<1+\frac{1}{n}-\frac{1}{r}$ and we recover the result of [22].

Motivated by applications to the theory of elasticity, recently Colombo and
Mingione [8],[9] (see also [2],[24],[18],[19]) studied the so-called double
phase integrals

$\int_{\Omega}|Du|^{p}+b(x)|Du|^{q}\,dx,\quad 1<p<q\,.$ (1.9)

The model case we have in mind here is different: we consider the degenerate
functional with non standard growth of the form

$I(u)=\int_{B_{1}(0)}a(x)(1+|Du|^{2})^{\frac{p}{2}}+b(x)(1+|Du|^{2})^{\frac{q}{2}}\,dx$
(1.10)

with $0\leq a(x)\leq b(x)\leq L$ for some $L>0$. The integrand of $I(u)$
satisfies (1.3)2 with
$k\left(x\right)=\left|Da\left(x\right)\right|+\left|Db\left(x\right)\right|$.
It is worth mentioning that in the literature $a(x)$ is usually assumed
positive and bounded away from zero, see e.g. [2],[8], which is not the case
here since $a(x)$ may vanish at some point. The counterpart is that we
consider the powers of $(1+|Du|^{2})^{\frac{1}{2}}$ instead of $|Du|$. We
notice that the regularity result of Theorem 1.1 is new also when $p=q\geq 2$,
for example for the energy integral

$F_{1}(u)=\int_{B_{1}(0)}a(x)(1+|Du|^{2})^{\frac{p}{2}}\,dx$ (1.11)

with $a(x)\geq 0$, $\frac{1}{a}\in L^{s}(\Omega)$ and $\left|Da\right|\in
L^{r}$ with $\frac{1}{s}+\frac{1}{r}<\frac{1}{n}$. As far as we know, the
results proposed here are the first approach to the study of the Lipschitz
continuity of the local minimizers in the setting of degenerate elliptic
integrals under $p,q-$growth.

As well known, weak solutions to the elliptic equation in divergence form of
the type

$-\mathrm{div}\left(A(x,Du)\right)=0\,\,\text{in}\,\,\Omega.$

are locally Lipschitz continuous provided the vector field
$A:\Omega\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}$ is differentiable with
respect to $\xi$ and satisfies the uniformly elliptic conditions

$\Lambda_{1}(1+|\xi|^{2})^{\frac{p-2}{2}}|\lambda|^{2}\leq\langle
A_{\xi}(x,\xi)\lambda,\lambda\rangle\leq\Lambda_{2}(1+|\xi|^{2})^{\frac{p-2}{2}}|\lambda|^{2}.$

Trudinger [49] started the study of the interior regularity of solutions to
linear elliptic equation of the form

$\sum_{i,j=1}^{n}\frac{\partial}{\partial
x_{i}}\left(a_{ij}(x)\,\frac{\partial u}{\partial{x_{j}}}(x)\right)=0\,,\qquad
x\in\Omega\subseteq\mathbb{R}^{n},$ (1.12)

where the measurable coefficients $a_{ij}$ satisfy the non-uniform condition

$\lambda(x)|\xi|^{2}\leq\sum_{i,j=1}^{n}a_{ij}(x)\xi_{i}\xi_{j}\leq
n^{2}\mu(x)|\xi|^{2}$ (1.13)

for a.e. $x\in\Omega$ and every $\xi\in\mathbb{R}^{n}$. Here $\lambda(x)$ is
the minimum eigenvalue of the symmetric matrix $A(x)=(a_{ij}(x))$ and
$\mu(x):=\sup_{ij}|a_{ij}|$. Trudinger proved that any weak solution of (1.12)
is locally bounded in $\Omega$, under the following integrability assumptions
on $\lambda$ and $\mu$

$\lambda^{-1}\in
L_{\mathrm{loc}}^{r}(\Omega)\quad\text{and}\quad\mu_{1}=\lambda^{-1}\mu^{2}\in
L_{\mathrm{loc}}^{\sigma}(\Omega)\quad\text{with
$\frac{1}{r}+\frac{1}{\sigma}<\frac{2}{n}$}.$ (1.14)

The equation (1.12) is usually called degenerate when $\lambda^{-1}\notin
L^{\infty}(\Omega)$, whereas it is called singular when $\mu\notin
L^{\infty}(\Omega)$. These names in this case refer to the degenerate and the
singular cases with respect to the $x-$variable, but in the mathematical
literature these names are often referred to the gradient variable; this
happens for instance with the $p-$Laplacian operator
$-\mathrm{div}\left(\left|Du\right|^{p-2}Du\right)$. We do not study in this
paper the degenerate case with respect to the gradient variable, but we refer
for instance to the analysis made by Duzaar and Mingione [21], who studied an
$L^{\infty}-$gradient bound for solutions to non-homogeneous $p-$Laplacian
type systems and equations; see also Cianchi and Maz’ya [7] and the references
therein for the rich literature on the subject.

The result by Trudinger was extended in many settings and directions: firstly,
by Trudinger himself in [50] and later by Fabes, Kenig and Serapioni in [28];
Pingen in [47] dealt with systems. More recently for the regularity of
solutions and minimizers we refer to [3],[5],[10][15], [16],[31]. For the
higher integrability of the gradient we refer to [32] (see also [6]). Very
recently Calderon-Zygmund’s estimates for the $p-$Laplace operator with
degenerate weights have been established in [1]. The literature concerning
non-uniformly elliptic problems is extensive and we refer the interested
reader to the references therein.

The study of the Lipschitz regularity in the $p,q-$growth context started with
the papers by Marcellini [34],[35] and, since then, many and various
contributions to the subject have been provided, see the references in
[41],[39]. The vectorial homogeneous framework was considered in [36],[40] and
by Esposito, Leonetti and Mingione [26],[27]. The condition (1.3)2 for general
non autonomous integrands $f=f(x,Du)$ has been first introduced in
[22],[23],[24]. It is worth to highlight that, due to the $x-$dependence, the
study of regularity is significantly harder and the techniques more complex.
The research on this subject is intense, as confirmed by the many articles
recently published, see e.g. [11],[13],[20],[29],
[37],[38],[39],[44],[45],[46].

Let us briefly sketch the tools to get our regularity result. First, for
Lipschitz and higher differentiable minimizers, we prove a weighted
summability result for the second order derivatives of minimizers of
functionals with possibly degenerate energy densities, see Proposition 3.2.
Next in Theorem 3.3 we get an a-priori estimate for the $L^{\infty}$-norm of
the gradient. To establish the a-priori estimate we use the Moser’s iteration
method [43] for the gradient and the ideas of Trudinger [49]. An approximation
procedure allows us to conclude. Actually, if $u$ is a local minimizer of
(1.1), we construct a sequence of suitable variational problems in a ball
$B_{R}\subset\subset\Omega$ with boundary value data $u$. In order to apply
the a-priori estimate to the minimizers of the approximating functionals we
prove a higher differentiability result (Theorem 4.1) for minimizers of the
class of functionals with $p,q-$growth studied in [22], where only the
Lipschitz continuity was proved. By applying the previous a-priori estimate to
the sequence of the solutions we obtain a uniform control in $L^{\infty}$ of
the gradient which allows to transfer the local Lipschitz continuity property
to the original minimizer $u$.

Another difficulty due to the $x-$dependence of the energy density is that the
Lavrentiev phenomenon may occur. A local minimizer of $F$ is a function $u\in
W_{\mathrm{loc}}^{1,1}(\Omega)$ such that $f(x,Du)\in
L_{\mathrm{loc}}^{1}(\Omega)$ and

$\int_{\Omega}f(x,Du)\,dx\leq\int_{\Omega}f(x,Du+D\varphi)\,dx$

for every $\varphi\in C_{0}^{1}(\Omega)$. If $u$ is a local minimizer of the
functional $F$, by virtue of (1.4) we have that $a(x)|Du|^{p}\in
L_{\mathrm{loc}}^{1}(\Omega)$ and, by (1.5), $u\in
W_{\mathrm{loc}}^{1,\frac{ps}{s+1}}(\Omega)$ since

$\int_{B_{R}}|Du|^{\frac{ps}{s+1}}\,dx\leq\left(\int_{B_{R}}a|Du|^{p}\,dx\right)^{\frac{s}{s+1}}\left(\int_{B_{R}}\frac{1}{a^{s}}\,dx\right)^{\frac{1}{s+1}}<+\infty$
(1.15)

for every ball $B_{R}\subset\Omega$. Therefore in our context a-priori the
presence of the Lavrentiev phenomenon cannot be excluded. Indeed, due to the
growth assumptions on the energy density, the integral in (1.1) is well
defined if $u\in W^{1,q\frac{r}{r-1}}$, but a-priori this is not the case if
$u\in W^{1,\frac{ps}{s+1}}(\Omega)\setminus
W_{\mathrm{loc}}^{1,q\frac{r}{r-1}}(\Omega)$. However, as a consequence of
Theorem 1.1, under the stated assumptions (1.2),(1.3),(1.5),(1.6) the
Lavrentiev phenomenon for the integral functional $F$ in (1.1) cannot occur.
For the gap in the Lavrentiev phenomenon we refer to [51],[4],[27],[25].

We conclude this introduction by observing that even in the one-dimensional
case the Lipschitz continuity of minimizers for non-uniformly elliptic
integrals is not obvious. Indeed, if we consider a minimizer $u$ to the one-
dimensional integral

$F\left(u\right)=\int_{-1}^{1}a\left(x\right)\left|u^{\prime}\left(x\right)\right|^{p}\,dx\,,\qquad
p>1,$ (1.16)

then the Euler’s first variation takes the form

$\int_{-1}^{1}a\left(x\right)p\left|u^{\prime}\left(x\right)\right|^{p-2}u^{\prime}\left(x\right)\varphi^{\prime}\left(x\right)\,dx\,=0,\;\;\;\forall\;\varphi\in
C_{0}^{1}\left(-1,1\right).$

This implies that the quantity
$a\left(x\right)\left|u^{\prime}\left(x\right)\right|^{p-2}u^{\prime}\left(x\right)$
is constant in $\left(-1,1\right)$; it is a nonzero constant, unless
$u\left(x\right)$ itself is constant in $\left(-1,1\right)$, a trivial case
that we do not consider here. In particular the sign of
$u^{\prime}\left(x\right)$ is constant and we get

$\left|u^{\prime}\left(x\right)\right|^{p-1}=\frac{c}{a\left(x\right)}\,,\;\;\;\text{a.e.}\;x\in\left(-1,1\right).$

Therefore if $a\left(x\right)$ vanishes somewhere in $\left(-1,1\right)$ then
$\left|u^{\prime}\left(x\right)\right|$ is unbounded (and viceversa),
independently of the exponent $p>1$. Thus for $n=1$ the local Lipschitz
regularity of the minimizers does not hold in general if the coefficient
$a\left(x\right)$ vanishes somewhere.

We can compare this one-dimensional fact with the general conditions
considered in the Theorem 1.1. In the case
$a\left(x\right)=\left|x\right|^{\alpha}$ for some $\alpha\in\left(0,1\right)$
then, taking into account the assumptions in (1.5), for the integral in (1.16)
we have
$k\left(x\right)=a^{\prime}\left(x\right)=\alpha\left|x\right|^{\alpha-2}x$
and

$\left\\{\begin{array}[]{ccccc}\frac{1}{a}\in
L_{\mathrm{loc}}^{s}\left(-1,1\right)&\Leftrightarrow&1-\alpha
s>0&\Leftrightarrow&\alpha<\frac{1}{s}\\\ k\left(x\right)=a^{\prime}\in
L_{\mathrm{loc}}^{r}\left(-1,1\right)&\Leftrightarrow&r\left(\alpha-1\right)>-1&\Leftrightarrow&\alpha>1-\frac{1}{r}.\end{array}\right.$

These conditions are compatible if and only if $1-\frac{1}{r}<\frac{1}{s}$.
Therefore, also in the one-dimensional case we have a counterexample to the
$L^{\infty}-$gradient bound in (1.7) if

$\frac{1}{r}+\frac{1}{s}>1\,.$ (1.17)

This is a condition that can be easily compared with the assumption (1.6) for
the validity of $L^{\infty}-$gradient bound (1.7) in the general
$n-$dimensional case. In fact, being $1\leq\frac{q}{p}$, (1.6) implies

$1<\frac{s}{s+1}\left(1+\frac{1}{n}-\frac{1}{r}\right)\;\;\;\;\Leftrightarrow\;\;\;\;\frac{1}{r}+\frac{1}{s}<\frac{1}{n}\,,$

which essentially is the complementary condition to (1.17) when $n=1$.

The plan of the paper is the following. In Section 2 we list some definitions
and preliminary results. In Section 3 we prove an a-priori estimates of the
$L^{\infty}$-norm of the gradient of local minimizers and an higher
differentiability result, see Theorem 3.3. In Section 4 we prove an estimate
for the second order derivatives of a minimizer of an auxiliary uniformly
elliptic functional, see Theorem 4.1. In the last section we complete the
proof of Theorem 1.1.

##  2\. Preliminary results

We shall denote by $C$ or $c$ a general positive constant that may vary on
different occasions, even within the same line of estimates. Relevant
dependencies will be suitably emphasized using parentheses or subscripts. In
what follows, $B(x,r)=B_{r}(x)=\\{y\in\mathbb{R}^{n}:\,\,|y-x|<r\\}$ will
denote the ball centered at $x$ of radius $r$. We shall omit the dependence on
the center and on the radius when no confusion arises.

To prove our higher differentiability result ( see Theorem 4.1 below) we use
the finite difference operator. For a function
$u:\Omega\rightarrow\mathbb{R}^{k}$, $\Omega$ open subset of $\mathbb{R}^{n}$,
given $s\in\\{1,\ldots,n\\}$, we define

$\tau_{s,h}u(x):=u(x+he_{s})-u(x),\qquad x\in\Omega_{|h|},$ (2.1)

where $e_{s}$ is the unit vector in the $x_{s}$ direction, $h\in\mathbb{R}$
and

$\Omega_{|h|}:=\\{x\in\Omega\,:\,\mathrm{dist\,}(x,\partial\Omega)<|h|\\}.$

We now list the main properties of this operator.

