# Lipschitz Bounds and
Nonautonomous Integrals

Cristiana De Filippis & Giuseppe Mingione<EMAIL_ADDRESS>
###### Abstract

We provide a general approach to Lipschitz regularity of solutions for a large
class of vector-valued, nonautonomous variational problems exhibiting
nonuniform ellipticity. The functionals considered here range from those with
unbalanced polynomial growth conditions to those with fast, exponential type
growth. The results obtained are sharp with respect to all the data considered
and yield new, optimal regularity criteria also in the classical uniformly
elliptic case. We give a classification of different types of nonuniform
ellipticity, accordingly identifying suitable conditions to get regularity
theorems.

###### Contents

  1. 1 Introduction
  2. 2 Basic notation
  3. 3 Potentials, functions spaces, iteration lemmas
  4. 4 Assumptions and general results
  5. 5 Approximation of integrands
  6. 6 A priori estimates
  7. 7 Proofs of Theorems 4.1 and 4.3
  8. 8 Uniform ellipticity and proof of Theorem 4.2
  9. 9 Proof of Theorem 4.4
  10. 10 Proof of Theorems 1.1-1.7
  11. 11 Obstacle problems and Theorem 1.8
  12. 12 Justification of a priori regularity

##  1 Introduction

The aim of this paper is to provide a comprehensive treatment of Lipschitz
regularity of solutions for a very large class of vector-valued nonautonomous
variational problems, involving integral functionals of the type

$W^{1,1}_{{\rm loc}}(\Omega;\mathbb{R}^{N})\ni
w\mapsto\mathcal{F}(w;\Omega):=\int_{\Omega}\left[F(x,Dw)-f\cdot
w\right]\,dx\,.$ (1.1)

Here $\Omega\subset\mathbb{R}^{n}$ is an open subset with $n\geq 2$ and $N\geq
1$. In the following we shall assume the structure condition
$F(x,Dw)\equiv\tilde{F}(x,|Dw|)$, which is natural in the vectorial case,
where $\tilde{F}:\Omega\times[0,\infty)\to[0,\infty)$ is a suitably regular
function (see Section 4.1 below for the precise assumptions). The vector field
$f\colon\Omega\mapsto\mathbb{R}^{N}$ will be at least $L^{n}$-integrable

$\displaystyle f\in L^{n}(\Omega;\mathbb{R}^{N})\,.$ (1.2)

The notion of local minimizer used in this paper is quite standard in the
literature.

###### Definition 1

A map $u\in W^{1,1}_{{\rm loc}}(\Omega;\mathbb{R}^{N})$ is a local minimizer
of the functional $\mathcal{F}$ in (1.1) with $f\in
L^{n}(\Omega;\mathbb{R}^{N})$ if, for every open subset
$\tilde{\Omega}\Subset\Omega$, we have $\mathcal{F}(u;\tilde{\Omega})<\infty$
and $\mathcal{F}(u;\tilde{\Omega})\leq\mathcal{F}(w;\tilde{\Omega})$ holds for
every competitor $w\in u+W^{1,1}_{0}(\tilde{\Omega};\mathbb{R}^{N})$.

In the rest of the paper we shall abbreviate local minimizer simply by
minimizer. We just remark that, thanks to (1.2) and Sobolev embedding,
requiring $\mathcal{F}(u;\tilde{\Omega})<\infty$ for every $\tilde{\Omega}$ in
Definition 1 is the same to requiring $F(\cdot,Du)\in L^{1}_{{\rm
loc}}(\Omega)$. In this paper we deal with the following, classical

Problem. Find minimal regularity assumptions on $f$ and $x\mapsto F(x,\cdot)$,
guaranteeing local Lipschitz continuity of minima of the functional
$\mathcal{F}$ in (1.1), provided this type of regularity holds when $f\equiv
0$ and no $x$-dependence occurs, i.e., $F(x,z)\equiv F(z)$.

Eventually, Lipschitz continuity opens the way to higher order regularity. In
the problem above, a situation where no dependence on $x$ typically occurs
when considering frozen integrands of the type $z\mapsto F(x_{0},z)$, for some
$x_{0}\in\Omega$. We recall that, under suitable growth conditions, the
analysis of (1.1) usually involves the related Euler-Lagrange system

$-\,{\rm div}(\tilde{a}(x,|Du|)Du)=f\,,\quad\
\tilde{a}(x,|Du|):=\frac{\tilde{F}^{\prime}(x,|Du|)}{|Du|}\,.$ (1.3)

Here $\tilde{F}^{\prime}$ denotes the partial derivative of $\tilde{F}$ with
respect to the second variable.

Recently, Cianchi & Maz’ya CM0 ; CM1 ; CM2 ; CM3 ; CM4  (global estimates) and
Kuusi and the second author KMstein ; KMbull ; KMjems  (local estimates),
investigated the above problem in the uniformly elliptic, autonomous case,
i.e. $\tilde{a}(x,|Du|)\equiv\tilde{a}(|Du|)$ and

$\left\\{\begin{array}[]{c}\displaystyle-1<i_{\textnormal{a}}\leq\frac{\tilde{a}^{\prime}(t)t}{\tilde{a}(t)}\leq
s_{\textnormal{a}}<\infty\quad\mbox{for every $t>0$}\\\\[10.0pt]
\mbox{$\tilde{a}\colon(0,\infty)\to[0,\infty)$ is of class $C^{1}_{{\rm
loc}}(0,\infty)$}\,.\end{array}\right.$ (1.4)

A special, yet important model, is given by the $p$-Laplacean system with
coefficients

$-\,{\rm div}(\mathfrak{c}(x)|Du|^{p-2}Du)=f\,,\quad p>1\,,\quad
0<\nu\leq\mathfrak{c}(\cdot)\leq L\,.$ (1.5)

For this we have

Nonlinear Stein Theorem KMstein . Let $u\in W^{1,p}_{{\rm
loc}}(\Omega;\mathbb{R}^{N})$ be a weak solution to (1.5). If $f\in
L(n,1)(\Omega;\mathbb{R}^{N})$, and $\mathfrak{c}(\cdot)$ is Dini continuous,
then $Du$ is continuous.

In particular, $Du$ is locally bounded. We recall that $f\in
L(n,1)(\Omega;\mathbb{R}^{N})$ means that

$\|f\|_{L(n,1)(\Omega)}:=\int_{0}^{\infty}|\\{x\in\Omega\,\colon|f(x)|>\lambda\\}|^{1/n}\,d\lambda<\infty\,,$
(1.6)

and also that $L^{q}\subset L(n,1)\subset L^{n}$ for every $q>n$. Moreover,
denoting by $\omega(\cdot)$ the modulus of continuity of
$\mathfrak{c}(\cdot)$, the Dini continuity of $\mathfrak{c}(\cdot)$ amounts to
require that

$\int_{0}\omega(\varrho)\,\frac{d\varrho}{\varrho}<\infty\,.$ (1.7)

The above theorem extends to general equations KMbull  and to systems
depending on forms sil ; it also extends classical results of Uhlenbeck Uh
and Uraltseva Ur ; we again refer to Cianchi & Maz’ya CM0 ; CM1 ; CM3  for
global statements. The terminology is motivated by the fact that, for
$\mathfrak{c}(\cdot)\equiv 1$ and $p=2$, this is another classical result of
Stein St . It is optimal both with respect to condition (1.6), as shown by
Cianchi CiGA , and with respect to (1.7), as shown by Jin, Maz’ya & Van
Schaftingen mazyaex . The relevant fact here is that the conditions on $f$ and
$\mathfrak{c}(\cdot)$ implying local Lipschitz continuity are independent of
$p$. In fact, when considering more general equations, they are independent of
the vector field the divergence operator is applied to; for this, see KMbull ,
and Baroni  for conditions (1.3)-(1.4).

In the case of nonuniformly elliptic operators, the problem of deriving sharp
conditions with respect to data for Lipschitz regularity is considerably more
difficult. This has been attacked only recently by Beck and the second author
BM , but only for the case of autonomous functionals in the principal part,
i.e., $F(x,z)\equiv F(z)$ (with some abuse of terminology, in this paper we
shall refer to $F(x,z)\equiv F(z)$ as to the autonomous case, no matter
$f(\cdot)$ can still be present in (1.1), to emphasize that coefficients $x$
do not appear in the part of the integrand containing gradient terms). The
outcome of BM  is that, when $n>2$, condition (1.6) is still sufficient to
guarantee the local Lipschitz regularity of minima, thereby revealing itself
as a sort of universal property. In the case $n=2$, the alternative, slightly
stonger borderline condition $L^{2}(\log
L)^{\mathfrak{a}}(\Omega;\mathbb{R}^{N})$ with $\mathfrak{a}>2$, implies
Lipschitz continuity:

$f\in L^{2}(\log
L)^{\mathfrak{a}}(\Omega;\mathbb{R}^{N})\Longleftrightarrow\int_{\Omega}|f|^{2}\log^{\mathfrak{a}}\left(\textnormal{e}+|f|\right)\,dx<\infty\,.$
(1.8)

Therefore, in the nonuniformly, autonomous case, the condition on $f$ can be
summarized as

$|f|\in\mathfrak{X}(\Omega)=\left\\{\begin{array}[]{cc}L(n,1)(\Omega)&\mbox{
if $n>2$}\\\\[4.0pt] L^{2}(\log L)^{\mathfrak{a}}(\Omega)\,,\
\mathfrak{a}>2&\mbox{ if $n=2$}\,.\end{array}\right.$ (1.9)

In this paper we deal with the general, fully nonautonomous case (1.1). This
is by no means a technical extension as, in fact, when passing to the
nonuniformly elliptic case, the role of coefficients drastically changes and
they can no longer be treated via perturbation as in KMstein . To give a
glimpse of the situation, let us consider the so called double phase
functional