  * (i)

if $u\in W^{1,t}(\Omega)$, $1\leq t\leq\infty$, then $\tau_{s,h}u\in
W^{1,t}(\Omega_{|h|})$

$D_{i}(\tau_{s,h}u)=\tau_{s,h}(D_{i}u),$

  * (ii)

if $f$ or $g$ has support in $\Omega_{|h|}$, then

$\int_{\Omega}f\tau_{s,h}g\,dx=\int_{\Omega}g\tau_{s,-h}f\,dx,$

  * (iii)

if $u,u_{x_{s}}\in L^{t}(B_{R})$, $1\leq t<\infty$, and $0<\rho<R$, then for
every $h$, $|h|\leq R-\rho$,

$\int_{B_{\rho}}|\tau_{s,h}u(x)|^{t}\,dx\leq|h|^{t}\int_{B_{R}}|u_{x_{s}}(x)|^{t}\,dx,$

  * (iv)

if $u\in L^{t}(B_{R})$, $1<t<\infty$, and for $0<\rho<R$ there exists $K>0$
such that for every $h$, $|h|<R-\rho$,

$\sum_{s=1}^{n}\int_{B_{\rho}}|\tau_{s,h}u(x)|^{t}\,dx\leq K|h|^{t},$ (2.2)

then letting $h$ go to $0$, $Du\in L^{t}(B_{\rho})$ and
$\|u_{x_{s}}\|_{L^{t}(B_{\rho})}\leq K$ for every $s\in\\{1,\ldots,n\\}$.

We recall the following estimate for the auxiliary function

$V_{p}(\xi):=\Bigl{(}1+|\xi|^{2}\Bigr{)}^{\frac{p-2}{4}}\xi,$ (2.3)

which is a convex function since $p\geq 2$ (see the Step 2 in [33] and the
proof of [30, Lemma 8.3]).

######  Lemma 2.1.

Let $1<p<\infty$. There exists a constant $c=c(n,p)>0$ such that

$c^{-1}\Bigl{(}1+|\xi|^{2}+|\eta|^{2}\Bigr{)}^{\frac{p-2}{2}}\leq\frac{|V_{p}(\xi)-V_{p}(\eta)|^{2}}{|\xi-\eta|^{2}}\leq
c\Bigl{(}1+|\xi|^{2}+|\eta|^{2}\Bigr{)}^{\frac{p-2}{2}}$

for any $\xi$, $\eta\in\mathbb{R}^{n}$.

In the next lemma we prove that (1.3)1 implies the, possibly degenerate,
$p,q-$growth condition stated in (1.4).

######  Lemma 2.2.

Let $f=f(x,\xi)$ be convex and of class $C^{2}$ with respect to the
$\xi-$variable.

Assume (1.2) and

$a(x)\,(1+|\xi|^{2})^{\frac{p-2}{2}}|\lambda|^{2}\leq\langle
D_{\xi\xi}f(x,\xi)\lambda,\lambda\rangle\leq
b(x)\,(1+|\xi|^{2})^{\frac{q-2}{2}}|\lambda|^{2}$ (2.4)

for some exponents $2\leq p\leq q$ and nonnegative functions $a,b$. Then there
exists a constant $c$ such that

$c\,a(x)(1+|\xi|^{2})^{\frac{p-2}{2}}|\xi|^{2}\leq f(x,\xi)\leq
b(x)(1+|\xi|^{2})^{\frac{q}{2}}+f(x,0).$

###### Proof.

For $x\in\Omega$ and $s\in\mathbb{R}$, let us set $\varphi(s)=g(x,st)$, where
we recall that $g$ is linked to $f$ by (1.2). The assumptions on $f$ imply
that $\varphi\in C^{2}(\mathbb{R})$ and that $g_{t}$ is increasing in the
gradient variable $t\in[0;+\infty)$ with $g_{t}(x,0)=0$. Since

$\varphi^{\prime}(s)=g_{t}(x,st)\cdot
t\,,\;\;\;\;\;\varphi^{\prime\prime}(s)=g_{tt}(x,st)\cdot t^{2},$

Taylor expansion formula yields that there exists $\vartheta\in(0,1)$ such
that

$\varphi(1)=\varphi(0)+\varphi^{\prime}(0)+\frac{1}{2}\varphi^{\prime\prime}(\vartheta)\,.$

Recalling the definition of $\varphi$, we get

$g(x,t)=g(x,0)+g_{t}(x,0)\cdot t+\frac{1}{2}g_{tt}(x,\vartheta t)\cdot
t^{2}=g(x,0)+\frac{1}{2}g_{tt}(x,\vartheta t)\cdot t^{2}.$ (2.5)

Assumption (2.4) translates into

$a(x)\,(1+t^{2})^{\frac{p-2}{2}}\leq g_{tt}(x,t)\leq
b(x)\,(1+t^{2})^{\frac{q-2}{2}}.$ (2.6)

Inserting (2.6) in (2.5), we obtain

$\frac{a(x)}{2}\,(1+(\vartheta t)^{2})^{\frac{p-2}{2}}t^{2}+g(x,0)\leq
g(x,t)\leq g(x,0)+\frac{b(x)}{2}\,[1+(\vartheta t)^{2}]^{\frac{q-2}{2}}t^{2}.$
(2.7)

Note that, since $\vartheta<1$ and $q>2$, the right hand side of (2.7) can be
controlled with

$g(x,t)\leq g(x,0)+b(x)\,[1+(\vartheta t)^{2}]^{\frac{q-2}{2}}t^{2}\leq
g(x,0)+b(x)\,[1+t^{2}]^{\frac{q-2}{2}}t^{2}$

Moreover since $g(x,0)\geq 0$ and $p\geq 2$, the left hand side of (2.7) can
be controlled from below as follows

$g(x,t)\geq\frac{a(x)}{2}\,(1+(\vartheta
t)^{2})^{\frac{p-2}{2}}t^{2}+g(x,0)\geq\frac{a(x)}{2}\,(1+(\vartheta
t)^{2})^{\frac{p-2}{2}}t^{2}$ $\geq\frac{a(x)}{2}\,(\vartheta^{2}+(\vartheta
t)^{2})^{\frac{p-2}{2}}t^{2}=\vartheta^{p-2}\frac{a(x)}{2}(1+t^{2})^{\frac{p-2}{2}}t^{2}.$

Combining the last two estimates and recalling that $f(x,\xi)=g(x,|\xi|)$, we
conclude that there exists a constant $c=c(\vartheta)$ such that

$c(\vartheta)a(x)(1+|\xi|^{2})^{\frac{p-2}{2}}|\xi|^{2}\leq f(x,\xi)\leq
b(x)\,(1+|\xi|^{2})^{\frac{q-2}{2}}|\xi|^{2}+f(x,0)$

and the conclusion follows. ∎

We end this preliminary section with a well known property. The following
lemma has important applications in the so called hole-filling method. Its
proof can be found for example in [30, Lemma 6.1] .

######  Lemma 2.3.

Let $h:[r,R_{0}]\rightarrow\mathbb{R}$ be a nonnegative bounded function and
$0<\vartheta<1$, $A,B\geq 0$ and $\beta>0$. Assume that

$h(s)\leq\vartheta h(t)+\frac{A}{(t-s)^{\beta}}+B,$

for all $r\leq s<t\leq R_{0}$. Then

$h(r)\leq\frac{cA}{(R_{0}-r)^{\beta}}+cB,$

where $c=c(\vartheta,\beta)>0$.

##  3\. The a-priori estimate

The main result in this section is an a-priori estimate of the
$L^{\infty}$-norm of the gradient of local minimizers of the functional $F$ in
(1.1) satisfying weaker assumptions than those in Theorem 3.3. Precisely, in
this section we consider the following growth conditions

$\left\\{\begin{array}[]{l}a(x)\,(1+|\xi|^{2})^{\frac{p-2}{2}}|\lambda|^{2}\leq\langle
f_{\xi\xi}(x,\xi)\lambda,\lambda\rangle\leq
b(x)\,(1+|\xi|^{2})^{\frac{q-2}{2}}|\lambda|^{2}\\\ |f_{\xi x}(x,\xi)|\leq
k(x)(1+|\xi|^{2})^{\frac{q-1}{2}},\end{array}\right.$ (3.1)

for a.e. $x\in\Omega$ and for every $\xi,\lambda\in\mathbb{R}^{N\times n}$.
Here, $a,b,k$ are non-negative measurable functions. We do not require $a,b\in
L^{\infty}$, but, in the main result of this section, see Theorem 3.3, we
assume the following summability properties:

$\frac{1}{a}\in L_{\mathrm{loc}}^{s}(\Omega),\qquad a\in
L_{\mathrm{loc}}^{\frac{rs}{2s+r}}(\Omega),\qquad b,\,k\in
L_{\mathrm{loc}}^{r}(\Omega),\qquad\text{with $r>n$}.$ (3.2)

Moreover, we assume (1.6). We use the following weighted Sobolev type
inequality, whose proof relies on the Hölder’s inequality, see e.g. [16].

######  Lemma 3.1.

Let $p\geq 2$, $s\geq 1$ and $w\in
W_{0}^{1,\frac{ps}{s+1}}(\Omega;\mathbb{R}^{N})$ ($w\in
W_{0}^{1,p}(\Omega;\mathbb{R}^{N})$ if $s=\infty$). Let
$\lambda:\Omega\to[0,+\infty)$ be a measurable function such that
$\lambda^{-1}\in L^{s}(\Omega)$. There exists a constant $c=c(n)$ such that

$\left(\int_{\Omega}|w|^{\sigma^{*}}\,dx\right)^{\frac{p}{\sigma^{*}}}\leq
c(n)\|\lambda^{-1}\|_{L^{s}(\Omega)}\int_{\Omega}\lambda|Dw|^{p}\,dx,$ (3.3)

where $\sigma=\frac{ps}{s+1}$ ($\sigma=p$ if $s=+\infty$).

In establishing the a-priori estimate, we need to deal with quantities that
involve the $L^{2}$-norm of the second derivatives of the minimizer weighted
with the function $a(x)$. Next result tells that a $W^{2,\frac{2s}{s+1}}$
assumption on the second derivatives implies that they belong to the weighted
space $L^{2}(a(x)dx)$. More precisely, we have

######  Proposition 3.2.

Consider the functional $F$ in (1.1) satisfying the assumption (3.1) with

$a,b\in L_{\mathrm{loc}}^{1}(\Omega),\,k\in
L_{\mathnormal{loc}}^{\frac{2s}{s-1}}(\Omega),$ (3.4)

for some $s\geq 1$. If $u\in W_{\mathnormal{loc}}^{1,\infty}(\Omega)\cap
W_{\mathnormal{loc}}^{2,\frac{2s}{s+1}}(\Omega)$ is a local minimizer of $F$
then

$a(x)|D^{2}u|^{2}\in L_{\mathrm{loc}}^{1}(\Omega).$

###### Proof.

Since $u$ is a local minimizer of the functional $F$, then $u$ satisfies the
Euler’s system

$\int_{\Omega}\sum_{i,\alpha}f_{\xi_{i}^{\alpha}}(x,Du)\varphi_{x_{i}}^{\alpha}(x)\,dx=0\qquad\forall\varphi\in
C_{0}^{\infty}(\Omega;\mathbb{R}^{N}),$

and, using the second variation, for every $s=1,\ldots,n$ it holds

$\int_{\Omega}\left\\{\sum_{i,j,\alpha,\beta}f_{\xi_{i}^{\alpha}\xi_{j}^{\beta}}(x,Du)\varphi_{x_{i}}^{\alpha}u_{x_{s}x_{j}}^{\beta}+\sum_{i,\alpha}f_{\xi_{i}^{\alpha}x_{s}}(x,Du)\varphi_{x_{i}}^{\alpha}\right\\}\,dx=0\qquad\forall\varphi\in
C_{0}^{\infty}(\Omega;\mathbb{R}^{N}).$ (3.5)

Fix $s=1,\ldots,n$, a cut off function $\eta\in C_{0}^{\infty}(\Omega)$ and
define for any $\gamma\geq 0$ the function

$\varphi^{\alpha}:=\eta^{4}u_{x_{s}}^{\alpha}\quad\alpha=1,\ldots,N.$

Thanks to our assumptions on the minimizer $u$, through a standard density
argument, we can use $\varphi$ as test function in the equation (3.5), thus
getting

$\displaystyle 0=$
$\displaystyle\int_{\Omega}4\eta^{3}\sum_{i,j,s,\alpha,\beta}f_{\xi_{i}^{\alpha}\xi_{j}^{\beta}}(x,Du)\eta_{x_{i}}u_{x_{s}}^{\alpha}u_{x_{s}x_{j}}^{\beta}\,dx$
$\displaystyle+\int_{\Omega}\eta^{4}\sum_{i,j,s,\alpha,\beta}f_{\xi_{i}^{\alpha}\xi_{j}^{\beta}}(x,Du)u_{x_{s}x_{i}}^{\alpha}u_{x_{s}x_{j}}^{\beta}\,dx$
$\displaystyle+\int_{\Omega}4\eta^{3}\sum_{i,s,\alpha}f_{\xi_{i}^{\alpha}x_{s}}(x,Du)\eta_{x_{i}}u_{x_{s}}^{\alpha}\,dx$
$\displaystyle+\int_{\Omega}\eta^{4}\sum_{i,s,\alpha}f_{\xi_{i}^{\alpha}x_{s}}(x,Du)u_{x_{s}x_{i}}^{\alpha}\,dx$
$\displaystyle=:$ $\displaystyle J_{1}+J_{2}+J_{3}+J_{4}.$

By the use of Cauchy-Schwartz and Young’s inequalities and by virtue of the
second inequality of (3.1), we can estimate the integral $I_{1}$ as follows

$\displaystyle|J_{1}|$ $\displaystyle\leq$ $\displaystyle
4\int_{\Omega}\\!\\!\Bigg{\\{}\eta^{2}\sum_{i,j,s,\alpha,\beta}f_{\xi_{i}^{\alpha}\xi_{j}^{\beta}}(x,Du)\eta_{x_{i}}u_{x_{s}}^{\alpha}\eta_{x_{j}}u_{x_{s}}^{\beta}\Bigg{\\}}^{\frac{1}{2}}\\!\\!\Bigg{\\{}\eta^{4}\sum_{i,j,s,\alpha,\beta}f_{\xi_{i}^{\alpha}\xi_{j}^{\beta}}(x,Du)u_{x_{s}x_{i}}^{\alpha}u_{x_{s}x_{j}}^{\beta}\Bigg{\\}}^{\frac{1}{2}}$
$\displaystyle\leq$ $\displaystyle
C\int_{\Omega}\eta^{2}|D\eta|^{2}b(x)(1+|Du|^{2})^{\frac{q}{2}}\,dx$
$\displaystyle\qquad+\frac{1}{2}\int_{\Omega}\eta^{4}\sum_{i,j,s,\alpha,\beta}f_{\xi_{i}^{\alpha}\xi_{j}^{\beta}}(x,Du)u_{x_{s}x_{i}}^{\alpha}u_{x_{s}x_{j}}^{\beta}\,dx.$

Moreover, by the last inequality in (3.1) we obtain

$\displaystyle|J_{3}|$ $\displaystyle\leq$ $\displaystyle
4\int_{\Omega}\eta^{3}k(x)(1+|Du|^{2})^{\frac{q-1}{2}}\sum_{i,s,\alpha}|\eta_{x_{i}}u_{x_{s}}^{\alpha}|\,dx$
$\displaystyle\leq$ $\displaystyle
4\int_{\Omega}\eta^{3}|D\eta|k(x)(1+|Du|^{2})^{\frac{q}{2}}\,dx.$

and also

$\displaystyle|J_{4}|$ $\displaystyle\leq$
$\displaystyle\int_{\Omega}\eta^{4}k(x)(1+|Du|^{2})^{\frac{q-1}{2}}|D^{2}u|\,dx.$

Therefore we get

$\displaystyle\int_{\Omega}\eta^{4}\sum_{i,j,s,\alpha,\beta}f_{\xi_{i}^{\alpha}\xi_{j}^{\beta}}(x,Du)u_{x_{s}x_{i}}^{\alpha}u_{x_{s}x_{j}}^{\beta}\,dx\leq\frac{1}{2}\int_{\Omega}\eta^{4}\sum_{i,j,s,\alpha,\beta}f_{\xi_{i}^{\alpha}\xi_{j}^{\beta}}(x,Du)u_{x_{s}x_{i}}^{\alpha}u_{x_{s}x_{j}}^{\beta}\,dx$
$\displaystyle+C\int_{\Omega}\eta^{2}|D\eta|^{2}b(x)(1+|Du|^{2})^{\frac{q}{2}}\,dx+4\int_{\Omega}\eta^{3}|D\eta|k(x)(1+|Du|^{2})^{\frac{q}{2}}\,dx$
$\displaystyle+\int_{\Omega}\eta^{4}k(x)(1+|Du|^{2})^{\frac{q-1}{2}}|D^{2}u|\,dx.$

Reabsorbing the first integral in the right hand side by the left hand side we
obtain

$\displaystyle\int_{\Omega}\eta^{4}\sum_{i,j,s,\alpha,\beta}f_{\xi_{i}^{\alpha}\xi_{j}^{\beta}}(x,Du)u_{x_{s}x_{i}}^{\alpha}u_{x_{s}x_{j}}^{\beta}\,dx$
$\displaystyle\leq$ $\displaystyle
C\int_{\Omega}\eta^{2}|D\eta|^{2}b(x)(1+|Du|^{2})^{\frac{q}{2}}\,dx+4\int_{\Omega}\eta^{3}|D\eta|k(x)(1+|Du|^{2})^{\frac{q}{2}}\,dx$
$\displaystyle+\int_{\Omega}\eta^{4}k(x)(1+|Du|^{2})^{\frac{q-1}{2}}|D^{2}u|\,dx.$

By the ellipticity assumption in (3.1) and since $u\in
W^{1,\infty}_{\mathrm{loc}}(\Omega)$ we get

$\displaystyle\int_{\Omega}\eta^{4}a(x)(1+|Du|^{2})^{\frac{p-2}{2}}|D^{2}u|^{2}\,dx$
$\displaystyle\leq$ $\displaystyle
C\|1+|Du|\|_{L^{\infty}(\mathrm{supp}\eta)}^{q}\int_{\Omega}\eta^{2}|D\eta|^{2}b(x)\,dx+C\|1+|Du|\|_{L^{\infty}(\mathrm{supp}\eta)}^{q}\int_{\Omega}\eta^{3}|D\eta|k(x)$
$\displaystyle+C\|1+|Du|\|_{L^{\infty}(\mathrm{supp}\eta)}^{q-1}\int_{\Omega}\eta^{4}k(x)|D^{2}u|\,dx.$

Hölder’s inequality yields

$\displaystyle\int_{\Omega}\eta^{4}a(x)(1+|Du|^{2})^{\frac{p-2}{2}}|D^{2}u|^{2}\,dx$
(3.6) $\displaystyle\leq$ $\displaystyle
C\|1+|Du|\|_{L^{\infty}(\mathrm{supp}\eta)}^{q}\int_{\Omega}\eta^{2}|D\eta|^{2}b(x)\,dx+C\|1+|Du|\|_{L^{\infty}(\mathrm{supp}\eta)}^{q}\int_{\Omega}\eta^{3}|D\eta|k(x)$
$\displaystyle+C\|1+|Du|\|_{L^{\infty}(\mathrm{supp}\eta)}^{q-1}\left(\int_{\Omega}\eta^{4}k^{\frac{2s}{s-1}}\,dx\right)^{\frac{s-1}{2s}}\left(\int_{\Omega}\eta^{4}|D^{2}u|^{\frac{2s}{s+1}}\,dx\right)^{\frac{s+1}{2s}}.$

Since $k\in L^{\frac{2s}{s-1}}_{\mathrm{loc}}(\Omega)$ and $u\in
W^{2,\frac{2s}{s+1}}_{\mathnormal{loc}}(\Omega)$ then estimate (3.6) implies
that

$a(x)|D^{2}u|^{2}\in L^{1}_{\mathrm{loc}}(\Omega).$

∎

We are now ready to establish the main result of this section.

######  Theorem 3.3.

Consider the functional $F$ in (1.1) satisfying the assumptions (3.1), (3.2),
(1.2) and (1.6). If $u\in W_{\mathnormal{loc}}^{1,\infty}(\Omega)\cap
W_{\mathnormal{loc}}^{2,{\frac{2s}{s+1}}}(\Omega)$ is a local minimizer of $F$
then for every ball $B_{R_{0}}\Subset\Omega$

$\|Du\|_{L^{\infty}(B_{R_{0}/2})}\leq
C\mathcal{K}_{R_{0}}^{\vartheta}\left(\int_{B_{R_{0}}}\left(1+f(x,Du)\right)\,dx\right)^{\vartheta}$
(3.7)
$\int_{B_{\rho}}a(1+|Du|^{2})^{\frac{p-2}{2}}\left|D^{2}u\right|^{2}\,dx\leq
c\left({\int_{B_{R_{0}}}}(1+f(x,Du)\,dx\right)^{\vartheta},$ (3.8)

hold, for any $\rho<\frac{R}{2}$. Here

$\mathcal{K}_{R_{0}}=1+\|a^{-1}\|_{L^{s}(B_{R_{0}})}\|k+b\|_{L^{r}(B_{R_{0}})}^{2}+\|a\|_{L^{\frac{rs}{2s+r}}(B_{R_{0}})},$

$\vartheta>0$ is depending on the data, $C$ is depending also on $R_{0}$ and
$c$ is depending also on $\rho$ and $\mathcal{K}_{R_{0}}$.

###### Proof.

Since $u$ is a local minimizer of the functional $F$, then $u$ satisfies the
Euler’s system

$\int_{\Omega}\sum_{i,\alpha}f_{\xi_{i}^{\alpha}}(x,Du)\varphi_{x_{i}}^{\alpha}(x)\,dx=0\qquad\forall\varphi\in
C_{0}^{\infty}(\Omega;\mathbb{R}^{N}),$

and, using the second variation, for every $s=1,\ldots,n$ it holds

$\int_{\Omega}\left\\{\sum_{i,j,\alpha,\beta}f_{\xi_{i}^{\alpha}\xi_{j}^{\beta}}(x,Du)\varphi_{x_{i}}^{\alpha}u_{x_{s}x_{j}}^{\beta}+\sum_{i,\alpha}f_{\xi_{i}^{\alpha}x_{s}}(x,Du)\varphi_{x_{i}}^{\alpha}\right\\}\,dx=0\qquad\forall\varphi\in
C_{0}^{\infty}(\Omega;\mathbb{R}^{N}).$ (3.9)

Fix $s=1,\ldots,n$, a cut off function $\eta\in C_{0}^{\infty}(\Omega)$ and
define for any $\gamma\geq 0$ the function

$\varphi^{\alpha}:=\eta^{4}u_{x_{s}}^{\alpha}(1+|Du|^{2})^{\frac{\gamma}{2}}\quad\alpha=1,\ldots,N.$

One can easily check that

$\displaystyle\varphi^{\alpha}_{x_{i}}=$ $\displaystyle
4\eta^{3}\eta_{x_{i}}u_{x_{s}}^{\alpha}(1+|Du|^{2})^{\frac{\gamma}{2}}+\eta^{4}u_{x_{s}x_{i}}^{\alpha}(1+|Du|^{2})^{\frac{\gamma}{2}}+\gamma\eta^{4}u_{x_{s}}^{\alpha}(1+|Du|^{2})^{\frac{\gamma-2}{2}}|Du|\big{(}|Du|\big{)}_{x_{i}}.$

Thanks to our assumptions on the minimizer $u$, through a standard density
argument, we can use $\varphi$ as test function in the equation (3.9), thus
getting

$\displaystyle 0=$
$\displaystyle\int_{\Omega}4\eta^{3}(1+|Du|^{2})^{\frac{\gamma}{2}}\sum_{i,j,s,\alpha,\beta}f_{\xi_{i}^{\alpha}\xi_{j}^{\beta}}(x,Du)\eta_{x_{i}}u_{x_{s}}^{\alpha}u_{x_{s}x_{j}}^{\beta}\,dx$
$\displaystyle+\int_{\Omega}\eta^{4}(1+|Du|^{2})^{\frac{\gamma}{2}}\sum_{i,j,s,\alpha,\beta}f_{\xi_{i}^{\alpha}\xi_{j}^{\beta}}(x,Du)u_{x_{s}x_{i}}^{\alpha}u_{x_{s}x_{j}}^{\beta}\,dx$
$\displaystyle+\gamma\int_{\Omega}\eta^{4}(1+|Du|^{2})^{\frac{\gamma-2}{2}}|Du|\sum_{i,j,s,\alpha,\beta}f_{\xi_{i}^{\alpha}\xi_{j}^{\beta}}(x,Du)u_{x_{s}}^{\alpha}u_{x_{s}x_{j}}^{\beta}(|Du|)_{x_{i}}\,dx$
$\displaystyle+\int_{\Omega}4\eta^{3}(1+|Du|^{2})^{\frac{\gamma}{2}}\sum_{i,s,\alpha}f_{\xi_{i}^{\alpha}x_{s}}(x,Du)\eta_{x_{i}}u_{x_{s}}^{\alpha}\,dx$
$\displaystyle+\int_{\Omega}\eta^{4}(1+|Du|^{2})^{\frac{\gamma}{2}}\sum_{i,s,\alpha}f_{\xi_{i}^{\alpha}x_{s}}(x,Du)u_{x_{s}x_{i}}^{\alpha}\,dx$
$\displaystyle+\gamma\int_{\Omega}\eta^{4}(1+|Du|^{2})^{\frac{\gamma-2}{2}}|Du|\sum_{i,s,\alpha}f_{\xi_{i}^{\alpha}x_{s}}(x,Du)u_{x_{s}}^{\alpha}\big{(}|Du|\big{)}_{x_{i}}\,dx$
$\displaystyle=:$ $\displaystyle I_{1}+I_{2}+I_{3}+I_{4}+I_{5}+I_{6}.$ (3.10)

Estimate of $I_{1}$

By the use of Cauchy-Schwartz and Young’s inequalities and by virtue of the
second inequality in (3.1), we can estimate the integral $I_{1}$ as follows

$\displaystyle|I_{1}|$ $\displaystyle\leq$ $\displaystyle
4\int_{\Omega}\\!\\!(1+|Du|^{2})^{\frac{\gamma}{2}}\Bigg{\\{}\\!\\!\eta^{2}\sum_{i,j,s,\alpha,\beta}f_{\xi_{i}^{\alpha}\xi_{j}^{\beta}}(x,Du)\eta_{x_{i}}u_{x_{s}}^{\alpha}\eta_{x_{j}}u_{x_{s}}^{\beta}\Bigg{\\}}^{\frac{1}{2}}\\!\\!\Bigg{\\{}\eta^{4}\sum_{i,j,s,\alpha,\beta}f_{\xi_{i}^{\alpha}\xi_{j}^{\beta}}(x,Du)u_{x_{s}x_{i}}^{\alpha}u_{x_{s}x_{j}}^{\beta}\Bigg{\\}}^{\frac{1}{2}}$
(3.11) $\displaystyle\leq$ $\displaystyle
C(\varepsilon)\int_{\Omega}\eta^{2}|D\eta|^{2}b(x)(1+|Du|^{2})^{\frac{q+\gamma}{2}}\,dx$
$\displaystyle\qquad+\varepsilon\int_{\Omega}\eta^{4}(1+|Du|^{2})^{\frac{\gamma}{2}}\sum_{i,j,s,\alpha,\beta}f_{\xi_{i}^{\alpha}\xi_{j}^{\beta}}(x,Du)u_{x_{s}x_{i}}^{\alpha}u_{x_{s}x_{j}}^{\beta}\,dx,$

where $\varepsilon>0$ will be chosen later.

Estimate of $I_{3}$

Since

$f_{\xi_{i}^{\alpha}\xi_{j}^{\beta}}(x,\xi)=\left(\frac{g_{tt}(x,|\xi|)}{|\xi|^{2}}-\frac{g_{t}(x,|\xi|)}{|\xi|^{3}}\right)\xi_{i}^{\alpha}\xi_{j}^{\beta}+\frac{g_{t}(x,|\xi|)}{|\xi|}\delta_{\xi_{i}^{\alpha}\xi_{j}^{\beta}}$

and

$(|Du|)_{x_{i}}=\frac{1}{|Du|}\sum_{\alpha,s}u_{x_{i}x_{s}}^{\alpha}u_{x_{s}}^{\alpha}$
(3.12)

then

$\displaystyle\sum_{i,j,s,\alpha,\beta}f_{\xi_{i}^{\alpha}\xi_{j}^{\beta}}(x,Du)u_{x_{s}}^{\alpha}u_{x_{s}x_{j}}^{\beta}(|Du|)_{x_{i}}$
$\displaystyle=$
$\displaystyle\left(\frac{g_{tt}(x,|Du|)}{|Du|^{2}}-\frac{g_{t}(x,|Du|)}{|Du|^{3}}\right)\sum_{i,j,s,\alpha,\beta}u_{x_{s}}^{\alpha}u_{x_{s}x_{j}}^{\beta}u_{x_{i}}^{\alpha}u_{x_{j}}^{\beta}(|Du|)_{x_{i}}$
$\displaystyle+\frac{g_{t}(x,|Du|)}{|Du|}\sum_{i,s,\alpha}u_{x_{s}}^{\alpha}u_{x_{s}x_{i}}^{\alpha}(|Du|)_{x_{i}}$
$\displaystyle=$
$\displaystyle\left(\frac{g_{tt}(x,|Du|)}{|Du|}-\frac{g_{t}(x,|Du|)}{|Du|^{2}}\right)\sum_{\alpha}\left(\sum_{i}u_{x_{i}}^{\alpha}(|Du|)_{x_{i}}\right)^{2}$
$\displaystyle+g_{t}(x,|Du|)|D(|Du|)|^{2}.$ (3.13)

Thus,

$\displaystyle
I_{3}=\gamma\int_{\Omega}\eta^{4}(1+|Du|^{2})^{\frac{\gamma-2}{2}}|Du|$
$\displaystyle\Big{\\{}\Big{(}\frac{g_{tt}(x,|Du|)}{|Du|}-\frac{g_{t}(x,|Du|)}{|Du|^{2}}\Big{)}\sum_{\alpha}\left(\sum_{i}u_{x_{i}}^{\alpha}(|Du|)_{x_{i}}\right)^{2}$
$\displaystyle+g_{t}(x,|Du|)|D(|Du|)|^{2}\Big{\\}}\,dx.$

Using the Cauchy-Schwartz inequality, i.e.