$\begin{cases}\displaystyle w\mapsto\int_{\Omega}\left[H(x,Dw)-f\cdot
w\right]\,dx\\\ \displaystyle
H(x,z):=\tilde{H}(x,|z|):=|z|^{p}/p+a(x)|z|^{q}/q\end{cases}$ (1.10)

with $1<p<q$, $0\leq a(\cdot)\in L^{\infty}(\Omega)$. This functional has been
originally introduced by Zhikov Z1 ; Z2  in the setting of Homogeneization of
strongly anisotropic materials, and the corresponding regularity theory has
been studied at length starting by BCM ; CoM0 ; CoM . The functional in (1.10)
changes its rate of ellipticity/coercivity - from $p$ to $q$ \- around the
zero set $\\{a(x)=0\\}$. As shown in sharp ; FMM , already when $f\equiv 0$,
local minima fail to be continuous if the ratio $q/p$ is too far from $1$, in
dependence on the rate of Hölder continuity $\alpha$. Specifically, the
condition

$\frac{q}{p}\leq 1+\frac{\alpha}{n}\,,\qquad a(\cdot)\in
C^{0,\alpha}(\Omega)\,,\qquad\alpha\in(0,1]$ (1.11)

is necessary sharp ; FMM  and sufficient BCM  to get gradient local
continuity, thereby linking growth conditions of the integrand with respect to
the gradient variable, to the smoothness of coefficients. In particular,
classical Schauder’s theory generally fails. This is the main theme of this
paper. Condition in (1.11) reveals a typical phenomenon occurring when
nonuniform ellipticity is directly generated by the presence of the
$x$-variable as in (1.10). In this case, it is indeed the very presence of $x$
making functionals as in (1.10) fail to meet the standard, two-sided
polynomial conditions with the same exponent, i.e.,
$H(x,z)\not\approx|z|^{p}$. We shall also deal with more drastic examples of
such an interplay, as for instance

$w\mapsto\int_{\Omega}\left[\mathfrak{c}_{1}(x)\exp(\mathfrak{c}_{2}(x)|Dw|^{p})-f\cdot
w\right]\,dx\,,\qquad p>1\,,$ (1.12)

where $0<\nu\leq\mathfrak{c}_{1}(\cdot),\mathfrak{c}_{2}(\cdot)\leq L$. Here
the dependence on $x$ becomes even more delicate as it makes the ellipticity
rate vary more drastically. Such integrands fail to satisfy the so-called
$\triangle_{2}$-condition, i.e., $\tilde{F}(x,2t)\not\lesssim\tilde{F}(x,t)$.
This reflects in a loss of related integrability conditions on minimizers as
one tries to use perturbation methods, that is, considering a specific point
$x_{0}\in\Omega$ and making small variations of $x$ around $x_{0}$. In other
words, $\exp(\mathfrak{c}_{2}(\cdot)|Dw|^{p})\in L^{1}$ does not necessarily
imply $\exp(\mathfrak{c}_{2}(x_{0})|Dw|^{p})\in L^{1}$, and vice versa, and
plain perturbation methods are again banned. Exponential type functionals are
classical in the Calculus of Variations starting by the work of Duc & Eells DE
and Marcellini M2 . In the nonautonomous version, they are treated for
instance in the setting of weak KAM-theory by Evans evans , but under special
assumptions and boundary conditions. More recent progress is in DM , for
$f\equiv 0$.

Nonuniform ellipticity is a very classical topic in partial differential
equations, and it is often motivated by geometric and physical problems.
Seminal papers on this subject are for instance DE ; LU ; Simon ; Z1 ; Z2 ; a
classical monograph is Ivanov’s ivanov . In the setting of the Calculus of
Variations there is a wide literature available, starting from the basic
papers of Uraltseva & Urdaletova UU  and Marcellini M1 ; M2 ; M3 ; M4 . More
recently, the study of the nonautonomous case has intensified; many papers
have been devoted to study specific structures as well as genereal non-
uniformly elliptic problems BS ; Bi ; BM ; BF1 ; BB2 ; BOR ; BO ; BY ;
Cellinastaicu ; DeMi1 ; DeMi ; DM ; HO ; HS . Connections to related function
spaces have been studied too diening ; HH ; RR .

The results obtained in this paper are very general and cover large classes of
different models cases simultaneously. For this, a number of technical
assumptions is necessary; see Section 4.5 below. Anyway, when applied to
single models, such assumptions reveal to be minimal and produce sharp
results. In the autonomous case $F(x,z)\equiv F(z)$, they coincide with the
sharp ones introduced in BM . For this reason, and also to ease the reading,
in this introductory part we shall present a few main corollaries of the
general theorems, in connection to some relevant instances of nonuniformly
elliptic functionals often considered in the literature. These models fall in
three different general classes, detailed in Sections 1.1-1.3 below. We refer
the reader to Section 2 for a full account of the notation used in this paper,
while more remarks on nonuniform ellipticity are in Section 4.6 below.

###  1.1 Nonuniform ellipticity at polynomial rates

We start considering functionals with unbalanced polynomial growth conditions,
of so-called $(p,q)$-type after Marcellini M1 ; M2 . The idea is to provide
general conditions on the partial map $x\mapsto F(x,\cdot)$ implying
regularity of minima and matching those suggested by counterexamples sharp ;
FMM . In this respect, we consider, for exponents $1<p\leq q$, the conditions

$\left\\{\begin{array}[]{c}\ F(x,z)=\tilde{F}(x,\lvert z\rvert)\quad\mbox{for
all}\ (x,z)\in\Omega\times\mathbb{R}^{N\times n}\\\\[3.0pt]
\nu(|z|^{2}+\mu^{2})^{p/2}\leq
F(x,z)\leq\Lambda(|z|^{2}+\mu^{2})^{q/2}+\Lambda(|z|^{2}+\mu^{2})^{p/2}\\\\[3.0pt]
(|z|^{2}+\mu^{2})|\partial_{zz}F(x,z)|\leq\Lambda(|z|^{2}+\mu^{2})^{q/2}+\Lambda(|z|^{2}+\mu^{2})^{p/2}\\\\[3.0pt]
\nu(|z|^{2}+\mu^{2})^{(p-2)/2}|\xi|^{2}\leq\partial_{zz}F(x,z)\xi\cdot\xi\,,\end{array}\right.$
(1.13)

for every choice of $z,\xi\in\mathbb{R}^{n}$ such that $|z|\not=0$. Here
$0<\nu\leq 1\leq\Lambda$ and $\mu\in[0,1]$ are fixed constants. $\tilde{F}$
satisfies (4.1)2,3 below. We moreover assume that

$t\mapsto\frac{\tilde{F}^{\prime}(x,t)}{(t^{2}+\mu^{2})^{(p-2)/2}t}\quad\mbox{is
non-decreasing}$ (1.14)

for every $x\in\Omega$; as in the rest of the paper, we denote
$\tilde{F}^{\prime}\equiv\partial_{t}\tilde{F}$. As for the crucial dependence
on $x$, we assume that

$\begin{cases}\ |\partial_{x}\tilde{F}^{\prime}(x,t)|\leq
h(x)\left[(t^{2}+\mu^{2})^{(q-2)/2}t+(t^{2}+\mu^{2})^{(p-2)/2}t\right]\\\ \
h(\cdot)\in L^{d}(\Omega),\ d>n\end{cases}$ (1.15)

holds for a.e. $x\in\Omega$ and every $t>0$. Of course,
$F,\tilde{F}^{\prime},$ are also assumed to be continuous on
$\Omega\times(0,\infty)$, while $\tilde{F}^{\prime\prime}$ and
$\partial_{x}\tilde{F}$ are Carathéodory regular. Using Sobolev regularity on
coefficients, is a natural approach, also considered elsewhere. See for
instance the paper KM  in the case of uniformly elliptic integrals. As for the
nonuniformly elliptic setting, this approach has been used in EMM ; see also
Marcellini’s survey M4  for a general overview. We also mention that, over the
last years, Sobolev coefficients have been systematically considered as a
replacement of usual Lipschitz ones to find optimal conditions in several
other fields of analysis and PDE; see for instance bcrippa ; FGS ;
crippadelellis .

###### Theorem 1.1

Let $u\in W^{1,1}_{{\rm loc}}(\Omega;\mathbb{R}^{N})$ be a minimizer of
$\mathcal{F}$ in (1.1), under assumptions (1.13)-(1.15). Assume that (1.9)
holds and

$\frac{q}{p}<1+\min\left\\{\frac{1}{n}-\frac{1}{d},\mathfrak{m}_{p}\right\\}\quad\mbox{with}\
\ \mathfrak{m}_{p}:=\left\\{\begin{array}[]{ccc}\frac{4(p-1)}{\vartheta
p(n-2)}&\mbox{if}&n\geq 3\\\\[6.0pt] \ \frac{2(p-1)}{\vartheta
p}&\mbox{if}&n=2\,,\end{array}\right.$ (1.16)

where $\vartheta=1$ if $p\geq 2$ and $\vartheta=2$ otherwise. Then $Du\in
L^{\infty}_{{\rm loc}}(\Omega;\mathbb{R}^{N\times n})$. When either $p\geq 2$
or $f\equiv 0$, (1.16) can be replaced by

$\frac{q}{p}<1+\frac{1}{n}-\frac{1}{d}\,.$ (1.17)

Theorem 1.1 actually follows from Theorem 4.1 in Section 4.5 below and, as all
the other results presented in this Introduction, comes along with explicit
local a priori estimates. In particular, for splitting structures as

$w\mapsto\int_{\Omega}\left[\mathfrak{c}(x)F(Dw)-f\cdot w\right]\,dx\,,\qquad
0<\nu\leq\mathfrak{c}(\cdot)\leq L\,,$ (1.18)

Theorem 1.1 becomes

###### Theorem 1.2

Let $u\in W^{1,1}_{{\rm loc}}(\Omega;\mathbb{R}^{N})$ be a minimizer of the
functional in (1.18), under assumptions (1.13)-(1.14) with $F(\cdot)\equiv
F(z)\equiv\tilde{F}(|z|)$, and $f$ as in (1.9). Assume that
$\mathfrak{c}(\cdot)\in W^{1,d}(\Omega)$ with $d>n$ and that (1.16) is
satisfied. Then $Du\in L^{\infty}_{{\rm loc}}(\Omega;\mathbb{R}^{N\times n})$.
When either $p\geq 2$ or $f\equiv 0$, (1.16) can be replaced by (1.17).

For double phase functionals in (1.10), condition (1.15) again amounts to
assume that $a(\cdot)\in W^{1,d}(\Omega)$, and indeed we have

###### Theorem 1.3

Let $u\in W^{1,1}_{{\rm loc}}(\Omega;\mathbb{R}^{N})$ be a minimizer of the
functional in (1.10), such that $0\leq a(\cdot)\in W^{1,d}(\Omega)$, $d>n$,
and that (1.9) holds together with

$q/p\leq 1+1/n-1/d\,,\ \mbox{if $n\geq 2$}\ \ \mbox{and, when $n=2$, also
$q/p<p$}\,.$ (1.19)

Then $Du\in L^{\infty}_{{\rm loc}}(\Omega;\mathbb{R}^{N\times n})$.

Theorem 1.3 allows to clarify in which sense assumptions (1.15) and (1.17) &
(1.19) are sharp. Indeed, Sobolev-Morrey embedding yields $a(\cdot)\in
C^{0,\alpha}$ with $\alpha=1-n/d$. This last identity makes conditions (1.11)
and (1.17) coincide. In turn, (1.11) is sharp by the counterexamples in sharp
; FMM . Therefore, (1.15) is the sharp differentiable version of (1.11), which
is stronger than (1.11), but weaker than assuming that $a(\cdot)$ is
Lipschitz. Lipschitz continuity of coefficients is typically assumed in the
literature in the nonautonomous case (see for instance M2 ; dark  and related
references).

###  1.2 Nonuniform ellipticity at fast rates and a different phenomenon

A prototype we have in mind here is given by (1.12). Looking at the case of
polynomial growth in Section 1.1, from (1.15) and (1.16) we see that the
required integrability rate of coefficients $d$ increases with the ratio
$q/p$. A naive, but seemingly natural bet, would then assert that the
exponential case needs more stringent conditions on the integrability exponent
$d$. On the contrary, the situation reverses, and any $d>n$ implies local
Lipschitz continuity:

###### Theorem 1.4

Let $u\in W^{1,1}_{{\rm loc}}(\Omega;\mathbb{R}^{N})$ be a minimizer of the
functional in (1.12), such that
$\mathfrak{c}_{1}(\cdot),\mathfrak{c}_{2}(\cdot)\in W^{1,d}(\Omega)$ with
$d>n$ and $f$ satisfies (1.9). Then $Du\in L^{\infty}_{{\rm
loc}}(\Omega;\mathbb{R}^{N\times n})$.

The same applies to more general functionals with faster growth, involving
arbitrary compositions of exponentials, and therefore even faster growth
conditions. Specifically, we fix exponent functions $\\{p_{k}(\cdot)\\}$ and
coefficients $\\{\mathfrak{c}_{k}(\cdot)\\}$, all defined on the open subset
$\Omega\subset\mathbb{R}^{n}$, such that

$\displaystyle\begin{cases}1<p_{\mathfrak{m}}\leq p_{0}(\cdot)\leq
p_{M}\,,\quad 0<p_{m}\leq p_{k}(\cdot)\leq p_{M}\,,\ \mbox{for $k\geq 1$}\\\
0<\nu_{\mathfrak{m}}\leq\mathfrak{c}_{k}(\cdot)\leq L\,,\quad
p_{k}(\cdot),\mathfrak{c}_{k}(\cdot)\in W^{1,d}(\Omega)\,,\ \ d>n\,,\
\mbox{for $k\geq 0$}\,.\end{cases}$ (1.20)

We then inductively define, for every $k\in\mathbb{N}$, the functions ${\bf
e}_{k}\colon\Omega\times[0,\infty)\to\mathbb{R}$ as

$\left\\{\begin{array}[]{ccc}{\bf
e}_{k+1}(x,t)&:=&\exp\left(\mathfrak{c}_{k+1}(x)\left[{\bf
e}_{k}(x,t)\right]^{p_{k+1}(x)}\right)\\\\[6.0pt] {\bf
e}_{0}(x,t)&:=&\exp\left(\mathfrak{c}_{0}(x)t^{p_{0}(x)}\right)\,,\end{array}\right.$
(1.21)

and consider the variational integrals

$w\mapsto\int_{\Omega}\left[{\bf e}_{k}(x,|Dw|)-f\cdot w\right]\,dx\,.$ (1.22)

Functionals as in (1.22) have been studied at length in the literature also
because they provide the best case study to test how far one can go in
relaxing the standard uniform ellipticity assumptions; see BM ; M2  and
related references. The nonautonomous case is of special interest as the
sensitivity to the $x$-dependence is magnified by taking multiple compositions
of exponentials; see comments after display (1.12). We have the following
result, which, as also Theorem 1.4, is completely new already in the case
$f\equiv 0$:

###### Theorem 1.5

Let $u\in W^{1,1}_{{\rm loc}}(\Omega;\mathbb{R}^{N})$ be a minimizer of the
functional in (1.22) for some $k\in\mathbb{N}$, under assumptions (1.20) and
such that $f$ satisfies (1.9). Then $Du\in L^{\infty}_{{\rm
loc}}(\Omega;\mathbb{R}^{N\times n})$.

In other words, this fact brings functionals as in (1.12) closer to the realm
of uniformly elliptic ones. The next step comes in fact in the subsequent
section.

###  1.3 New results in the uniformly elliptic setting

New results follow in the classical uniformly elliptic setting too. This time
the model is

$w\mapsto\int_{\Omega}\left[A(x,|Dw|)-f\cdot w\right]\,dx\,,\quad
A(x,t):=\mathfrak{c}(x)\int_{0}^{t}\tilde{a}(s)s\,ds\,,$ (1.23)

for $t\geq 0$, with (1.4) being in force and such that
$0<\nu\leq\mathfrak{c}(\cdot)\leq L$. Under such conditions, every solution to
the system in (1.3) is a minimizer of the functional in (1.23) and the second
identity in (1.3) is automatically satisfied.

###### Theorem 1.6

Let $u\in W^{1,1}_{{\rm loc}}(\Omega;\mathbb{R}^{N})$ be a minimizer of the
functional in (1.23), under assumptions (1.4). If
$|f|,|D\mathfrak{c}|\in\mathfrak{X}(\Omega)$ as defined in (1.9), then $Du\in
L^{\infty}_{{\rm loc}}(\Omega;\mathbb{R}^{N\times n})$. Moreover, there exists
a positive radius $R_{*}\equiv
R_{*}(n,N,i_{\textnormal{a}},s_{\textnormal{a}},\linebreak\mathfrak{c}(\cdot))\leq
1$ such that if $\mathcal{B}\Subset\Omega$ is a ball with
$\texttt{r}(\mathcal{B})\leq R_{*}$, then the following estimate holds for
$c\equiv c(n,N,\nu,L,i_{\textnormal{a}},s_{\textnormal{a}})$:

$\|A(\cdot,\lvert Du\rvert)\|_{L^{\infty}(\mathcal{B}/2)}\leq
c\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{\mathcal{B}}A(\cdot,\lvert
Du\rvert)\,dx+c\|f\|_{\mathfrak{X}(\mathcal{B})}^{\frac{i_{\textnormal{a}}+2}{i_{\textnormal{a}}+1}}+c\,.$
(1.24)

In other words, $f$ and $D\mathfrak{c}$ are this time required to have the
same degree of regularity. Theorem 1.6 applies to (1.5) by taking
$A(x,t)\equiv\mathfrak{c}(x)t^{p}/p$ and it is sufficient to require that
$D\mathfrak{c}\in L(n,1)(\Omega)$ for $n>2$. This is, already when $f\equiv
0$, a new regularity criterion on coefficients, which goes beyond the known
and classical one in (1.7). Indeed, $D\mathfrak{c}\in L(n,1)$ implies that
$\mathfrak{c}(\cdot)$ is continuous St , but not necessarily with a modulus of
continuity $\omega(\cdot)$ satisfying (1.7). Moreover, this criterion works
for the general cases as in (1.23), to which methods from KMstein  do not
apply under the only considered structure assumption (1.4). When considered in
the special case (1.5), it is $i_{\textnormal{a}}=p-2$ and estimate (1.24)
gives back the classical one valid for the $p$-Laplacean system in (1.5); see
for instance KMstein ; KMjems .