$\sum_{\alpha}\left(\sum_{i}u_{x_{i}}^{\alpha}(|Du|)_{x_{i}}\right)^{2}\leq|Du|^{2}|D(|Du|)|^{2}$

and observing that

$g_{t}(x,|Du|)\geq 0,$

we conclude

$I_{3}\geq\gamma\int_{\Omega}\eta^{4}(1+|Du|^{2})^{\frac{\gamma-2}{2}}|Du|\frac{g_{tt}(x,|Du|)}{|Du|}\sum_{\alpha}\left(\sum_{i}u_{x_{i}}^{\alpha}(|Du|)_{x_{i}}\right)^{2}\,dx\geq
0.$ (3.14)

Estimate of $I_{4}$

By using the last inequality in (3.1) we obtain

$\displaystyle|I_{4}|$ $\displaystyle\leq$ $\displaystyle
4\int_{\Omega}\eta^{3}k(x)(1+|Du|^{2})^{\frac{q-1+\gamma}{2}}\sum_{i,s,\alpha}|\eta_{x_{i}}u_{x_{s}}^{\alpha}|\,dx$
(3.15) $\displaystyle\leq$ $\displaystyle
4\int_{\Omega}\eta^{3}|D\eta|k(x)(1+|Du|^{2})^{\frac{q+\gamma}{2}}\,dx.$

Estimate of $I_{5}$

Using the last inequality in (3.1) and Young’s inequality we have that

$\displaystyle|I_{5}|$ $\displaystyle\leq$
$\displaystyle\int_{\Omega}\eta^{4}k(x)(1+|Du|^{2})^{\frac{q-1+\gamma}{2}}|D^{2}u|\,dx$
(3.16) $\displaystyle\leq$
$\displaystyle\sigma\int_{\Omega}\eta^{4}a(x)(1+|Du|^{2})^{\frac{p-2+\gamma}{2}}|D^{2}u|^{2}\,dx$
$\displaystyle+C_{\sigma}\int_{\Omega}\eta^{4}\frac{k^{2}(x)}{a(x)}(1+|Du|^{2})^{\frac{2q-p+\gamma}{2}}\,dx,$

where $\sigma\in(0,1)$ will be chosen later and $a$ is the function appearing
in (3.1).

Estimate of $I_{6}$

Using the last inequality in (3.1) and (3.12), we get

$\displaystyle|I_{6}|$ $\displaystyle\leq$
$\displaystyle\gamma\int_{\Omega}\eta^{4}k(x)(1+|Du|^{2})^{\frac{q-1+\gamma}{2}}|D(|Du|)|\,dx$
(3.17) $\displaystyle\leq$
$\displaystyle\gamma\int_{\Omega}\eta^{4}(1+|Du|^{2})^{\frac{q-1+\gamma}{2}}k(x)|D^{2}u|\,dx$
$\displaystyle\leq$
$\displaystyle\sigma\int_{\Omega}\eta^{4}a(x)(1+|Du|^{2})^{\frac{p-2+\gamma}{2}}|D^{2}u|^{2}\,dx$
$\displaystyle+C_{\sigma}\gamma^{2}\int_{\Omega}\eta^{4}\frac{k^{2}(x)}{a(x)}(1+|Du|^{2})^{\frac{2q-p+\gamma}{2}}\,dx,$

where we used Young’s inequality again.
Since the equality (3.10) can be written as follows

$I_{2}+I_{3}=-I_{1}-I_{4}-I_{5}-I_{6}\,,$

by virtue of (3.14), we get

$I_{2}\leq|I_{1}|+|I_{4}|+|I_{5}|+|I_{6}|\,$

and therefore, recalling the estimates (3.11), (3.15), (3.16) and (3.17), we
obtain

$\displaystyle\int_{\Omega}\eta^{4}(1+|Du|^{2})^{\frac{\gamma}{2}}\sum_{i,j,s,\alpha,\beta}f_{\xi_{i}^{\alpha}\xi_{j}^{\beta}}(x,Du)u_{x_{s}x_{i}}^{\alpha}u_{x_{s}x_{j}}^{\beta}\,dx$
(3.18) $\displaystyle\leq$
$\displaystyle\varepsilon\int_{\Omega}\eta^{4}(1+|Du|^{2})^{\frac{\gamma}{2}}\sum_{i,j,s,\alpha,\beta}f_{\xi_{i}^{\alpha}\xi_{j}^{\beta}}(x,Du)u_{x_{s}x_{i}}^{\alpha}u_{x_{s}x_{j}}^{\beta}\,dx$
$\displaystyle+4\int_{\Omega}\eta^{3}|D\eta|k(x)(1+|Du|^{2})^{\frac{q+\gamma}{2}}\,dx$
$\displaystyle+2\sigma\int_{\Omega}\eta^{4}a(x)(1+|Du|^{2})^{\frac{p-2+\gamma}{2}}|D^{2}u|^{2}\,dx$
$\displaystyle+C_{\sigma}(1+\gamma^{2})\int_{\Omega}\eta^{4}\frac{k^{2}(x)}{a(x)}(1+|Du|^{2})^{\frac{2q-p+\gamma}{2}}\,dx$
$\displaystyle+C_{\varepsilon}\int_{\Omega}\eta^{2}|D\eta|^{2}b(x)(1+|Du|^{2})^{\frac{q+\gamma}{2}}\,dx.$

Choosing $\varepsilon=\frac{1}{2}$, we can reabsorb the first integral in the
right hand side by the left hand side thus getting

$\displaystyle\int_{\Omega}\eta^{4}(1+|Du|^{2})^{\frac{\gamma}{2}}\sum_{i,j,s,\alpha,\beta}f_{\xi_{i}^{\alpha}\xi_{j}^{\beta}}(x,Du)u_{x_{s}x_{i}}^{\alpha}u_{x_{s}x_{j}}^{\beta}\,dx$
(3.19) $\displaystyle\leq$ $\displaystyle
4\sigma\int_{\Omega}\eta^{4}a(x)(1+|Du|^{2})^{\frac{p-2+\gamma}{2}}|D^{2}u|^{2}\,dx$
$\displaystyle+C\int_{\Omega}\eta^{3}|D\eta|k(x)(1+|Du|^{2})^{\frac{q+\gamma}{2}}\,dx+C_{\sigma}(1+\gamma^{2})\int_{\Omega}\eta^{4}\frac{k^{2}(x)}{a(x)}(1+|Du|^{2})^{\frac{2q-p+\gamma}{2}}\,dx$
$\displaystyle+C\int_{\Omega}\eta^{2}|D\eta|^{2}b(x)(1+|Du|^{2})^{\frac{q+\gamma}{2}}\,dx.$

Now, using the ellipticity condition in (3.1) to estimate the left hand side
of (3.19), we get

$\displaystyle
c_{2}\int_{\Omega}\eta^{4}a(x)(1+|Du|^{2})^{\frac{p-2+\gamma}{2}}|D^{2}u|^{2}\,dx$
$\displaystyle\leq$ $\displaystyle
4\sigma\int_{\Omega}\eta^{4}a(x)(1+|Du|^{2})^{\frac{p-2+\gamma}{2}}|D^{2}u|^{2}\,dx$
$\displaystyle+C\int_{\Omega}\eta^{3}|D\eta|k(x)(1+|Du|^{2})^{\frac{q+\gamma}{2}}\,dx+C_{\sigma}(1+\gamma^{2})\int_{\Omega}\eta^{4}\frac{k^{2}(x)}{a(x)}(1+|Du|^{2})^{\frac{2q-p+\gamma}{2}}\,dx$
$\displaystyle+C\int_{\Omega}\eta^{2}|D\eta|^{2}b(x)(1+|Du|^{2})^{\frac{q+\gamma}{2}}\,dx.$

We claim that $k\in L^{\frac{2s}{s-1}}_{\mathrm{loc}}(\Omega)$. Since by
assumption (3.2), $k\in L^{r}_{\mathrm{loc}}(\Omega)$, we need to prove that
$\frac{2s}{s-1}\leq r$ that is equivalent to $\frac{2}{r}+\frac{1}{s}\leq 1$.
This holds true, because by $\eqref{gap}$ and $q\geq p$ we get

$\frac{s}{s+1}\left(1+\frac{1}{n}-\frac{1}{r}\right)>1\Leftrightarrow\frac{n}{r}+\frac{n}{s}<1$

and we conclude, because $n\geq 2$. Therefore, since by the assumption $u\in
W^{1,\infty}_{\mathrm{loc}}(\Omega)\cap
W^{2,\frac{2s}{s+1}}_{\mathrm{loc}}(\Omega)$, we can use Proposition 3.2, that
implies that the first integral in the right hand side of previous estimate is
finite. By choosing $\sigma=\frac{c_{2}}{8}$, we can reabsorb the first
integral in the right hand side by the left hand side thus getting

$\displaystyle\int_{\Omega}\eta^{4}a(x)(1+|Du|^{2})^{\frac{p-2+\gamma}{2}}|D^{2}u|^{2}\,dx$
(3.20) $\displaystyle\leq$ $\displaystyle
C\int_{\Omega}\eta^{3}|D\eta|k(x)(1+|Du|^{2})^{\frac{q+\gamma}{2}}\,dx+C(1+\gamma^{2})\int_{\Omega}\eta^{4}\frac{k^{2}(x)}{a(x)}(1+|Du|^{2})^{\frac{2q-p+\gamma}{2}}\,dx$
$\displaystyle+C\int_{\Omega}\eta^{2}|D\eta|^{2}b(x)(1+|Du|^{2})^{\frac{q+\gamma}{2}}\,dx$
$\displaystyle\leq$ $\displaystyle
C\int_{\Omega}\left(\eta^{2}|D\eta|^{2}+|D\eta|^{4}\right)a(x)(1+|Du|^{2})^{\frac{p+\gamma}{2}}\,dx$
$\displaystyle\quad+C(\gamma+1)^{2}\int_{\Omega}\eta^{4}\frac{k^{2}(x)+b^{2}(x)}{a(x)}(1+|Du|^{2})^{\frac{2q-p+\gamma}{2}}\,dx,$

where we used Young’s inequality again. Now, we note that

$\eta^{4}a(x)\left|D\big{(}(1+|Du|^{2})^{\frac{p+\gamma}{4}}\big{)}\right|^{2}\leq
c(p+\gamma)^{2}a(x)\eta^{4}(1+|Du|^{2})^{\frac{p-2+\gamma}{2}}|D^{2}u|^{2}$

and so fixing $\frac{R_{0}}{2}\leq\rho<t^{\prime}<t<R<R_{0}$ with $R_{0}$ such
that $B_{R_{0}}\Subset\Omega$, and choosing $\eta\in C_{0}^{\infty}(B_{t})$ a
cut off function between $B_{t^{\prime}}$ and $B_{t}$, by the assumption
$a^{-1}\in L^{s}_{\mathrm{loc}}(\Omega)$ we can use Sobolev type inequality of
Lemma 3.1 with $w=\eta^{2}(1+|Du|^{2})^{\frac{p+\gamma}{4}}$, $\lambda=a$ and
$p=2$, thus obtaining

$\displaystyle\left(\int_{B_{t}}\left(\eta^{2}(1+|Du|^{2})^{\frac{p+\gamma}{4}}\right)^{\left(\frac{2s}{s+1}\right)^{*}}\,dx\right)^{\frac{2}{\left(\frac{2s}{s+1}\right)^{*}}}\leq\frac{c(n)}{(t-t^{\prime})^{2}}\int_{B_{t}}a(x)(1+|Du|^{2})^{\frac{p+\gamma}{2}}\,dx$
$\displaystyle\qquad\qquad+c(n)(p+\gamma)^{2}\int_{B_{t}}\eta^{4}a\,(1+|Du|^{2})^{\frac{p-2+\gamma}{2}}|D^{2}u|^{2}\,dx,$

with a constant $c(n)$ depending only on $n$.

Using (3.20) to estimate the last integral in previous inequality, we obtain

$\displaystyle\left(\int_{B_{t}}\eta^{\frac{4ns}{n(s+1)-2s}}(1+|Du|^{2})^{\frac{(p+\gamma)ns}{2(n(s+1)-2s)}}\,dx\right)^{\frac{n(s+1)-2s}{ns}}$
$\displaystyle\leq$ $\displaystyle
c\left(\frac{(p+\gamma)^{2}}{(t-t^{\prime})^{2}}+\frac{(p+\gamma)^{2}}{(t-t^{\prime})^{4}}\right)\int_{B_{t}}a(x)(1+|Du|^{2})^{\frac{p+\gamma}{2}}\,dx$
$\displaystyle\qquad+c(p+\gamma)^{4}\int_{B_{t}}\frac{k^{2}(x)+b^{2}(x)}{a(x)}(1+|Du|^{2})^{\frac{2q-p+\gamma}{2}}\,dx$
$\displaystyle\leq$ $\displaystyle
c\left(\frac{(p+\gamma)^{2}}{(t-t^{\prime})^{2}}+\frac{(p+\gamma)^{2}}{(t-t^{\prime})^{4}}\right)\int_{B_{t}}a(x)(1+|Du|^{2})^{\frac{p+\gamma}{2}}\,dx$
$\displaystyle\quad+c(p+\gamma)^{4}\left(\int_{B_{t}}\frac{1}{a^{s}}\,dx\right)^{\frac{1}{s}}\left(\int_{B_{t}}(k^{r}+b^{r})\,dx\right)^{\frac{2}{r}}$
$\displaystyle\qquad\times\left(\int_{B_{t}}(1+|Du|^{2})^{\frac{(2q-p+\gamma)rs}{2(rs-2s-r)}}dx\right)^{\frac{rs-2s-r}{rs}},$

where we used assumptions (3.2) and Hölder’s inequality with exponents $s$,
$\frac{r}{2}$ and $\frac{rs}{rs-2s-r}$.