###  1.4 Calderón-Zygmund theory

In Theorems 1.1-1.6, we can replace (1.9) by the weaker $f\in
L^{n}(\Omega;\mathbb{R}^{N})$, getting, as a corresponding outcome, that
$Du\in L^{p}_{{\rm loc}}(\Omega;\mathbb{R}^{N\times n})$ for every $p\geq 1$;
see Theorem 4.3 below. This result is new in the nonuniformly elliptic case
and is in perfect accordance with the Nonlinear Calderón-Zygmund theory known
for the uniformly elliptic one KMbull . For instance, considering the system
in (1.5), $D\mathfrak{c}\in L^{n}$ implies that $\mathfrak{c}(\cdot)\in$ VMO,
the space of functions with vanishing mean oscillations sarason . At this
point, $Du\in L^{p}_{{\rm loc}}(\Omega)$, for every $p\geq 1$, follows from
the standard theory (see BOR ; DoK  and references). In fact, we provide the
first Calderón-Zygmund type estimates in problems with non-polynomial growth
conditions. An example is the following result, which is completely new
already in the autonomous case:

###### Theorem 1.7

Let $u\in W^{1,1}_{{\rm loc}}(\Omega;\mathbb{R}^{N})$ be a minimizer of the
functional in (1.22) for some $k\in\mathbb{N}$, under assumptions (1.20) with
$n>2$, and such that $f\in L^{n}(\Omega;\mathbb{R}^{N})$. Then ${\bf
e}_{k}(\cdot,|Dw|)\in L^{p}_{{\rm loc}}(\Omega)$ for every $p\geq 1$.

###  1.5 Obstacles

Applications follow to obstacle problems, leading to completely new and sharp
results, already in the classical, uniformly elliptic case. For instance, we
give the first results for fast growth functionals as in (1.12), and these are
new already in the case of smooth obstacles. For this we consider the
functional

$\mathcal{F}_{0}(w;\Omega):=\int_{\Omega}F(x,Dw)\,dx\;$ (1.25)

defined on $W^{1,1}(\Omega)$, where $F$ is for instance one of the integrands
considered in Theorems 1.1-1.6; here we of course consider the scalar case
$N=1$. Next we consider a measurable function $\psi\colon\Omega\to\mathbb{R}$
and the convex set $\mathcal{K}_{\psi}(\Omega):=\\{w\in W^{1,1}_{{\rm
loc}}(\Omega)\colon w(x)\geq\psi(x)\ \mbox{for a.e.}\ x\in\Omega\\}$. We then
say that a function $u\in W^{1,1}_{{\rm
loc}}(\Omega)\cap\mathcal{K}_{\psi}(\Omega)$ is a constrained local minimizer
of $\mathcal{F}_{0}$ if, for every open subset $\tilde{\Omega}\Subset\Omega$,
we have $\mathcal{F}_{0}(u;\tilde{\Omega})<\infty$ and if
$\mathcal{F}_{0}(u;\tilde{\Omega})\leq\mathcal{F}_{0}(w;\tilde{\Omega})$ holds
for every competitor $w\in u+W^{1,1}_{0}(\tilde{\Omega})$ such that
$w\in\mathcal{K}_{\psi}(\tilde{\Omega})$. We then have the following, far
reaching extension of classical theorems from choe ; CL ; fuobs ; KS ; liebe2
:

###### Theorem 1.8

Let $u\in W^{1,1}_{{\rm loc}}(\Omega)$ be a constrained local minimizer of
$\mathcal{F}_{0}$ in (1.25), where
$F\colon\Omega\times\mathbb{R}^{n}\to\mathbb{R}$ is one of the integrands from
Theorems 1.1-1.6 with $p\geq 2,i_{\textnormal{a}}\geq 0$ (whenever such
parameters are involved). If $\psi\in W^{2,1}_{{\rm loc}}(\Omega)$ and $\lvert
D^{2}\psi\rvert\in\mathfrak{X}(\Omega)$, then $Du\in L^{\infty}_{{\rm
loc}}(\Omega;\mathbb{R}^{n})$.

This last result is new already in the classical $p$-Laplacean case
$F(x,z)\equiv|z|^{p}/p$, for which it offers a criterion which is alternative
to those given in choe ; CL  – see also KL  for double phase type functionals.
In such papers Lipschitz estimates are obtained assuming that $D\psi$ is
locally Hölder continuous. Here we trade this last condition with $\lvert
D^{2}\psi\rvert\in\mathfrak{X}(\Omega)$, that in turn implies the mere
continuity of $D\psi$. This is essentially the same phenomenon seen in Theorem
1.6, where the condition $\lvert D\mathfrak{c}\rvert\in\mathfrak{X}(\Omega)$
replaces the Dini-continuity of $\mathfrak{c}(\cdot)$. We note that in the
constrained versions of Theorems 1.1 and 1.2 we can allow for $p\geq 1$
provided $\mu>0$; for this see Remark 11 below.

###  1.6 Remarks and organization of the paper

Some of the methods here also extend to general scalar functionals, i.e., when
minima and competitors are real valued functions. In this case there is no
need to assume the radial structure $F(x,Dw)\equiv\tilde{F}(x,|Dw|)$. On the
other hand, additional conditions ensuring the absence of the so-called
Lavrentiev phenomenon are needed to build suitable approximation arguments,
see for instance DeMi ; sharp . The radial structure is usually assumed in the
vectorial case, otherwise singular minimizers might occur, even when data are
smooth dark ; SY1 . Again in the scalar case, we mention the recent, very
interesting paper HO , where gradient regularity results are obtained for
minimizers of functionals as in (1.25). These results cover functionals with
polynomial growth and special structure - the double phase functional is an
instance - under Hölder continuity assumptions on coefficients. Anyway, they
miss to cover all the classes of integrands described in Sections 1.1-1.2.

The rest of the paper goes as follows. After Sections 2 and 3, containing
notations and preliminaries, respectively, in Section 4 we describe in detail
the assumptions and the main results of the paper, that is Theorems 4.1-4.4 in
Section 4.5. These will imply, directly or with a few additional arguments,
Theorems 1.1-1.8 above. We then proceed to Section 5, that contains the
necessary approximation tools for the proofs. One word here: this is a
delicate point, as the approximations considered must carefully match the
shape of the a priori estimates found later, on one side, and reflect the
original structure assumptions on the other. The core of the paper is Section
6, where we derive all the main a priori estimates. The proofs here involve a
series of ingredients. First, we employ a delicate version of Moser’s
iteration scheme developed in Proposition 1. This is based on a peculiar
choice of test maps suited to the structural assumptions considered. It goes
via a finite step procedure taking advantage of suitable smallness conditions;
by the way, this is sufficient to get the basic $L^{p}$-estimates of Theorem
4.3. In turn, this is a preliminary ingredient used to make a nonlinear
potential theoretic approach work; this last one is encoded in the abstract
iteration Lemma 1 below. This approach works in the case $n>2$, but breaks
down in two dimensions $n=2$, where more difficulties appear, as for instance
already noted in BM ; CM0 ; CM1 ; CM2 ; CM3 . In this case we take another
path, as devised in Section 6.7 below. We use a different interpolation type
approach, eventually culminating in the use of Lemma 1 again. Sections 7
features the proofs of Theorems 4.1 and 4.3, combining the approximation
scheme of Section 5 and the estimates in Section 6. Section 8 contains the
results and the proofs for the uniformly elliptic case, i.e., the proof of
Theorem 4.2; additional a priori estimates are included here. In Section 10 we
demonstrate the derivation of Theorems 1.1-1.7. These are all direct
corollaries of the main results, but Theorem 1.3. This in fact requires some
additional arguments to reach the equality borderline case in the first bound
from (1.19), thereby reconnecting Theorem 1.3 to the known literature BCM ;
CoM0 ; CoM  in the case $f\equiv 0$; the details can be found in Section 10.2.
Section 11 contains applications to obstacle problems and the proof of Theorem
1.8. Finally, Section 12 features some auxiliary technical facts aimed at
making certain computations in Sections 6 and 8 legal.

##  2 Basic notation

In the following, $\Omega\subset\mathbb{R}^{n}$ denotes an open domain, and
$n\geq 2$ and there is no loss of generality, in assuming that $\Omega$ is
also bounded, as all our results are local in nature. We denote by $c$ a
general constant larger than $1$. Different occurrences from line to line will
be still denoted by $c$. Special occurrences will be denoted by
$c_{*},\tilde{c}$ or likewise. Relevant dependencies on parameters will be as
usual emphasized by putting the corresponding parameters in parentheses. We
denote by $B_{r}(x_{0}):=\\{x\in\mathbb{R}^{n}:|x-x_{0}|<r\\}$ the open ball
with center $x_{0}$ and radius $r>0$; we omit denoting the center when it is
not necessary, i.e., $B\equiv B_{r}\equiv B_{r}(x_{0})$; this especially
happens when various balls in the same context will share the same center.
Given a ball $B$, we denote by $\texttt{r}(B)$ its radius; with $\gamma$ being
a positive number, we denote by $\gamma B$ the ball concentric to $B$, with
radius $\gamma\texttt{r}(B)$, and set $B/\gamma\equiv(1/\gamma)B$. In denoting
several function spaces like $L^{p}(\Omega),W^{1,p}(\Omega)$, we shall denote
the vector valued version by
$L^{p}(\Omega;\mathbb{R}^{k}),W^{1,p}(\Omega;\mathbb{R}^{k})$ in the case the
maps considered take values in $\mathbb{R}^{k}$, $k\in\mathbb{N}$. We shall
often abbreviate $L^{p}(\Omega;\mathbb{R}^{k})\equiv
L^{p}(\Omega),W^{1,p}(\Omega;\mathbb{R}^{k})\equiv W^{1,p}(\Omega)$. We denote
$\\{e^{\alpha}\\}_{\alpha=1}^{N}$ and $\\{e_{i}\\}_{i=1}^{n}$ standard bases
for $\mathbb{R}^{N}$ and $\mathbb{R}^{n}$, respectively; we shall always
assume $n\geq 2$ and $N\geq 1$. The general second-order tensor of size
$(N,n)$ as $\zeta=\zeta_{i}^{\alpha}e^{\alpha}\otimes e_{i}$ is identified
with an element of $\mathbb{R}^{N\times n}$. The Frobenius product of second-
order tensors $z$ and $\xi$ is defined as
$z\cdot\xi=z_{i}^{\alpha}\xi_{i}^{\alpha}$ so that $\xi\cdot\xi=|\xi|^{2}$,
and this is the norm we use here for tensors, vectors, matrixes; needless to
say, we are using Einstein’s convention on repeated indexes. For instance, if
$v,u\in\mathbb{R}^{k}$, then $v\cdot u=v_{i}u_{i}$. The gradient of a map
$u=u^{\alpha}e^{\alpha}$ is thus defined as
$Du=\partial_{x_{i}}u^{\alpha}e^{\alpha}\otimes e_{i}\equiv
D_{i}u^{\alpha}e^{\alpha}\otimes e_{i}\,,$ and the divergence of a tensor
$\zeta=\zeta_{i}^{\alpha}e^{\alpha}\otimes e_{i}$ as $\,{\rm
div}\,\zeta=\partial_{x_{i}}\zeta_{i}^{\alpha}e^{\alpha}$. When dealing with
the integrands of the type $F\colon\Omega\times\mathbb{R}^{N\times
n}\to[0,\infty)$, as the one considered in (1.1), second differential of
$\partial_{zz}F(\cdot,z)$ is interpreted as
$\partial_{zz}F(\cdot,z)=\partial_{z_{j}^{\beta}z_{i}^{\alpha}}F(\cdot,z)(e^{\alpha}\otimes
e_{i})\otimes(e^{\beta}\otimes e_{j})$, whenever $z\in\mathbb{R}^{N\times n}$.
For the rest of the paper we shall keep the following notation:

$E(t):=\sqrt{t^{2}+\mu^{2}}\,,\qquad
E_{\varepsilon}(t):=\sqrt{t^{2}+(\mu+\varepsilon)^{2}}$ (2.1)

for $t>0$, $\mu\in[0,1]$, $\varepsilon>0$. With
$\mathcal{B}\subset\mathbb{R}^{n}$ being a measurable subset with bounded
positive measure $0<|\mathcal{B}|<\infty$, and with
$g\colon\mathcal{B}\to\mathbb{R}^{k}$, $k\geq 1$, being a measurable map, we
denote

$(g)_{\mathcal{B}}\equiv\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{\mathcal{B}}g(x)\,dx:=\frac{1}{|\mathcal{B}|}\int_{\mathcal{B}}g(x)\,dx\,.$

Finally, in the following we denote

$\displaystyle 2^{*}:=\begin{cases}\ \frac{2n}{n-2}\quad&\mbox{if}\ \ n>2\\\ \
\mbox{any number larger than}\ \ 2\quad&\mbox{if}\ \ n=2\,.\end{cases}$ (2.2)

The actual value of $2^{*}$ when $n=2$ will be clear from the context.

##  3 Potentials, functions spaces, iteration lemmas

With $g\in L^{2}(B_{r}(x_{0});\mathbb{R}^{k})$ and
$B_{r}(x_{0})\subset\mathbb{R}^{n}$ being any ball, $n\geq 2$, we consider the
following nonlinear potential, that will play a crucial role in this paper:

${\bf
P}_{1}^{g}(x_{0},r):=\int_{0}^{r}\left(\varrho^{2}\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{B_{\varrho}(x_{0})}|g|^{2}\,dx\right)^{1/2}\frac{d\varrho}{\varrho}\,.$

This quantity naturally relates to the standard, truncated Riesz potential in
the sense that

$\int_{B_{r}(x_{0})}\frac{|g(x)|}{|x-x_{0}|^{n-1}}\,dx\lesssim{\bf
P}_{1}^{g}(x_{0},2r)\,.$

As a matter of fact, ${\bf P}_{1}^{g}$ can be used as an effective replacement
of the original Riesz potential when dealing with nonlinear problems.
Actually, its mapping properties coincide with those of the classical Riesz
potentials on those function spaces that are in a sense smaller than $L^{2}$.
We refer to KMbull ; MH  for more information on such nonlinear potentials and
for recent results on Nonlinear Potential Theory in the setting of this paper.
The space $\mathfrak{X}(\cdot)$ in (1.9) plays a special role to ensure the
local boundedness of ${\bf P}_{1}$. Indeed, given concentric balls
$B_{R}\subset B_{R+r}\subset\mathbb{R}^{n}$, with $r>0$ and $R+r\leq 2$, and
$g\in L^{2}(B_{R+r};\mathbb{R}^{k})$, the following inequalities hold:

$\begin{cases}\left\|{\bf P}_{1}^{g}(\cdot,r)\right\|_{L^{\infty}(B_{R})}\leq
c(n)\|g\|_{L(n,1)(B_{R+r})}\quad n>2\\\ \left\|{\bf
P}_{1}^{g}(\cdot,r)\right\|_{L^{\infty}(B_{R})}\leq
c(\mathfrak{a})\|g\|_{L^{2}(\log L)^{\mathfrak{a}}(B_{R+r})}\quad
n=2\,,\end{cases}$ (3.1)

where, in the last of the two inequalities, it is $c(\mathfrak{a})\to\infty$
when $\mathfrak{a}\searrow 2$. For this we refer to BM ; KMbull . Note that in
the right-hand side of (3.1)2 we find the Luxemburg norm of the Orlicz space
$L^{2}(\log L)^{\mathfrak{a}}$ in (1.8), which is defined as

$\|g\|_{{L^{2}(\log
L)^{\mathfrak{a}}(B_{R+r})}}:=\inf\left\\{\lambda>0\,\colon\int_{B_{R+r}}\left(\frac{|g|^{2}}{\lambda^{2}}\right)\log^{\mathfrak{a}}\left(\textnormal{e}+\frac{|g|}{\lambda}\right)\,dx\leq
1\right\\}.$

The two following inequalities are well-known and hold for any $\tau>0$:

$\begin{cases}\|g\|_{L(n,1)(B_{R+r})}\leq
c(n,\tau)\|g\|_{L^{n+\tau}(B_{R+r})}\\\ \|g\|_{{L^{2}(\log
L)^{\mathfrak{a}}(B_{R+r})}}\leq
c(\mathfrak{a},\tau)\|g\|_{L^{2+\tau}(B_{R+r})}\,.\end{cases}$ (3.2)

Recalling the standard notation $(v-\kappa)_{+}:=\max\\{v-\kappa,0\\}$, we
select the following result from (BM, , Lemma 3.1), that incorporates the
basic elements of De Giorgi’s iteration, as used in Nonlinear Potential Theory
since the basic work of Kilpeläinen & Malý kilmal :

###### Lemma 1

Let $B_{r_{0}}(x_{0})\subset\mathbb{R}^{n}$, $n\geq 2$,
$\delta\in\left(0,1/2\right)$ and $v\in W^{1,2}(B_{r_{0}}(x_{0}))$ be non-
negative and $f_{1},f_{2}\in L^{2}(B_{2r_{0}}(x_{0}))$. Assume that there
exist positive constants $\tilde{c},M_{1},M_{2},M_{3}\geq 1$ and a number
$\kappa_{0}\geq 0$ such that for all $\kappa\geq\kappa_{0}$ and every ball
$B_{r}(x_{0})\subset B_{r_{0}}(x_{0})$ the inequality

$\displaystyle\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{B_{r/2}(x_{0})}\lvert
D(v-\kappa)_{+}\rvert^{2}\,dx\leq\frac{\tilde{c}M_{1}^{2}}{r^{2}}\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{B_{r}(x_{0})}(v-\kappa)_{+}^{2}\,dx$
$\displaystyle\qquad\qquad\qquad+\tilde{c}M_{2}^{2}\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{B_{r}(x_{0})}\lvert
f_{1}\rvert^{2}\,dx+\tilde{c}M_{3}^{2}\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{B_{r}(x_{0})}\lvert
f_{2}\rvert^{2}\,dx$ (3.3)

holds. If $x_{0}$ is a Lebesgue point for $v$, then

$\displaystyle v(x_{0})$
$\displaystyle\leq\kappa_{0}+cM_{1}^{1+\max\left\\{\delta,\frac{n-2}{2}\right\\}}\left(\mathop{\int\hskip-10.50005pt-\,\\!\\!\\!}\nolimits_{B_{r_{0}}(x_{0})}(v-\kappa_{0})_{+}^{2}\,dx\right)^{1/2}$
$\displaystyle\qquad\
+cM_{1}^{\max\left\\{\delta,\frac{n-2}{2}\right\\}}\left[M_{2}\mathbf{P}_{1}^{f_{1}}(x_{0},2r_{0})+M_{3}\mathbf{P}_{1}^{f_{2}}(x_{0},2r_{0})\right]$
(3.4)

holds too, with $c\equiv c(n,\tilde{c},\delta)$.

We conclude with another, more classical iteration lemma (see references of BM
).

###### Lemma 2

Let $\mathcal{Z}\colon[\varrho,\xi]\to[0,\infty)$ be a bounded function. Let
$\varepsilon\in(0,1)$, $a_{1},a_{2},\mathfrak{b}\geq 0$ be numbers. If

$\mathcal{Z}(\tau_{1})\leq\varepsilon\mathcal{Z}(\tau_{2})+\frac{a_{1}}{(\tau_{2}-\tau_{1})^{\mathfrak{b}}}+a_{2}$

holds whenever $\xi\leq\tau_{1}<\tau_{2}\leq\varrho$, then

$\mathcal{Z}(\varrho)\leq\frac{ca_{1}}{(\varrho-\xi)^{\mathfrak{b}}}+ca_{2}$

holds too, with $c\equiv c(\varepsilon,\mathfrak{b})$.