Using the properties of $\eta$ we obtain

$\displaystyle\left(\int_{B_{t^{\prime}}}(1+|Du|^{2})^{\frac{(p+\gamma)ns}{2(n(s+1)-2s)}}\,dx\right)^{\frac{n(s+1)-2s}{ns}}$
$\displaystyle\leq$ $\displaystyle
c(p+\gamma)^{4}\|a^{-1}\|_{L^{s}(B_{R_{0}})}\|k+b\|_{L^{r}(B_{R_{0}})}^{2}\left(\int_{B_{t}}(1+|Du|^{2})^{\frac{(2q-p+\gamma)rs}{2(rs-2s-r)}}dx\right)^{\frac{rs-2s-r}{rs}}$
$\displaystyle\qquad+c\left(\frac{(p+\gamma)^{2}}{(t-t^{\prime})^{2}}+\frac{(p+\gamma)^{2}}{(t-t^{\prime})^{4}}\right)\int_{B_{t}}a(x)(1+|Du|^{2})^{\frac{p+\gamma}{2}}\,dx$
$\displaystyle\leq$ $\displaystyle
c(p+\gamma)^{4}\|a^{-1}\|_{L^{s}(B_{R_{0}})}\|k+b\|_{L^{r}(B_{R_{0}})}^{2}\left(\int_{B_{t}}(1+|Du|^{2})^{\frac{(2q-p+\gamma)rs}{2(rs-2s-r)}}dx\right)^{\frac{rs-2s-r}{rs}}$
$\displaystyle\qquad+c\left(\frac{(p+\gamma)^{2}}{(t-t^{\prime})^{2}}+\frac{(p+\gamma)^{2}}{(t-t^{\prime})^{4}}\right)\|a\|_{L^{\frac{rs}{2s+r}}(B_{R_{0}})}\left(\int_{B_{t}}(1+|Du|^{2})^{\frac{(p+\gamma)rs}{2(rs-2s-r)}}\,dx\right)^{\frac{rs-2s-r}{rs}},$

where we used the assumption $a\in
L_{\mathrm{loc}}^{\frac{rs}{2s+r}}(\Omega)$. Setting

$\mathcal{K}_{R_{0}}=1+\|a^{-1}\|_{L^{s}(B_{R_{0}})}\|k+b\|_{L^{r}(B_{R_{0}})}^{2}+\|a\|_{L^{\frac{rs}{2s+r}}}(B_{R_{0}})$
(3.21)

and assuming without loss of generality that $t-t^{\prime}<1$, we can write
the previous estimate as follows

$\displaystyle\left(\int_{B_{t^{\prime}}}(1+|Du|^{2})^{\frac{(p+\gamma)ns}{2(n(s+1)-2s)}}\,dx\right)^{\frac{n(s+1)-2s}{ns}}$
$\displaystyle\leq
c(p+\gamma)^{4}\mathcal{K}_{R_{0}}\left(\int_{B_{t}}(1+|Du|^{2})^{\frac{(2q-p+\gamma)rs}{2(rs-2s-r)}}dx\right)^{\frac{rs-2s-r}{rs}}$
$\displaystyle\qquad+c(p+\gamma)^{2}\frac{\mathcal{K}_{R_{0}}}{(t-t^{\prime})^{4}}\left(\int_{B_{t}}(1+|Du|^{2})^{\frac{(p+\gamma)rs}{2(rs-2s-r)}}\,dx\right)^{\frac{rs-2s-r}{rs}},$

and, using the a-priori assumption $u\in W_{\mathrm{loc}}^{1,\infty}(\Omega)$,
we get

$\displaystyle\left(\int_{B_{t^{\prime}}}(1+|Du|^{2})^{\frac{(p+\gamma)ns}{2(n(s+1)-2s)}}\,dx\right)^{\frac{n(s+1)-2s}{ns}}$
$\displaystyle\leq
c(p+\gamma)^{4}\mathcal{K}_{R_{0}}\left(\|Du\|_{L^{\infty}(B_{R})}^{2(q-p)}+\frac{1}{(t-t^{\prime})^{4}}\right)\left(\int_{B_{t}}(1+|Du|^{2})^{\frac{(p+\gamma)rs}{2(rs-2s-r)}}dx\right)^{\frac{rs-2s-r}{rs}}.$
(3.22)

Setting now

$m=\frac{rs}{rs-2s-r}$

and noting that

$\frac{ns}{n(s+1)-2s}=\frac{1}{2}\left(\frac{2s}{s+1}\right)^{*}=:\frac{2_{s}^{*}}{2}$

we can write (3.22) as follows

$\displaystyle\left(\int_{B_{t^{\prime}}}\left((1+|Du|^{2})^{\frac{(p+\gamma)m}{2}}\right)^{\frac{2_{s}^{*}}{2m}}\,dx\right)^{\frac{2m}{2_{s}^{*}}}$
(3.23) $\displaystyle\leq$ $\displaystyle
c(p+\gamma)^{4m}\mathcal{K}^{m}_{R_{0}}\frac{\|Du\|^{2(q-p)m}_{L^{\infty}(B_{R})}}{(t-t^{\prime})^{4m}}\int_{B_{t}}(1+|Du|^{2})^{\frac{(p+\gamma)m}{2}}dx,$

where, without loss of generality, we supposed
$\|Du\|^{2(q-p)m}_{L^{\infty}(B_{R})}\geq 1$. Define now the decreasing
sequence of radii by setting

$\rho_{i}=\rho+\frac{R-\rho}{2^{i}}$

and the increasing sequence of exponents

$p_{0}=pm\qquad\qquad{p_{i}}={p_{i-1}}\left(\frac{2_{s}^{*}}{2m}\right)=p_{0}\left(\frac{2_{s}^{*}}{2m}\right)^{i}$

As we will prove (see (3) below) the right hand side of (3.23) is finite for
$\gamma=0$ . Then for every $\rho<\rho_{i+1}<\rho_{i}<R$, we may iterate it on
the concentric balls $B_{\rho_{i}}$ with exponents $p_{i}$, thus obtaining

$\displaystyle\left(\int_{B_{\rho_{i+1}}}(1+|Du|^{2})^{\frac{p_{i+1}}{2}}\,dx\right)^{\frac{1}{p_{i+1}}}$
(3.24) $\displaystyle\leq$
$\displaystyle\displaystyle{\prod_{j=0}^{i}}\left(C^{m}\mathcal{K}^{m}_{R_{0}}{\frac{p_{j}^{4m}\|Du\|^{2m(q-p)}_{L^{\infty}(B_{R})}}{(\rho_{j}-\rho_{j+1})^{4m}}}\right)^{\frac{1}{p_{j}}}\left(\int_{B_{R}}(1+|Du|^{2})^{\frac{p_{0}}{2}}\,dx\right)^{\frac{1}{p_{0}}}$
$\displaystyle=$
$\displaystyle\displaystyle{\prod_{j=0}^{i}}\left(C^{m}\mathcal{K}^{m}_{R_{0}}{\frac{4^{jm}p_{j}^{4m}\|Du\|^{2(q-p)m}_{L^{\infty}(B_{R})}}{(R-\rho)^{4m}}}\right)^{\frac{1}{p_{j}}}\left(\int_{B_{R}}(1+|Du|^{2})^{\frac{p_{0}}{2}}\,dx\right)^{\frac{1}{p_{0}}}$
$\displaystyle=$
$\displaystyle\displaystyle{\prod_{j=0}^{i}}\left(4^{jm}p_{j}^{4m}\right)^{\frac{1}{p_{j}}}\displaystyle{\prod_{j=0}^{i}}\left({\frac{C^{m}\mathcal{K}^{m}_{R_{0}}\|Du\|^{2(q-p)m}_{L^{\infty}(B_{R})}}{(R-\rho)^{4m}}}\right)^{\frac{1}{p_{j}}}\left(\int_{B_{R}}(1+|Du|^{2})^{\frac{p_{0}}{2}}\,dx\right)^{\frac{1}{p_{0}}}.$

Since

$\displaystyle{\prod_{j=0}^{i}}\left(4^{jm}p_{j}^{4m}\right)^{\frac{1}{p_{j}}}=\exp\left(\sum_{j=0}^{i}\frac{1}{p_{j}}\log(4^{jm}p_{j}^{4m})\right)\leq\exp\left(\sum_{j=0}^{+\infty}\frac{1}{p_{j}}\log(4^{jm}p_{j}^{4m})\right)\leq
c(n,r)$

and

$\displaystyle\displaystyle{\prod_{j=0}^{i}}\left(\frac{C^{m}\mathcal{K}^{m}_{R_{0}}\|Du\|^{2(q-p)m}_{L^{\infty}(B_{t})}}{{(R-\rho)^{4m}}}\right)^{\frac{1}{p_{j}}}=\left(\frac{C^{m}\mathcal{K}^{m}_{R_{0}}\|Du\|^{2(q-p)m}_{L^{\infty}(B_{R})}}{{(R-\rho)^{4m}}}\right)^{\sum_{j=0}^{i}\frac{1}{p_{j}}}$
$\displaystyle\leq$
$\displaystyle\left(\frac{C^{m}\mathcal{K}^{m}_{R_{0}}\|Du\|^{2(q-p)m}_{L^{\infty}(B_{R})}}{{(R-\rho)^{4m}}}\right)^{\sum_{j=0}^{+\infty}\frac{1}{p_{j}}}=\left(\frac{C\mathcal{K}_{R_{0}}\|Du\|^{2(q-p)}_{L^{\infty}(B_{R})}}{{(R-\rho)^{4}}}\right)^{\frac{2^{*}_{s}}{p(2^{*}_{s}-2m)}}$

we can let $i\to\infty$ in (3.24) thus getting

$\displaystyle\|Du\|_{L^{\infty}(B_{\rho})}\leq
C(n,r,p)\left(\frac{\mathcal{K}_{R_{0}}}{{(R-\rho)^{4}}}\right)^{\frac{2^{*}_{s}}{p(2^{*}_{s}-2m)}}\|Du\|^{\frac{2(q-p)2^{*}_{s}}{p(2^{*}_{s}-2m)}}_{L^{\infty}(B_{R})}\left(\int_{B_{R}}(1+|Du|^{2})^{\frac{pm}{2}}\,dx\right)^{\frac{1}{pm}},$

where we used that $p_{0}=pm$. Since assumption (1.6) implies that

$\frac{2(q-p)2^{*}_{s}}{p(2^{*}_{s}-2m)}<1,$ (3.25)

we can use Young’s inequality with exponents

$\frac{p(2^{*}_{s}-2m)}{2(q-p)2^{*}_{s}}>1\qquad\text{and}\qquad\frac{p(2^{*}_{s}-2m)}{p(2^{*}_{s}-2m)-2(q-p)2^{*}_{s}}$

to deduce that

$\displaystyle\|Du\|_{L^{\infty}(B_{\rho})}\leq\frac{1}{2}\|Du\|_{L^{\infty}(B_{R})}+C(n,r,p,s)\left(\frac{\mathcal{K}_{R_{0}}}{{(R-\rho)^{4}}}\right)^{\vartheta}\left(\int_{B_{R}}(1+|Du|^{2})^{\frac{pm}{2}}\,dx\right)^{\varsigma},$
(3.26)

with $\vartheta=\vartheta(p,q,n,s)$ and $\varsigma=\varsigma(p,q,n,s)$.

We now estimate the last integral. By definition of $m$ and by the assumption
on $s$, i.e., $s>\frac{nr}{r-n}$, we get

$m<\frac{ns}{n(s+1)-2s}.$

Thus, by Hölder’s inequality,

$\int_{B_{R}}(1+|Du|^{2})^{\frac{pm}{2}}\,dx\leq
c(R_{0},n,r,s)\left(\int_{B_{R}}(1+|Du|^{2})^{\frac{pns}{2(n(s+1)-2s)}}\,dx\right)^{\frac{r(n(s+1)-2s)}{n(rs-2s-r)}}.$
(3.27)

This last integral can be estimated by using (3.22) with $\gamma=0$. Indeed,
let us re-define $t^{\prime},t$ and $\eta$ as follows: consider $R\leq
t^{\prime}<t\leq 2R-\rho\leq R_{0}$ and $\eta$ a cut off function, $\eta\equiv
1$ on $B_{t^{\prime}}$ and $\mathrm{supp\,}\eta\subset B_{t}$. By (3.22) with
$\gamma=0$,

$\displaystyle\left(\int_{B_{t^{\prime}}}(1+|Du|^{2})^{\frac{pns}{2(n(s+1)-2s)}}\,dx\right)^{\frac{n(s+1)-2s}{ns}}$
(3.28) $\displaystyle\leq$ $\displaystyle
c\left(\frac{p^{2}}{(t-t^{\prime})^{2}}+\frac{p^{2}}{(t-t^{\prime})^{4}}\right)\int_{B_{R_{0}}}a(x)(1+|Du|^{2})^{\frac{p}{2}}\,dx$
$\displaystyle\qquad+cp^{4}\mathcal{K}_{B_{R_{0}}}\left(\int_{B_{R_{0}}}(k^{r}+b^{r})\,dx\right)^{\frac{2}{r}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\times\left(\int_{B_{t}}(1+|Du|^{2})^{\frac{(2q-p)rs}{2(rs-2s-r)}}dx\right)^{\frac{rs-2s-r}{rs}}.$

If we denote

$\tau:=\frac{(2q-p)rs}{rs-2s-r},\quad\tau_{1}:=\frac{nps}{n(s+1)-2s},\quad\tau_{2}:=\frac{ps}{s+1},$

by (1.6) and $s>\frac{rn}{r-n}$, we get

$\frac{\tau}{\tau_{1}}<1<\frac{\tau}{\tau_{2}}.$

Therefore there exists $\theta\in(0,1)$ such that

$1=\theta\frac{\tau}{\tau_{1}}+(1-\theta)\frac{\tau}{\tau_{2}}.$

The precise value of $\theta$ is

$\theta=\frac{ns(qr-pr+p)+qrn}{rs(2q-p)}.$ (3.29)

By Hölder’s inequality with exponents $\frac{\tau_{1}}{\theta\tau}$ and
$\frac{\tau_{2}}{(1-\theta)\tau}$ we get

$\displaystyle\left(\int_{B_{t}}(1+|Du|^{2})^{\frac{(2q-p)rs}{2(rs-2s-r)}}dx\right)^{\frac{rs-2s-r}{rs}}=\left(\int_{B_{t}}(1+|Du|^{2})^{\theta\frac{\tau}{2}+(1-\theta)\frac{\tau}{2}}dx\right)^{\frac{2q-p}{\tau}}$
$\displaystyle\leq\left(\int_{B_{t}}(1+|Du|^{2})^{\frac{\tau_{1}}{2}}dx\right)^{\frac{(2q-p)\theta}{\tau_{1}}}\left(\int_{B_{t}}(1+|Du|^{2})^{\frac{\tau_{2}}{2}}dx\right)^{\frac{(2q-p)(1-\theta)}{\tau_{2}}}.$

Hence, we can use the inequality above to estimate the last integral of (3.28)
to deduce that