##  4 Assumptions and general results

In this section we are going to describe a number of conditions on the
integrand $F$ in (1.1) implying our main results, that is Theorems 4.1-4.4 in
Section 4.5 below; in turn, these will imply Theorems 1.1-1.8 from the
Introduction.

###  4.1 Basic structural assumptions, and consequences

We assume that $F$ has radial structure, i.e., there exists
$\tilde{F}:\Omega\times[0,\infty)\to[0,\infty)$

$\displaystyle\begin{cases}\ F(x,z)=\tilde{F}(x,\lvert z\rvert)\ \mbox{for all
$(x,z)\in\Omega\times\mathbb{R}^{N\times n}$}\\\ \ t\mapsto\tilde{F}(x,t)\in
C^{1}_{\textnormal{loc}}[0,\infty)\cap
C^{2}_{\textnormal{loc}}(0,\infty)\quad\mbox{for all $x\in\Omega$}\\\ \
x\mapsto\tilde{F}^{\prime}(x,t)\in W^{1,n}(\Omega)\ \ \mbox{for every
$t>0$}\,.\end{cases}$ (4.1)

We assume in general that $\tilde{F},\tilde{F}^{\prime}$ are continuous on
$\Omega\times(0,\infty)$, while
$\tilde{F}^{\prime\prime},\partial_{x}\tilde{F}^{\prime}$ are instead
Carathéodory regular on the same set. Of course we also assume that
$\sup_{\Omega\times K}\partial_{zz}{F}$ is finite for every compact subset
$K\subset\mathbb{R}^{N\times n}\setminus\\{0_{R^{N\times n}}\\}$. Now we
describe the minimal and standard assumptions qualifying the functional in
(1.1) as elliptic. For this, we use three locally bounded functions
$g_{i}\colon\Omega\times(0,\infty)\to[0,\infty)$, for $i\in\\{1,2,3\\}$, in
the sense that $\sup_{\Omega\times K}g_{i}$ is finite, for every compact
subset $K\subset(0,\infty)$. The first two $g_{1},g_{2}$ are continuous and
serve to bound the lowest and the largest eigenvalues of $\partial_{zz}F$,
respectively. The third one $g_{3}$ is Carathéodory regular and controls the
growth of derivatives with respect to $x$. Specifically, we assume that there
exists $T>0$ such that

$\displaystyle\begin{cases}\ z\mapsto F(x,z)\ \ &\mbox{is convex for all
$x\in\Omega$}\\\ \ \lvert\partial_{zz}F(x,z)\rvert\leq g_{2}(x,\lvert
z\rvert)\ \ &\mbox{for all}\ x\in\Omega\ \mbox{on}\ \\{\lvert z\rvert\geq
T\\}\\\ \ g_{1}(x,\lvert
z\rvert)\lvert\xi\rvert^{2}\leq\partial_{zz}F(x,z)\xi\cdot\xi\ \ &\mbox{on}\
\\{\lvert z\rvert\geq T\\}\ \mbox{and for all $x\in\Omega$}\\\ \
|\partial_{x}\tilde{F}^{\prime}(x,t)|\leq h(x)g_{3}(x,t)\ \ &\mbox{for
a.e.\,$x\in\Omega$ and every $t>0$}\,,\end{cases}$ (4.2)

where (4.2)3 holds for every $\xi\in\mathbb{R}^{N\times n}$, $h(\cdot)\in
L^{n}(\Omega)$ and we assume also that

$\inf_{x\in\Omega}g_{1}(x,T)>0\,.$ (4.3)

Using as in (1.3) the notation $\tilde{a}(x,t):=\tilde{F}^{\prime}(x,t)/t$,
defined for $(x,t)\in\Omega\times(0,\infty)$, we assume that for fixed numbers
$\gamma>1$, $\mu\in[0,1]$, and for every $x\in\Omega$, the minimal
$\gamma$-superlinear growth of the lowest eigenvalue of
$\partial_{zz}F(x,\cdot)$, that is

$t\mapsto\frac{\tilde{a}(x,t)}{(t^{2}+\mu^{2})^{\frac{\gamma-2}{2}}}\
\mbox{and}\ t\mapsto\frac{g_{1}(x,t)}{(t^{2}+\mu^{2})^{\frac{\gamma-2}{2}}}\ \
\mbox{are non-decreasing on}\ (0,\infty).$ (4.4)

Note that the global convexity of $F(\cdot)$ in (4.2)1 implies, arguing as in
(4.9)-(4.11) below, that $\tilde{a}(\cdot)\geq 0$, that is,
$t\mapsto\tilde{F}(x,t)$ is non-decreasing on $[0,\infty)$, for every
$x\in\Omega$.

###### Remark 1

When considering partial derivatives of $F$ and $g_{1}$ in the $x$-variable,
as well as the function $g_{3}$, properties will be given a.e. with respect to
$x$ in the Carath́eodory sense. For instance, (4.2)4 means there exists a
negligible set $N\subset\Omega$ such that (4.2)4 holds for every $t>0$,
whenever $x\in\Omega\setminus N$. The same will apply later when considering
for instance (4.16)-(4.17) and (5.13).

###### Remark 2

In most of the relevant models it turns out to be $\tilde{a}(\cdot)\equiv
g_{1}(\cdot)$, and this justifies the double assumption in (4.4). Assumptions
in (4.2) are bound to quantify ellipticity of $\partial_{zz}F(\cdot,z)$ only
outside the ball $\\{|z|<T\\}$ (recall we are assuming $\nu>0$), and this
allows to cover functionals loosing their ellipticity properties on a bounded
set of $z\in\mathbb{R}^{N\times n}$. This condition not only adds more
generality, but helps simplifying the treatment in the case of nonuniformly
elliptic problems. We could do the same also with respect to
$\partial_{x}\tilde{F}^{\prime}$ in (4.2)4, but we prefer not, as this would
only add technical difficulties, while not covering more examples.

Let us derive a few consequences of (4.1)-(4.4). First, for every $x\in\Omega$
it holds that

$0<s\leq t\Longrightarrow\begin{cases}g_{1}(x,s)s\leq g_{1}(x,t)t\\\
a(x,s)s\leq a(x,t)t\,.\end{cases}$ (4.5)

Indeed, as $t\mapsto(t^{2}+\mu^{2})^{(\gamma-2)/2}t$ is non-decreasing, we
have

$\displaystyle g_{1}(x,s)s$
$\displaystyle=\frac{g_{1}(x,s)}{(s^{2}+\mu^{2})^{\frac{\gamma-2}{2}}}(s^{2}+\mu^{2})^{\frac{\gamma-2}{2}}s$
$\displaystyle\stackrel{{\scriptstyle\eqref{00}}}{{\leq}}\frac{g_{1}(x,t)}{(t^{2}+\mu^{2})^{\frac{\gamma-2}{2}}}(t^{2}+\mu^{2})^{\frac{\gamma-2}{2}}t=g_{1}(x,t)t\,.$

The same argument works for $t\mapsto a(\cdot,t)t$. Moreover, again from
(4.4), it readily follows that

$\begin{cases}\nu(t^{2}+\mu^{2})^{\frac{\gamma-2}{2}}\leq g_{1}(x,t)\
\mbox{for}\ t\geq T\\\
0\stackrel{{\scriptstyle(\ref{infima})}}{{<}}\nu:=\displaystyle\min\left\\{\frac{\inf_{x\in\Omega}g_{1}(x,T)}{(T^{2}+\mu^{2})^{\frac{\gamma-2}{2}}},1\right\\}\leq
1\,.\end{cases}$ (4.6)

From the very definition of $\tilde{a}(\cdot)$, for
$(x,t)\in\Omega\times(0,\infty)$ we have

$F(x,z)-F(x,0_{\mathbb{R}^{N\times
n}})=\tilde{F}(x,|z|)-\tilde{F}(x,0)=\int_{0}^{|z|}\tilde{a}(x,s)s\,ds$ (4.7)

and therefore $\partial_{xz}F(x,|z|)=\partial_{x}\tilde{a}(x,|z|)\otimes
z=[\partial_{x}\tilde{F}^{\prime}(x,|z|)/|z|]\otimes z$ holds for $|z|\not=0$.
It follows that (4.1)3 and (4.2)4 imply that $x\mapsto\partial_{z}F(x,z)\in
W^{1,n}(\Omega;\mathbb{R}^{N\times n})$ for all $z\in\mathbb{R}^{N\times n}$
such that $|z|\not=0$, and

$|\partial_{xz}F(x,z)|,|\partial_{x}\tilde{a}(x,\lvert z\rvert)||z|\leq
h(x)g_{3}(x,\lvert z\rvert)\ \mbox{on $\\{|z|>0\\}$, a.e. $x\in\Omega$}\,.$
(4.8)

Again from the second-last display it follows that

$\displaystyle\partial_{zz}F(x,z)=\partial_{z}[\tilde{a}(x,\lvert
z\rvert)z]=\tilde{a}(x,\lvert z\rvert)\mathds{I}_{N\times
n}+\tilde{a}^{\prime}(x,\lvert z\rvert)\lvert z\rvert\frac{z\otimes z}{\lvert
z\rvert^{2}}$ (4.9)

holds for $|z|\not=0$, so that, using (4.2)2,3 with $\xi=z$ and $\xi\perp z$
in (4.9), we obtain

$\displaystyle\begin{cases}\ \tilde{a}(x,\lvert
z\rvert)+\tilde{a}^{\prime}(x,\lvert z\rvert)\lvert z\rvert\geq g_{1}(x,\lvert
z\rvert)\\\ \ \tilde{a}(x,\lvert z\rvert)\geq g_{1}(x,\lvert
z\rvert)\end{cases}\ \mbox{for $(x,z)\in\Omega\times\left\\{\lvert z\rvert\geq
T\right\\}$}$ (4.10)

and

$\displaystyle\begin{cases}\ \tilde{a}(x,\lvert
z\rvert)+\tilde{a}^{\prime}(x,\lvert z\rvert)\lvert z\rvert\leq g_{2}(x,\lvert
z\rvert)\\\ \ \tilde{a}(x,\lvert z\rvert)\leq g_{2}(x,\lvert
z\rvert)\end{cases}\ \mbox{for $(x,z)\in\Omega\times\left\\{\lvert z\rvert\geq
T\right\\}$}\,.$ (4.11)

In particular, it follows that

$\begin{cases}\ 0<g_{1}(\cdot,t)\leq g_{2}(\cdot,t)&\mbox{for $t\geq T$}\\\ \
\tilde{a}(\cdot,t)\leq c(t^{2}+\mu^{2})^{(\gamma-2)/2}&\mbox{for $0<t\leq
T$}.\end{cases}$ (4.12)

The first inequality in (4.12) is a consequence of (4.6). In (4.12)2, as a
constant $c$ we can take
$c=\|\tilde{a}(\cdot,T)\|_{L^{\infty}(\Omega)}(T^{2}+\mu^{2})^{(2-\gamma)/2}\leq\|g_{2}(\cdot,T)\|_{L^{\infty}(\Omega)}(T^{2}+\mu^{2})^{(2-\gamma)/2}$.
This follows from (4.4) and again from (4.11).

###  4.2 The energy functions

These are two functions bound to quantify the minimal energy controlled by the
functional in (1.1); they will play a crucial role in the rest of the paper.
For every $(x,t)\in\Omega\times(0,\infty)$, we set

$\displaystyle\begin{cases}\displaystyle
G(x,t):=\int_{T}^{\max\\{t,T\\}}g_{1}(x,s)s\,ds\\\ \
\bar{G}(x,t):=G(x,t)+(T^{2}+1)^{\gamma/2}\,.\end{cases}$ (4.13)

###  4.3 Quantifying nonuniform ellipticity

Here we quantify ellipticity and growth of $F$ in the nonuniformly elliptic
case. For this, we consider numbers $d,\sigma,\hat{\sigma}\geq 0$ such that

$h(\cdot)\in L^{d}(\Omega)\,,\qquad d>n$ (4.14)

and

$\sigma+\hat{\sigma}<\left\\{\begin{array}[]{ccc}\min\left\\{\frac{1}{n}-\frac{1}{d},\frac{4}{\vartheta(n-2)}\left(1-\frac{1}{\gamma}\right)\right\\}&\mbox{if}&n>2\\\\[8.0pt]
\min\left\\{\frac{1}{2}-\frac{1}{d},\frac{2}{\vartheta}\left(1-\frac{1}{\gamma}\right)\right\\}&\mbox{if}&n=2\,.\end{array}\right.$
(4.15)

Here it is $\vartheta\equiv\vartheta(\gamma)=1$ when $\gamma\geq 2$, and
$\vartheta=2$ otherwise. We assume that $x\mapsto g_{1}(x,t)\in
W^{1,d}(\Omega)$ for all $t\geq T$ and that $\partial_{x}g_{1}(\cdot)$ is
Carathéodory regular on $\Omega\times[T,\infty)$. Then, for a fixed constant
$c_{\textnormal{b}}\geq 1$, we consider the assumptions

$\lvert\partial_{x}g_{1}(x,t)\rvert\leq
h(x)[\bar{G}(x,t)]^{\hat{\sigma}}g_{1}(x,t)\,,$ (4.16)
$\left\\{\begin{array}[]{c}\ g_{3}(x,t)t\leq
c_{\textnormal{b}}[\bar{G}(x,t)]^{1+\sigma}\\\\[9.0pt] \displaystyle\
\frac{[g_{3}(x,t)]^{2}}{g_{1}(x,t)}\leq
c_{\textnormal{b}}[\bar{G}(x,t)]^{1+2\sigma}\,,\end{array}\right.$ (4.17)
$\frac{g_{2}(x,t)}{g_{1}(x,t)}\leq c_{\textnormal{b}}[\bar{G}(x,t)]^{\sigma}$
(4.18)

for every $t\geq T$, every $x\in\Omega$ in the case of (4.18), and for a.e.
$x\in\Omega$ in the case of (4.16)-(4.17). Comments on the meaning of
(4.16)-(4.18) can be found in Section 4.6 below.

###  4.4 The uniformly elliptic case

Here we instead describe conditions relevant to the uniformly elliptic case,
and therefore to models as in (1.23). Such conditions are a slightly more
general version of those considered in CM0 ; CM1 ; liebe1 , that are
classical. We retain the structure conditions (4.1)-(4.4) from Section 4.1
but, instead of using (4.14)-(4.18) from Section 4.3, for a fixed constant
$c_{\textnormal{u}}\geq 1$ this time we consider

$\displaystyle\begin{cases}\ t>0\Longrightarrow g_{i}(x,t)\equiv g_{i}(t),\ \
i\in\\{1,2,3\\}\\\ \ T\leq t\Longrightarrow g_{2}(t)\leq
c_{\textnormal{u}}g_{1}(t)\\\ \ T\leq t\Longrightarrow g_{1}(t)t\leq
g_{3}(t)\leq c_{\textnormal{u}}g_{1}(t)t\,.\end{cases}$ (4.19)

###  4.5 General results

In the following, we abbreviate the assumptions considered as

$\begin{cases}\texttt{set}:=\left\\{(\ref{0ass})-(\ref{00}),(\ref{sigmad})-(\ref{9})\right\\}\\\
\texttt{set}_{\textnormal{u}}:=\left\\{(\ref{0ass})-(\ref{00}),(\ref{uni0})\right\\}\end{cases}$
(4.20)

and these are going to be used in the nonuniformly elliptic setting, and in
the uniformly elliptic one, respectively. Accordingly, we also gather the
parameters influencing the constants in the a priori estimates as

$\begin{cases}\texttt{data}:=\left(n,N,\nu,\gamma,T,c_{\textnormal{b}},d,\sigma,\hat{\sigma},\mathfrak{a}\right)\\\
\texttt{data}_{\textnormal{u}}:=\left(n,N,\nu,\gamma,T,c_{\textnormal{u}},g_{1}(1),\mathfrak{a}\right).\end{cases}$
(4.21)

The presence of $\mathfrak{a}$ only occurs in the two-dimensional case $n=2$.

###### Theorem 4.1

Let $u\in W^{1,1}_{{\rm loc}}(\Omega;\mathbb{R}^{N})$ be a minimizer of
$\mathcal{F}$ in (1.1), under assumptions set in (4.20), with
$f\in\mathfrak{X}(\Omega)$ as defined in (1.9). Then $Du\in
L^{\infty}_{\textnormal{loc}}(\Omega;\mathbb{R}^{N\times n})$. Moreover, there
exists a positive radius $R_{*}\equiv R_{*}(\texttt{data},f(\cdot))\leq 1$
such that if $\mathcal{B}\Subset\Omega$ is a ball with
$\texttt{r}(\mathcal{B})\leq R_{*}$, then

$\displaystyle\lVert G(\cdot,\lvert
Du\rvert)\rVert_{L^{\infty}(s\mathcal{B})}$
$\displaystyle\qquad\leq\frac{c}{(1-s)^{\beta}[\texttt{r}(\mathcal{B})]^{\beta}}\left[\|F(\cdot,Du)\|_{L^{1}(\mathcal{B})}^{\theta}+\lVert
f\rVert_{\mathfrak{X}(\mathcal{B})}^{\theta}+1\right]$ (4.22)

holds for every $s\in(0,1)$, where $c\equiv c(\texttt{data},\lVert
h\rVert_{L^{d}(\mathcal{B})})\geq 1$,
$\beta,\theta\equiv\beta,\theta(n,\gamma,\linebreak d,\sigma,\hat{\sigma})>0$.

###### Remark 3

When either $\gamma\geq 2$ or $f\equiv 0$, in Theorem 4.1 we can replace the
upper bound on $\sigma+\hat{\sigma}$ in (4.15) by the less restrictive
$\sigma+\hat{\sigma}<1/n-1/d$. See Remark 9 below.

The main Lipschitz regularity result in the uniformly elliptic case is in the
next

###### Theorem 4.2

Let $u\in W^{1,1}_{{\rm loc}}(\Omega;\mathbb{R}^{N})$ be a minimizer of
$\mathcal{F}$ in (1.1), under assumptions $\texttt{set}_{\textnormal{u}}$ in
(4.20), and with $f,h\in\mathfrak{X}(\Omega)$ as defined in (1.9). Then $Du\in
L^{\infty}_{\textnormal{loc}}(\Omega;\mathbb{R}^{N\times n})$. Moreover, there
exists a positive radius $R_{*}\equiv
R_{*}(\texttt{data}_{\textnormal{u}},h(\cdot))\leq 1$ such that if
$\mathcal{B}\Subset\Omega$ is a ball with $\texttt{r}(\mathcal{B})\leq R_{*}$,
then

$\displaystyle\|F(\cdot,Du)\|_{L^{\infty}(s\mathcal{B})}$
$\displaystyle\qquad\leq\frac{c}{(1-s)^{n}[\texttt{r}(\mathcal{B})]^{n}}\|F(\cdot,Du)\|_{L^{1}(\mathcal{B})}+c\|f\|_{\mathfrak{X}(\mathcal{B})}^{\gamma/(\gamma-1)}+c$
(4.23)

holds for every $s\in(0,1)$, where $c\equiv
c(\texttt{data}_{\textnormal{u}})$.