$\displaystyle\left(\int_{B_{t^{\prime}}}(1+|Du|^{2})^{\frac{pns}{2(n(s+1)-2s)}}\,dx\right)^{\frac{n(s+1)-2s}{ns}}$
(3.30) $\displaystyle\leq$ $\displaystyle
c\left(\frac{p^{2}}{(t-t^{\prime})^{2}}+\frac{p^{2}}{(t-t^{\prime})^{4}}\right)\int_{B_{R_{0}}}a(x)(1+|Du|^{2})^{\frac{p}{2}}\,dx$
$\displaystyle+C\mathcal{K}_{B_{R_{0}}}\left(\int_{B_{t}}(1+|Du|^{2})^{\frac{ps}{2(s+1)}}\,dx\right)^{\frac{(1-\theta)(2q-p)(s+1)}{ps}}$
$\displaystyle\qquad\times\left(\int_{B_{t}}(1+|Du|^{2})^{\frac{pns}{2(n(s+1)-2s)}}\,dx\right)^{\frac{\theta(ns+n-2s)(2q-p)}{nps}}.$

Note that, again by (1.6) and (3.29), we have

$\frac{\theta(2q-p)}{p}<1.$

We can use Young’s inequality in the last term of (3.30) with exponents
$\frac{p}{p-\theta(2q-p)}$ and $\frac{p}{\theta(2q-p)}$ to obtain that for
every $\sigma<1$

$\displaystyle\left(\int_{B_{t^{\prime}}}(1+|Du|^{2})^{\frac{pns}{2(n(s+1)-2s)}}\,dx\right)^{\frac{n(s+1)-2s}{ns}}$
$\displaystyle\leq$ $\displaystyle
C\left(\frac{p^{2}}{(t-t^{\prime})^{2}}+\frac{p^{2}}{(t-t^{\prime})^{4}}\right)\int_{B_{R_{0}}}a(x)(1+|Du|^{2})^{\frac{p}{2}}\,dx$
$\displaystyle+C_{\sigma}\mathcal{K}_{B_{R_{0}}}^{\frac{p}{p-\theta(2q-p)}}\left(\int_{B_{t}}(1+|Du|^{2})^{\frac{ps}{2(s+1)}}\,dx\right)^{\frac{(1-\theta)(2q-p)}{p-\theta(2q-p)}\frac{s+1}{s}}$
$\displaystyle\qquad+\sigma\left(\int_{B_{t}}(1+|Du|^{2})^{\frac{pns}{2(n(s+1)-2s)}}\,dx\right)^{\frac{n(s+1)-2s}{ns}}.$

By applying Lemma 2.3, and noting that $2R-\rho-R=R-\rho$, we conclude that

$\displaystyle\left(\int_{B_{R}}(1+|Du|^{2})^{\frac{pns}{2(n(s+1)-2s)}}\,dx\right)^{\frac{n(s+1)-2s}{ns}}$
$\displaystyle\leq
C\left(\frac{p^{2}}{(R-\rho)^{2}}+\frac{p^{2}}{(R-\rho)^{4}}\right)\int_{B_{R_{0}}}a(x)(1+|Du|^{2})^{\frac{p}{2}}\,dx$
$\displaystyle+C\mathcal{K}_{B_{R_{0}}}^{\frac{p}{p-\theta(2q-p)}}\left(\int_{B_{R_{0}}}(1+|Du|^{2})^{\frac{ps}{2(s+1)}}\,dx\right)^{\frac{(1-\theta)(2q-p)}{p-\theta(2q-p)}\frac{s+1}{s}}.$
(3.31)

Collecting (3.27) and (3) we obtain

$\displaystyle\int_{B_{R}}(1+|Du|^{2})^{\frac{pm}{2}}\,dx$ $\displaystyle\leq$
$\displaystyle
C\left(\frac{p^{2}}{(R-\rho)^{2}}+\frac{p^{2}}{(R-\rho)^{4}}\right)\int_{B_{R_{0}}}a(x)(1+|Du|^{2})^{\frac{p}{2}}\,dx$
(3.32)
$\displaystyle+C\mathcal{K}_{B_{R_{0}}}^{\frac{p}{p-\theta(2q-p)}}\left(\int_{B_{R_{0}}}(1+|Du|^{2})^{\frac{ps}{2(s+1)}}\,dx\right)^{\frac{(1-\theta)(2q-p)}{p-\theta(2q-p)}\frac{s+1}{s}}.$

Notice that the right hand side is finite, because $u$ is a local minimizer
and (1.4) and (1.15) hold. This inequality, together with (3.26), implies

$\displaystyle\|Du\|_{L^{\infty}(B_{\rho})}\leq$
$\displaystyle\frac{1}{2}\|Du\|_{L^{\infty}(B_{R})}+C\left(\frac{\mathcal{K}_{B_{R_{0}}}}{(R-\rho)^{8}}\right)^{\theta}\left(\int_{B_{R_{0}}}a(x)(1+|Du|^{2})^{\frac{p}{2}}\,dx\right)^{\tilde{\theta}}$
$\displaystyle+C\left(\frac{\mathcal{K}_{B_{R_{0}}}}{(R-\rho)^{8}}\right)^{\theta}\left(\int_{B_{R_{0}}}(1+|Du|^{2})^{\frac{ps}{2(s+1)}}\,dx\right)^{\tilde{\varsigma}}$

with the constant $C$ depending on the data. Applying Lemma 2.3 we conclude
the proof of estimate (3.7). Now, we write the estimate (3.20) for $\gamma=0$
and for a cut off function $\eta\in C_{0}^{\infty}(B_{\frac{R}{2}})$, $\eta=1$
on $B_{\rho}$ for some $\rho<\frac{R}{2}$. This yields

$\displaystyle\int_{B_{\rho}}a(x)(1+|Du|^{2})^{\frac{p-2}{2}}|D^{2}u|^{2}\,dx$
$\displaystyle\leq$ $\displaystyle
C(R)\int_{B_{\frac{R}{2}}}a(x)(1+|Du|^{2})^{\frac{p}{2}}\,dx+C\int_{B_{\frac{R}{2}}}\frac{k^{2}(x)+b^{2}(x)}{a(x)}(1+|Du|^{2})^{\frac{2q-p}{2}}\,dx$
$\displaystyle\leq$ $\displaystyle
C(R)\int_{B_{\frac{R}{2}}}f(x,Du)\,dx+C\|1+|Du|\|_{L^{\infty}(B_{\frac{R}{2}})}^{2q-p}\int_{B_{\frac{R}{2}}}\frac{k^{2}(x)+b^{2}(x)}{a(x)}\,dx$
$\displaystyle\leq$ $\displaystyle C(R)\int_{B_{\frac{R}{2}}}f(x,Du)\,dx$
$\displaystyle\quad+C(R)\|1+|Du|\|_{L^{\infty}(B_{\frac{R}{2}})}^{2q-p}\left(\int_{B_{\frac{R}{2}}}(k^{r}(x)+b^{r}(x))\,dx\right)^{\frac{2}{r}}\left(\int_{B_{\frac{R}{2}}}\frac{1}{a^{s}(x)}\,dx\right)^{\frac{1}{s}},$

where we used Hölder’s inequality, since $\frac{1}{s}+\frac{2}{r}<1$ by
assumptions. Using (3.7) to estimate the $L^{\infty}$ norm of $|Du|$ and
recalling the definition of $\mathcal{K}_{R_{0}}$ at (3.21), we get

$\int_{B_{\rho}}a(1+|Du|^{2})^{\frac{p-2}{2}}|D^{2}u|^{2}\,dx\leq
c\left(\int_{B_{R}}1+f(x,Du)\,dx\right)^{\tilde{\varrho}},$

i.e. (3.8), with $c$ depending on $p,r,s,n,\rho,R,\mathcal{\ K}_{R_{0}}$. ∎

##  4\. An auxiliary functional: higher differentiability estimate

Consider the functional

$H(v)=\int_{\Omega}h(x,Dv)\,dx$

where $\Omega\subset\mathbb{R}^{n}$, $n\geq 2$, is a Sobolev map and
$h:\Omega\times\mathbb{R}^{N\times n}\rightarrow[0,+\infty)$ is a Carathéodory
function, convex and of class $C^{2}$ with respect to the second variable. We
assume that there exists
$\tilde{h}:\Omega\times[0,+\infty)\rightarrow[0,+\infty)$, increasing in the
last variable such that

$h(x,\xi)=\tilde{h}(x,|\xi|).$ (4.1)

Moreover assume that there exist $p,q$, $1<p<q$, and constant
$\ell,\nu,L_{1},L_{2}$ such that

$\ell(1+|\xi|^{2})^{\frac{p}{2}}\leq h(x,\xi)\leq
L_{1}\,(1+|\xi|^{2})^{\frac{q}{2}}$ (4.2)

for a.e. $x\in\Omega$ and for every $\xi\in\mathbb{R}^{N\times n}$. We assume
that $h$ is of class $C^{2}$ with respect to the $\xi-$variable, and that the
following conditions hold

$\nu\,(1+|\xi|^{2})^{\frac{p-2}{2}}|\lambda|^{2}\leq\langle
h_{\xi\xi}(x,\xi)\lambda,\lambda\rangle\leq
L_{2}\,(1+|\xi|^{2})^{\frac{q-2}{2}}|\lambda|^{2}$ (4.3)

for a.e. $x\in\Omega$ and for every $\xi,\lambda\in\mathbb{R}^{N\times n}$.
Moreover, we assume that there exists a non-negative function $k\in
L_{\mathrm{loc}}^{r}(\Omega)$ such that

$|D_{\xi x}h(x,\xi)|\leq k(x)(1+|\xi|^{2})^{\frac{q-1}{2}}$ (4.4)

for a.e. $x\in\Omega$ and for every $\xi\in\mathbb{R}^{N\times n}$.

The following is a higher differentiability result for minimizers of $H$,
that, by the result in [22], are locally Lipschitz continuos.

######  Theorem 4.1.

Let $v\in W^{1,\infty}_{\mathrm{loc}}(\Omega)$ be a local minimizer of the
functional $H$. Assume (4.1)–(4.4) for a couple of exponents $p,q$ such that

$\frac{q}{p}<1+\frac{1}{n}-\frac{1}{r}.$ (4.5)

Then $u\in W^{2,2}_{\mathrm{loc}}(\Omega)$ and the following estimate holds

$\displaystyle\int_{B_{\rho}}|DV_{p}(Du)|^{2}\,dx$ $\displaystyle\leq$
$\displaystyle\frac{c}{(R-\rho)^{2}}\|1+|Du|\|^{2q-p}_{\infty}\int_{B_{2R}}|Du|^{2}\,dx+c\|1+|Du|\|^{2q-p}_{\infty}\int_{B_{R}}k^{2}(x)\,dx$
$\displaystyle+c\|1+|Du|\|^{q-1}_{\infty}\left(\int_{B_{R}}k^{\frac{p}{p-1}}(x)\,dx\right)^{\frac{p-1}{p}}\left(\int_{B_{2R}}|Du|^{p}\,dx\right)^{\frac{1}{p}}$

for every ball $B_{\rho}\subset B_{R}\subset B_{2R}\Subset\Omega.$

###### Proof.

Since $v$ is a local minimizer of the functional $H$, then $v$ satisfies the
Euler’s system

$\int_{\Omega}\sum_{i,\alpha}h_{\xi_{i}^{\alpha}}(x,Du)\varphi_{x_{i}}^{\alpha}(x)\,dx=0\qquad\forall\varphi\in
C_{0}^{\infty}(\Omega;\mathbb{R}^{N}).$

Let $B_{2R}\Subset\Omega$ and let $\eta\in C_{0}^{\infty}(B_{R})$ be a cut off
function between $B_{\rho}$ and $B_{R}$ for some $\rho<R$. Fixed $1\leq s\leq
n$, and denoted $e_{s}$ is the unit vector in the $x_{s}$ direction, consider
the finite differential operator $\tau_{s,h}$, see (2.1), from now on simply
denoted $\tau_{h}$. Choosing $\varphi=\tau_{-h}(\eta^{2}\tau_{h}u)$ as test
function in the Euler’s system, we get, by properties (i) and (ii) of
$\tau_{h}$,

$\int_{\Omega}\sum_{i,\alpha}\tau_{h}(h_{\xi_{i}^{\alpha}}(x,Du))D_{x_{i}}(\eta^{2}\tau_{h}u^{\alpha})\,dx=0$

and so

$\int_{\Omega}\sum_{i,\alpha}\tau_{h}(h_{\xi_{i}^{\alpha}}(x,Du))(2\eta\eta_{x_{i}}\tau_{h}u^{\alpha}+\eta^{2}\tau_{h}u_{x_{i}}^{\alpha})\,dx=0.$

Exploiting the definition of $\tau_{h}$, we get

$\int_{\Omega}\big{(}h_{\xi_{i}^{\alpha}}(x+he_{s},Du(x+he_{s}))-h_{\xi_{i}^{\alpha}}(x,Du(x))\big{)}(2\eta\eta_{x_{i}}\tau_{h}u^{\alpha}+\eta^{2}\tau_{h}u_{x_{i}}^{\alpha})\,dx=0$

i.e.

$\displaystyle\int_{\Omega}\eta^{2}[h_{\xi_{i}^{\alpha}}(x+he_{s},Du(x+he_{s}))-h_{\xi_{i}^{\alpha}}(x+he_{s},Du(x))]\tau_{h}u_{x_{i}}^{\alpha}\,dx$
$\displaystyle=$
$\displaystyle\int_{\Omega}\eta^{2}[h_{\xi_{i}^{\alpha}}(x,Du(x))-h_{\xi_{i}^{\alpha}}(x+he_{s},Du(x))]\tau_{h}u_{x_{i}}^{\alpha}\,dx$
$\displaystyle-2\int_{\Omega}\eta[h_{\xi_{i}^{\alpha}}(x+he_{s},Du(x+he_{s}))-h_{\xi_{i}^{\alpha}}(x+he_{s},Du(x))]\eta_{x_{i}}\tau_{h}u^{\alpha}$
$\displaystyle+2\int_{\Omega}\eta[h_{\xi_{i}^{\alpha}}(x,Du(x))-h_{\xi_{i}^{\alpha}}(x+he_{s},Du(x))]\eta_{x_{i}}\tau_{h}u^{\alpha}.$

We can write previous equality as follows

$\displaystyle I_{0}=:$
$\displaystyle\int_{\Omega}\eta^{2}\int_{0}^{1}\sum_{i,j,\alpha,\beta}h_{\xi_{i}^{\alpha}\xi_{j}^{\beta}}(x+he_{s},Du(x)+\sigma\tau_{h}Du(x))d\sigma\tau_{h}u_{x_{i}}^{\alpha}\tau_{h}u_{x_{j}}^{\beta}\,dx$
$\displaystyle=$ $\displaystyle
h\int_{\Omega}\eta^{2}\int_{0}^{1}\sum_{i,\alpha}h_{x_{s}\xi_{i}^{\alpha}}(x+\sigma
he_{s},Du(x))d\sigma\tau_{h}u_{x_{i}}^{\alpha}\,dx$
$\displaystyle-2\int_{\Omega}\eta\int_{0}^{1}\sum_{i,j,\alpha,\beta}h_{\xi_{i}^{\alpha}\xi_{j}^{\beta}}(x+he_{s},Du(x)+\sigma\tau_{h}Du(x))d\sigma\eta_{x_{i}}\tau_{h}u^{\alpha}\tau_{h}u_{x_{j}}^{\beta}$
$\displaystyle+2h\int_{\Omega}\eta\int_{0}^{1}\sum_{i,\alpha}h_{x_{s}\xi_{i}^{\alpha}}(x+\sigma
he_{s},Du(x))d\sigma\eta_{x_{i}}\tau_{h}u^{\alpha}$ $\displaystyle=:$
$\displaystyle I_{1}+I_{2}+I_{3},$

that implies

$I_{0}\leq|I_{1}|+|I_{2}|+|I_{3}|$ (4.6)

The ellipticity assumption (4.3) yields

$I_{0}\geq\nu\int_{\Omega}\eta^{2}(1+|Du(x)|^{2}+|Du(x+he_{s})|^{2})^{\frac{p-2}{2}}|\tau_{h}Du|^{2}\,dx$
(4.7)

By assumption (4.4) we get

$\displaystyle|I_{1}|$ $\displaystyle\leq$
$\displaystyle|h|\int_{\Omega}\eta^{2}\int_{0}^{1}k(x+\sigma
he_{s})d\sigma(1+|Du(x)|^{2})^{\frac{q-1}{2}}|\tau_{h}Du|\,dx$ (4.8)
$\displaystyle\leq$
$\displaystyle|h|\|1+|Du|\|^{\frac{2q-p}{2}}_{\infty}\int_{\Omega}\eta^{2}\int_{0}^{1}k(x+\sigma
he_{s})d\sigma(1+|Du(x)|^{2})^{\frac{p-2}{4}}|\tau_{h}Du|\,dx$
$\displaystyle\leq$
$\displaystyle\frac{\nu}{4}\int_{\Omega}\eta^{2}(1+|Du(x)|^{2})^{\frac{p-2}{2}}|\tau_{h}Du|^{2}\,dx$
$\displaystyle+c_{\nu}|h|^{2}\|1+|Du|\|^{2q-p}_{\infty}\int_{\Omega}\eta^{2}\left(\int_{0}^{1}k(x+\sigma
he_{s})d\sigma\right)^{2}\,dx,$

where in the last line we used Young’s inequality. The right inequality in
(4.3) yields

$\displaystyle|I_{2}|$ $\displaystyle\leq$ $\displaystyle
c(L_{2})\int_{\Omega}\eta|D\eta|(1+|Du(x)|^{2}+|Du(x+he_{s})|^{2})^{\frac{q-2}{2}}|\tau_{h}Du||\tau_{h}u|\,dx$
(4.9) $\displaystyle\leq$ $\displaystyle
c(L_{2})\|1+|Du|\|^{\frac{2q-p}{2}}_{\infty}\int_{\Omega}\eta|D\eta|(1+|Du(x)|^{2}+|Du(x+he_{s})|^{2})^{\frac{p-2}{4}}|\tau_{h}Du||\tau_{h}u|\,dx$
$\displaystyle\leq$
$\displaystyle\frac{\nu}{4}\int_{\Omega}\eta^{2}(1+|Du(x)|^{2}+|Du(x+he_{s})|^{2})^{\frac{p-2}{2}}|\tau_{h}Du|^{2}\,dx$
$\displaystyle+c_{\nu,L_{2}}\|1+|Du|\|^{2q-p}_{\infty}\int_{\Omega}|D\eta|^{2}|\tau_{h}u|^{2}\,dx.$

Finally, using again assumption (4.4) and Hölder’s inequality, we obtain

$\displaystyle|I_{3}|$ $\displaystyle\leq$ $\displaystyle
2|h|\int_{\Omega}\eta|D\eta|\int_{0}^{1}k(x+\sigma
he_{s})d\sigma(1+|Du(x)|^{2})^{\frac{q-1}{2}}|\tau_{h}u|\,dx$ (4.10)
$\displaystyle\leq$
$\displaystyle|h|\|1+|Du|\|^{q-1}_{\infty}\int_{\Omega}\eta|D\eta|\int_{0}^{1}k(x+\sigma
he_{s})d\sigma|\tau_{h}u|\,dx$ $\displaystyle\leq$
$\displaystyle|h|\|1+|Du|\|^{q-1}_{\infty}\left(\int_{\Omega}\eta|D\eta|\left(\int_{0}^{1}k(x+\sigma
he_{s})d\sigma\right)^{\frac{p}{p-1}}\,dx\right)^{\frac{p-1}{p}}$
$\displaystyle\qquad\times\left(\int_{\Omega}\eta|D\eta||\tau_{h}u|^{p}\,dx\right)^{\frac{1}{p}}.$

Inserting (4.7), (4.8), (4.9) and (4.10) in (4.6), we get

$\displaystyle\nu\int_{\Omega}\eta^{2}(1+|Du(x)|^{2}+|Du(x+he_{s})|^{2})^{\frac{p-2}{2}}|\tau_{h}Du|^{2}\,dx$
$\displaystyle\leq$
$\displaystyle\frac{\nu}{2}\int_{\Omega}\eta^{2}(1+|Du(x)|^{2}+|Du(x+he_{s})|^{2})^{\frac{p-2}{2}}|\tau_{h}Du|^{2}\,dx$
$\displaystyle+c_{\nu,L_{2}}\|1+|Du|\|_{\infty}^{2q-p}\int_{\Omega}|D\eta|^{2}|\tau_{h}u|^{2}\,dx$
$\displaystyle+c_{\nu}|h|^{2}\|1+|Du|\|_{\infty}^{2q-p}\int_{\Omega}\eta^{2}\left(\int_{0}^{1}k(x+\sigma
he_{s})d\sigma\right)^{2}\,dx$
$\displaystyle+|h|\|1+|Du|\|_{\infty}^{q-1}\left(\int_{\Omega}\eta|D\eta|\left(\int_{0}^{1}k(x+\sigma
he_{s})d\sigma\right)^{\frac{p}{p-1}}\,dx\right)^{\frac{p-1}{p}}$
$\displaystyle\qquad\times\left(\int_{\Omega}\eta|D\eta||\tau_{h}u|^{p}\,dx\right)^{\frac{1}{p}}.$

Reabsorbing the first integral in the right hand side by the left hand side
and recalling the properties of the cut off function $\eta$, we obtain

$\displaystyle\int_{B_{\rho}}(1+|Du(x)|^{2}+|Du(x+he_{s})|^{2})^{\frac{p-2}{2}}|\tau_{h}Du|^{2}\,dx$
$\displaystyle\leq$
$\displaystyle\frac{c}{(R-\rho)^{2}}\|1+|Du|\|_{\infty}^{2q-p}\int_{B_{R}}|\tau_{h}u|^{2}\,dx$
$\displaystyle+c|h|^{2}\|1+|Du|\|_{\infty}^{2q-p}\int_{B_{R}}\left(\int_{0}^{1}k(x+\sigma
he_{s})d\sigma\right)^{2}\,dx$
$\displaystyle+c|h|\|1+|Du|\|_{\infty}^{q-1}\left(\int_{B_{R}}\left(\int_{0}^{1}k(x+\sigma
he_{s})d\sigma\right)^{\frac{p}{p-1}}\,dx\right)^{\frac{p-1}{p}}$
$\displaystyle\qquad\times\left(\int_{B_{R}}|\tau_{h}u|^{p}\,dx\right)^{\frac{1}{p}},$

where $c=c(\nu,L_{2},n,N)$. By property (iii) of the finite difference
operator we deduce that

$\displaystyle\int_{B_{\rho}}(1+|Du(x)|^{2}+|Du(x+he_{s})|^{2})^{\frac{p-2}{2}}|\tau_{h}Du|^{2}\,dx$
(4.11) $\displaystyle\leq$
$\displaystyle\frac{c}{(R-\rho)^{2}}|h|^{2}\|1+|Du|\|^{2q-p}_{\infty}\int_{B_{2R}}|Du|^{2}\,dx$
$\displaystyle+c|h|^{2}\|1+|Du|\|^{2q-p}_{\infty}\int_{B_{R}}\left(\int_{0}^{1}k(x+\sigma
he_{s})d\sigma\right)^{2}\,dx$
$\displaystyle+c|h|^{2}\|1+|Du|\|^{q-1}_{\infty}\left(\int_{B_{R}}\left(\int_{0}^{1}k(x+\sigma
he_{s})d\sigma\right)^{\frac{p}{p-1}}\,dx\right)^{\frac{p-1}{p}}$
$\displaystyle\qquad\times\left(\int_{B_{2R}}|Du|^{p}\,dx\right)^{\frac{1}{p}}.$

Dividing (4.11) by $|h|^{2}$ and using Lemma 2.1 in the left hand side, we get

$\displaystyle\int_{B_{\rho}}\frac{|\tau_{h}V_{p}(Du)|^{2}}{|h|^{2}}\,dx$
$\displaystyle\leq$
$\displaystyle\frac{c}{(R-\rho)^{2}}\|1+|Du|\|_{\infty}^{2q-p}\int_{B_{2R}}|Du|^{2}\,dx$
$\displaystyle+c\|1+|Du|\|_{\infty}^{2q-p}\int_{B_{R}}\left(\int_{0}^{1}k(x+\sigma
he_{s})d\sigma\right)^{2}\,dx$
$\displaystyle+c\|1+|Du|\|_{\infty}^{q-1}\left(\int_{B_{R}}\left(\int_{0}^{1}k(x+\sigma
he_{s})d\sigma\right)^{\frac{p}{p-1}}\,dx\right)^{\frac{p-1}{p}}$
$\displaystyle\qquad\times\left(\int_{B_{2R}}|Du|^{p}\,dx\right)^{\frac{1}{p}}.$

Letting $h$ go to $0$, by property (iv) of the finite difference operator, we
conclude

$\displaystyle\int_{B_{\rho}}|DV_{p}(Du)|^{2}\,dx$ (4.12) $\displaystyle\leq$
$\displaystyle\frac{c}{(R-\rho)^{2}}\|1+|Du|\|^{2q-p}_{\infty}\int_{B_{2R}}|Du|^{2}\,dx+c\|1+|Du|\|^{2q-p}_{\infty}\int_{B_{R}}k^{2}(x)\,dx$
$\displaystyle+c\|1+|Du|\|^{q-1}_{\infty}\left(\int_{B_{R}}k^{\frac{p}{p-1}}(x)\,dx\right)^{\frac{p-1}{p}}\left(\int_{B_{2R}}|Du|^{p}\,dx\right)^{\frac{1}{p}}.$

Since $k\in L^{r}$, with $r>n\geq 2\geq\frac{p}{p-1}$ , estimate (4.12)
implies the conclusion. ∎

##  5\. Proof of Theorem 1.1

Using the previous results and an approximation procedure, we can prove of our
main result.

###### Proof of Theorem 1.1.

For $f(x,\xi)$ satisfying the assumptions (1.3)–(1.6), let us introduce the
sequence

$f_{h}(x,\xi)=f(x,\xi)+\frac{1}{h}(1+|\xi|^{2})^{\frac{ps}{2(s+1)}}.$ (5.1)

Note that $f_{h}(x,\xi)$ satisfies the following set of conditions

$\frac{1}{h}(1+|\xi|^{2})^{\frac{ps}{2(s+1)}}\leq
f_{h}(x,\xi)\leq(1+L)\,(1+|\xi|^{2})^{\frac{q}{2}},$ (5.2)
$\frac{c_{1}}{h}\,(1+|\xi|^{2})^{\frac{ps}{2(s+1)}-2}|\lambda|^{2}\leq\langle
D_{\xi\xi}f_{h}(x,\xi)\lambda,\lambda\rangle,$ (5.3)
$|D_{\xi\xi}f_{h}(x,\xi)|\leq c_{2}(1+L)\,(1+|\xi|^{2})^{\frac{q-2}{2}},$
(5.4) $|D_{\xi x}f_{h}(x,\xi)|\leq k(x)(1+|\xi|^{2})^{\frac{q-1}{2}}$ (5.5)

for some constants $c_{1},c_{2}>0$, for a.e. $x\in\Omega$ and for every
$\xi\in\mathbb{R}^{N\times n}$.

Now, fix a ball $B_{R}\Subset\Omega$, and let $v_{h}\in
W^{1,\frac{ps}{s+1}}(B_{R},\mathbb{R}^{N})$ be the unique solution to the
problem

$\min\left\\{\int_{B_{R}}f_{h}(x,Dv)\,dx:\,\,v_{h}\in
u+W^{1,\frac{ps}{s+1}}_{0}(B_{R},\mathbb{R}^{N})\right\\}.$ (5.6)

Since $f_{h}(x,\xi)$ satisfies (5.2), (5.3), (5.4) (5.5) with $k\in L^{r}$,
$r>n$, and (1.6) holds, then by the result in [22] we have that $v_{h}\in
W^{1,\infty}_{\mathrm{loc}}(B_{R})$ and by Theorem 4.1, used with $p$ replaced
by $p\frac{s}{s+1}$, we also have $v_{h}\in W^{2,2}_{\mathrm{loc}}(B_{R})$.

Since $f_{h}(x,\xi)$ satisfies (5.2), by the minimality of $v_{h}$ we get

$\displaystyle\int_{B_{R}}|Dv_{h}|^{\frac{ps}{s+1}}\,dx\leq
c_{s}\int_{B_{R}}a(x)|Dv_{h}|^{p}+c_{s}\int_{B_{R}}\frac{1}{a^{s}(x)}\,dx$
$\displaystyle\leq$ $\displaystyle
c_{s}\int_{B_{R}}f_{h}(x,Dv_{h})\,dx+c_{s}\int_{B_{R}}\frac{1}{a^{s}(x)}\,dx$
$\displaystyle\leq$ $\displaystyle
c_{s}\int_{B_{R}}f_{h}(x,Du)\,dx+c_{s}\int_{B_{R}}\frac{1}{a^{s}(x)}\,dx$
$\displaystyle=$ $\displaystyle
c_{s}\int_{B_{R}}f(x,Du)\,dx+\frac{c_{s}}{h}\int_{B_{R}}(1+|Du|)^{\frac{ps}{s+1}}\,dx+c_{s}\int_{B_{R}}\frac{1}{a^{s}(x)}\,dx$
$\displaystyle\leq$ $\displaystyle
c_{s}\int_{B_{R}}f(x,Du)\,dx+c_{s}\int_{B_{R}}(1+|Du|)^{\frac{ps}{s+1}}\,dx+c_{s}\int_{B_{R}}\frac{1}{a^{s}(x)}\,dx.$

Therefore the sequence $v_{h}$ is bounded in $W^{1,\frac{ps}{s+1}}(B_{R})$, so
there exists $v\in u+W_{0}^{1,\frac{ps}{s+1}}(B_{R})$ such that, up to
subsequences,

$v_{h}\rightharpoonup v\qquad\text{weakly in}\,\,W^{\frac{ps}{s+1}}(B_{R}).$
(5.7)

On the other hand, we can apply Theorem 3.3 to $f_{h}(x,\xi)$ since the
assumptions are satisfied, with $b$ replaced by $1+L$. Thus, we are legitimate
to apply estimates (3.7) and (3.8) to the solutions $v_{h}$ to obtain

$\displaystyle\|Dv_{h}\|_{L^{\infty}(B_{\rho})}\leq
C\mathcal{K}_{R}^{\tilde{\vartheta}}\left(\int_{B_{R}}(1+f_{h}(x,Dv_{h}))\,dx\right)^{\tilde{\varsigma}}$
(5.8) $\displaystyle\leq$ $\displaystyle
C\mathcal{K}_{R}^{\tilde{\vartheta}}\left(\int_{B_{R}}(1+f_{h}(x,Du))\,dx\right)^{\tilde{\varsigma}}$
$\displaystyle=$ $\displaystyle
C\mathcal{K}_{R}^{\tilde{\vartheta}}\left(\int_{B_{R}}(1+f(x,Du)+\frac{1}{h}(1+|Du|^{2})^{\frac{ps}{2(s+1)}})\,dx\right)^{\tilde{\varsigma}}$
$\displaystyle\leq$ $\displaystyle
C\mathcal{K}_{R}^{\tilde{\vartheta}}\left(\int_{B_{R}}(1+f(x,Du)+(1+|Du|^{2})^{\frac{ps}{2(s+1)}})\,dx\right)^{\tilde{\varsigma}},$

with $C,\tilde{\vartheta},\tilde{\varsigma}$ independent of $h$ and
$0<\rho<R$. Therefore, up to subsequences,

$v_{h}\rightharpoonup v\qquad\text{weakly* in}\,\,W^{1,\infty}(B_{\rho}).$
(5.9)

Our next aim is to show that $v=u$. The lower semicontinuity of
$u\mapsto\int_{B_{R}}f(x,Du)$ and the minimality of $v_{h}$ imply

$\displaystyle\int_{B_{R}}f(x,Dv)\,dx\leq\liminf_{h}\int_{B_{R}}f(x,Dv_{h})\,dx\leq\liminf_{h}\int_{B_{R}}f_{h}(x,Dv_{h})\,dx$
$\displaystyle\leq$ $\displaystyle\liminf_{h}\int_{B_{R}}f_{h}(x,Du)\,dx$
$\displaystyle=$
$\displaystyle\liminf_{h}\int_{B_{R}}(f(x,Du)+\frac{1}{h}(1+|Du|^{2})^{\frac{ps}{2(s+1)}})\,dx$
$\displaystyle=$ $\displaystyle\int_{B_{R}}f(x,Du)\,dx.$

The strict convexity of $f$ yields that $u=v$. Therefore passing to the limit
as $h\rightarrow\infty$ in (5.8) we get

$\|Du\|_{L^{\infty}(B_{\rho})}\leq
C\mathcal{K}_{R}^{\tilde{\vartheta}}\left(\int_{B_{R}}(1+f(x,Du)+(1+|Du|^{2})^{\frac{ps}{2(s+1)}})\,dx\right)^{\tilde{\varsigma}},$

i.e. (1.7). Moreover, we are legitimate to apply estimate (3.8) to each
$v_{h}$ thus getting

$\displaystyle\int_{B_{\rho}}a(x)(1+|Dv_{h}|^{2})^{\frac{p-2}{2}}|D^{2}v_{h}|^{2}\,dx$
$\displaystyle\leq$ $\displaystyle
c\left(\int_{B_{R}}(1+f_{h}(x,Dv_{h}))\,dx\right)^{\tilde{\varrho}}$
$\displaystyle\leq$ $\displaystyle
c\left(\int_{B_{R}}(1+f_{h}(x,Du))\,dx\right)^{\tilde{\varrho}}$
$\displaystyle=$ $\displaystyle
c\left(\int_{B_{R}}\left(1+f(x,Du)+\frac{1}{h}(1+|Du|^{2})^{\frac{ps}{2(s+1)}}\right)\,dx\right)^{\tilde{\varrho}},$

where we used the minimality of $v_{h}$ and the definition of $f_{h}(x,\xi)$.
Since $v_{h}\rightarrow u$ a.e. up to a subsequence, we conclude that

$\displaystyle\int_{B_{\rho}}a(x)(1+|Du|^{2})^{\frac{p-2}{2}}|D^{2}u|^{2}\,dx\leq\liminf_{h}\int_{B_{\rho}}a(x)(1+|Dv_{h}|^{2})^{\frac{p-2}{2}}|D^{2}v_{h}|^{2}\,dx$
$\displaystyle\quad\leq
c\liminf_{h}\left(\int_{B_{R}}\left(1+f(x,Du)+\frac{1}{h}(1+|Du|^{2})^{\frac{ps}{2(s+1)}}\right)\,dx\right)^{\tilde{\varrho}}$
$\displaystyle\quad=\left(\int_{B_{R}}\left(1+f(x,Du)\right)\,dx\right)^{\tilde{\varrho}},$

i.e. (1.8). ∎

## References

  * [1] A.K. Balci, L. Diening, R. Giova, A. Passarelli di Napoli: Elliptic equations with degenerate weights, (2020), arXiv:2003.10380v1.
  * [2] P. Baroni, M. Colombo, G. Mingione: Regularity for general functionals with double phase,_Calc. Var. Partial Differential_ , 57 (2018), no. 2, Paper No. 62, 48 pp.
  * [3] P. Bella, M. Schäffner: Local boundedness and Harnack inequality for solutions of linear non-uniformly elliptic equations, _Comm. Pure App. Math._ (2019) to appear.
  * [4] M. Belloni, G. Buttazzo: A survey of old and recent results about the gap phenomenon in the calculus of variations,_R. Lucchetti, J. Revalski (Eds.), Recent developements in well-posed variational problems, Mathematical Applications,_ $\mathbf{331}$ (1995), 1-27.
  * [5] S. Biagi, G. Cupini, E. Mascolo: Regularity of quasi-minimizers for non-uniformly elliptic integrals, _J. Math Anal. Appl._ , 485 (2020), 123838, 20 pp.
  * [6] M. Carozza, G. Moscariello, A. Passarelli di Napoli: Higher integrability for minimizers of anisotropic functionals _Discrete Contin. Dyn. Syst. Ser. B_ , $\mathbf{11}$ (2009), 43-55.
  * [7] A. Cianchi, V. Maz’ya: Global Lipschitz regularity for a class of quasilinear elliptic equations, Comm. Partial Differential Equations, 36 (2011), 100-133.
  * [8] M. Colombo, G. Mingione: Regularity for double phase variational problems,_Arch. Rat. Mech. Anal._ , 215 (2015), 443-496.
  * [9] M. Colombo, G. Mingione: Bounded minimisers of double phase variational integrals, Arch. Rat. Mech. Anal., 218 (2015), 219-273.
  * [10] D. Cruz-Uribe, P. Di Gironimo, C. Sbordone: On the continuity of solutions to degenerate elliptic equations, _J. Differential Equations_ 250 (2011), 2671-2686.
  * [11] G. Cupini, F. Giannetti, R. Giova, A. Passarelli di Napoli: Regularity results for vectorial minimizers of a class of degenerate convex integrals, J. Differential Equations, 265 (2018), 4375-4416.
  * [12] G. Cupini, M. Guidorzi, E. Mascolo: Regularity of minimizers of vectorial integrals with $p,q$${-}$growth, Nonlinear Anal., 54 (2003), 591-616.
  * [13] G. Cupini, P. Marcellini, E. Mascolo: Local boundedness of solutions to quasilinear elliptic systems, Manuscripta Math., 137 (2012), 287-315.
  * [14] G. Cupini, P. Marcellini, E. Mascolo: Existence and regularity for elliptic equations under $p,q$${-}$growth, Adv. Differential Equations, 19 (2014), 693-724.
  * [15] G. Cupini, P. Marcellini, E. Mascolo: Regularity of minimizers under limit growth conditions, _Nonlinear Anal._ , 153 (2017), 294-310.
  * [16] G. Cupini, P. Marcellini, E. Mascolo: Nonuniformly elliptic energy integrals with $p,q$-growth, _Nonlinear Anal._ , 177 (2018), part A, 312-324.
  * [17] E. De Giorgi: Un esempio di estremali discontinue per un problema variazionale di tipo ellittico,_Boll. Un. Mat. Ital.,_ 1 (1968), 135-137.
  * [18] C. De Filippis, G. Mingione: On the regularity of non-autonomous functionals, _J. Geom. Anal._ 30 (2020), 1584-1626.
  * [19] C. De Filippis, J. Oh: Regularity for multi-phase variational problems _J. Differential Equations_ , 267 (2019), 1631-1670.
  * [20] T. Di Marco, P. Marcellini: A-priori gradient bound for elliptic systems under either slow or fast growth conditions, Calc. Var. Partial Differential Equations, 59 (2020), 26 pp.
  * [21] F. Duzaar, G. Mingione: Local Lipschitz regularity for degenerate elliptic systems, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 1361-1396.
  * [22] M. Eleuteri, P. Marcellini, E. Mascolo. Lipschitz estimates for systems with ellipticity conditions at infinity. _Ann. Mat. Pura Appl._ , 195 (2016), 1575-1603.
  * [23] M. Eleuteri, P. Marcellini, E. Mascolo: Lipschitz continuity for functionals with variable exponents _Rend. Lincei Mat. Appl.,_ 27 (2016), 61-87.
  * [24] M. Eleuteri, P. Marcellini, E. Mascolo: Regularity for scalar integrals without structure conditions, Advances in Calculus of Variations, 13 (2020), 279-300. https://doi.org/10.1515/acv-2017-0037
  * [25] A. Esposito, F. Leonetti, P.V. Petricca: Absence of Lavrentiev gap for non-autonomous functionals with $(p,q)-$growth, _Adv. Nonlinear Anal.,_ 8 (2019), 73-78.
  * [26] L. Esposito, F. Leonetti, G. Mingione: Regularity results for minimizers of irregular integrals with $(p,q)$ growth, _Forum Mathematicum_ , 14 (2002), 245-272.
  * [27] L. Esposito, F. Leonetti, G. Mingione: Sharp regularity for functionals with $(p,q)$ growth, _J. Differential Equations_ , 204 (2004), 5-55.
  * [28] E. Fabes, C. Kenig, R. Serapioni: The local regularity of solutions of degenerate elliptic equations, _Comm. Partial Diff. Eq._ , 7 (1982), 77-116.
  * [29] R. Giova, A. Passarelli di Napoli: Regularity results for a priori bounded minimizers of non-autonomous functionals with discontinuous coefficients, Adv. Calc. Var., 12 (2019), 85-110.
  * [30] E. Giusti: Direct methods in the calculus of variations. World scientific publishing Co. (2003) 50.
  * [31] T. Iwaniec, L. Migliaccio, G. Moscariello, A. Passarelli di Napoli: A priori estimates for nonlinear elliptic complexes, Adv. Differential Equations, 8 (2003), 513-546.
  * [32] T. Iwaniec, C. Sbordone: Quasiharmonic fields, _Ann. Inst. H. Poincaré Anal. Non Linéaire_ , 18 (2001), 519-572.
  * [33] P. Marcellini: Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals, _Manuscripta Math.,_ 51 (1985), no. 1-3, 1-28.
  * [34] P. Marcellini: Regularity of minimizers of integrals in the calculus of variations with non standard growth conditions,_Arch. Rational Mech. Anal._ , 105 (1989) 267-284.
  * [35] P. Marcellini: Regularity and existence of solutions of elliptic equations with $p,q$-growth conditions, _J. Differential Equations_ , 90 (1991), 1-30.
  * [36] P. Marcellini: Everywhere regularity for a class of elliptic systems without growth conditions ,_Ann. Scuola Norm. Sup. Pisa Cl. Sci._ , 23 (1996), 1-25.
  * [37] P. Marcellini: Regularity under general and $p,q-$growth conditions, Discrete Cont. Dinamical Systems Series S, 13 (2020), 2009-2031.
  * [38] P. Marcellini: A variational approach to parabolic equations under general and $p,q-$growth conditions, Nonlinear Anal., 194 (2020), 111456, 17 pp.
  * [39] P. Marcellini: Growth conditions and regularity for weak solutions to nonlinear elliptic pdes, J. Math. Anal. Appl., 2020, to appear. https://doi.org/10.1016/j.jmaa.2020.124408
  * [40] P. Marcellini, G. Papi: Nonlinear elliptic systems with general growth,_J. Differential Equations_ , 221 (2006), 412-443.
  * [41] G. Mingione: Regularity of minima: an invitation to the dark side of the calculus of variations,_Appl. Math.,_ 51 (2006), 355-426.
  * [42] C. Mooney, O. Savin : Some singular minimizers in low dimensions in the calculus of variations,_Arch. Ration. Mech. Anal._ , 22 (2016), 1-22
  * [43] J. Moser: A new proof of De Giorgi’s theorem concerning the regularity problem for elliptic differential equations, _Comm. Pure Appl. Math.,_ 13 (1960), 457-468.
  * [44] A. Passarelli di Napoli: Higher differentiability of minimizers of variational integrals with Sobolev coefficients, _Adv. Calc. Var._ , 7 (2014), 59-89.
  * [45] A. Passarelli di Napoli: Higher differentiability of solutions of elliptic systems with Sobolev coefficients: the case $p=n=2$, _Pot. Anal._ , 41 (2014), 715-735.
  * [46] A. Passarelli di Napoli: Regularity results for non-autonomous variational integrals with discontinuous coefficients, _Atti Accad. Naz. Lincei, Rend. Lincei Mat. Appl._ , 26 (2015), (4), 475-496.
  * [47] M. Pingen: Regularity results for degenerate elliptic systems,__ Ann. Inst. H. Poincaré Anal. Non Linéaire, 25 (2008), 369-380.
  * [48] V. $\check{\text{S}}$verák, X.Yan: A singular minimizer of a smooth strongly convex functional in three dimensions,_Calc. Var. Partial Differential Equations_ , 10 (2000) 213-221
  * [49] N.S. Trudinger: On the regularity of generalized solutions of linear, non-uniformly elliptic equations, _Arch. Rational Mech. Anal._ , 42 (1971), 42-50.
  * [50] N.S. Trudinger: Linear elliptic operators with measurable coefficients, _Ann. Scuola Norm. Sup. Pisa Cl. Sci._ , 27 (1973), 265-308.
  * [51] V.V. Zhikov: On Lavrentiev phenomenon, _Russian J. Math. Phys._ , 3 (1995), 249-269.