In deriving Theorem 4.1 we need to prove higher integrability bounds for the
gradient, that are worth being singled out in the following:

###### Theorem 4.3

Let $u\in W^{1,1}_{{\rm loc}}(\Omega;\mathbb{R}^{N})$ be a minimizer of
$\mathcal{F}$ in (1.1), under assumptions set in (4.20) with $n>2$, and with
$f\in L^{n}(\Omega)$. Then $Du\in
L^{p}_{\textnormal{loc}}(\Omega;\mathbb{R}^{N\times n})$ for every
$p\in[1,\infty)$. Moreover, for every $p\in[1,\infty)$ there exists a positive
radius $R_{*}\equiv R_{*}(\texttt{data},f(\cdot),p)\leq 1$ such that, if
$\mathcal{B}\Subset\Omega$ is a ball with $\texttt{r}(\mathcal{B})\leq R_{*}$,
then

$\displaystyle\|G(\cdot,\lvert Du\rvert)\|_{L^{p}(s\mathcal{B})}$
$\displaystyle\qquad\leq\frac{c}{(1-s)^{\beta_{p}}[\texttt{r}(\mathcal{B})]^{\beta_{p}}}\left[\|F(\cdot,Du)\|_{L^{1}(\mathcal{B})}^{\theta_{p}}+\lVert
f\rVert_{L^{n}(\mathcal{B})}^{\theta_{p}}+1\right]$ (4.24)

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

###### Remark 4

In order to prove Theorem 4.3 the full strength of (4.15) is not actually
needed. Assuming $\sigma+\hat{\sigma}<1/n-1/d$ suffices.

###### Remark 5

In Theorems 4.1-4.3, as well as in the other estimates in this paper, the
constants depending on data and $\texttt{data}_{\textnormal{u}}$ blow-up when
$T\to\infty$ (complete loss of ellipticity). On the contrary, they remain
bounded when $T\to 0$ (full recovery of ellipticity), as long as the quantity
$\nu\equiv\nu(T)$ in (4.6) stays bounded away from zero. This is for instance
the case in Theorems 1.1-1.5; see Section 10 below. As a matter of fact, there
is an additional dependence of the constants on the specific operator
considered. Indeed, the dependence on $T$ typically shows up via quantities
controlled by $\|\tilde{a}(\cdot,T)\|_{L^{\infty}}(T^{2}+\mu^{2})+T^{\gamma}$;
this can be in turn controlled as
$\lesssim\|\tilde{a}(\cdot,1)\|_{L^{\infty}}(T^{\gamma}+\mu^{\gamma})$ for
$T\leq 1$ via (4.4). All in all, the dependence of the constants on the
specific operator (i.e., $\partial_{x}F$) appears only through $\nu$ defined
in (4.6), and $\|\tilde{a}(\cdot,1)\|_{L^{\infty}}$ when $T$ approaches zero.
See also the appearance of $g_{1}(1)$ in $\texttt{data}_{\textnormal{u}}$ in
the uniformly elliptic case.

Finally, we give another general result concerning an intermediate form of
nonuniform ellipticity. This for instance applies to the case of functionals
with special structure as the double phase one in (1.10). More comments on
this will follow in Section 4.6 below.

###### Theorem 4.4

Let $u\in W^{1,1}_{{\rm loc}}(\Omega;\mathbb{R}^{N})$ be a minimizer of
$\mathcal{F}$ in (1.1), under assumptions set in (4.20) with
$\sigma=\hat{\sigma}$ in (4.16)-(4.17) and $f$ as in (1.9). Moreover, replace
(4.15) and (4.18) by

$\hat{\sigma}<\left\\{\begin{array}[]{ccc}1/n-1/d&\mbox{if}&n>2\\\\[6.0pt]
\min\\{1/2-1/d,\gamma-1\\}&\mbox{if}&n=2\end{array}\right.\quad\mbox{and}\quad\frac{g_{2}(x,t)}{g_{1}(x,t)}\leq
c_{\textnormal{b}}\,,$ (4.25)

respectively. Then $Du\in
L^{\infty}_{\textnormal{loc}}(\Omega;\mathbb{R}^{N\times n})$ and (4.22)
holds.

###### Remark 6

Replacing the integrand $F$ in (1.1) with the new one
$F_{0}(x,z):=F(x,z)-\tilde{F}(x,0)$, does not change the set of minimizers of
$\mathcal{F}$, and gets another functional still satisfying the conditions of
Theorems 4.1-4.4. Moreover, as $\tilde{F}(x,0)\geq 0$, once the estimates in
(4.22)-(4.24) are proved in the case $F_{0}(x,z)$ is considered, then they
also hold in the original case. Therefore for the rest of the paper we can
always assume that $\tilde{F}(x,0)=0$ and by (4.7) it follows that

$\displaystyle\tilde{F}(x,t)=\int_{0}^{t}\tilde{a}(x,s)s\,ds\quad\mbox{for}\
(x,t)\in\Omega\times(0,\infty)\,.$ (4.26)

Such a replacement makes sense provided $\tilde{F}(x,0)\in L^{1}_{{\rm
loc}}(\Omega)$, but this is always the case in our situation. Specifically,
this is true as soon as the functional $\mathcal{F}$ in (1.1) admits a
minimizer $u$, which is therefore such that $\tilde{F}(\cdot,|Du|)\in
L^{1}_{{\rm loc}}(\Omega)$. This implies that
$0\leq\tilde{F}(x,0)\leq\tilde{F}(x,|Du(x)|)$ (recall that
$t\mapsto\tilde{F}(x,t)$ is non-decreasing), so that $\tilde{F}(\cdot,0)\in
L^{1}_{{\rm loc}}(\Omega)$.

###  4.6 Different notions of nonuniform ellipticity

What do we call nonuniform ellipticity of the integrand $F$ here? To provide a
measure, in the autonomous case $F(x,z)\equiv F(z)$, it is rather standard to
use the ellipticity ratio

$\mathcal{R}_{F}(z):=\frac{\mbox{highest eigenvalue of}\
\partial_{zz}F(z)}{\mbox{lowest eigenvalue of}\ \partial_{zz}F(z)}\,.$ (4.27)

The occurrence of nonuniform ellipticity then refers to the fact that
$\mathcal{R}_{F}(z)\to\infty$ as $|z|\to\infty$; accordingly, the uniformly
elliptic case occurs when $\mathcal{R}_{F}(\cdot)\approx 1$. In the
nonautonomous case this leads to define the same pointwise quantity
$\mathcal{R}_{F}(x,z)$, by taking into account the $x$-dependence in the
right-hand side of (4.27). In this paper we have considered assumptions aimed
at bounding $\mathcal{R}_{F}(x,z)$ pointwise, i.e.,

$\mathcal{R}_{F}(x,z)\lesssim\frac{g_{2}(x,|z|)}{g_{1}(x,|z|)}\stackrel{{\scriptstyle(\ref{9})}}{{\lesssim}}\left(\int^{|z|}_{T}g_{1}(x,s)s\,ds\right)^{\sigma}+1\,.$
(4.28)

On the other hand, the ratio $\mathcal{R}_{F}(x,z)$ misses to properly encode
full information in the nonautonomous one. This leads to consider the nonlocal
quantity

$\mathfrak{R}_{F}(z,B):=\frac{\sup_{x\in B}\,\mbox{highest eigenvalue of}\
\partial_{zz}F(x,z)}{\inf_{x\in B}\,\mbox{lowest eigenvalue of}\
\partial_{zz}F(x,z)}\,,$ (4.29)

with $B\subset\Omega$ being any ball. The new ratio $\mathfrak{R}_{F}(z,B)$
naturally occurs in integral estimates and, in fact, turns out to be the right
quantity to describe intermediate notions of nonuniform ellipticity in the
nonautonomous case. Obviously, from the definitions it follows that
$\sup_{x\in B}\,\mathcal{R}_{F}(x,z)\leq\mathfrak{R}_{F}(z,B)$, with equality
in the autonomous case.

######  Remark 7 (Double face of the double phase functional (1.10))

To further motivate (4.29), recall that for general uniformly elliptic
problems one typically recovers classical theories as Schauder’s liebe1  and
Calderón-Zygmund’s BLO . This is in general not the case for the double phase
integral in (1.10), due to the counterexamples in sharp ; FMM . Nevertheless
the integrand $H(\cdot)$ turns out to be uniformly elliptic when using the
quantity in (4.27) as test, in the sense that $\mathcal{R}_{H}(\cdot)\approx
1$. On the other hand, when considering a ball $B$ such that
$B\cap\\{a(x)=0\\}\not=0$, we have that
$\mathfrak{R}_{H}(z,B)\approx|B|^{1/n-1/d}|z|^{q-p}+1$, therefore
$\mathfrak{R}_{H}(z,B)\to\infty$ when $|z|\to\infty$. In such sense, the use
of quantity in (4.29) resolves the above ambiguity. In this case, the
assumptions (1.19) provide a way to correct the growth of
$\mathfrak{R}_{H}(z,B)$ with respect to $|z|$ with the smallness of $|B|$, as
pointed out in the Introduction.

In this paper we consider different assumptions, playing with the parameters
$\sigma,\hat{\sigma}$ in (4.16)-(4.18), in order to tune different degrees of
nonuniform ellipticity, involving both $\mathcal{R}_{F}(\cdot)$ and
$\mathfrak{R}_{F}(\cdot)$. In Theorems 4.1-4.2, we prescribe a direct
pointwise bound on $\mathcal{R}_{F}(\cdot)$ as in (4.28), and then control the
behaviour with respect to $x$ of the derivatives of $F$ via (4.16)-(4.17).
This has the overall effect of providing an indirect control on
$\mathfrak{R}_{F}(\cdot)$ too. In Theorem 4.4, we bound
$\mathcal{R}_{F}(\cdot)\lesssim 1$, that is, uniform ellipticity is assumed in
the pointwise sense, still allowing for nonuniform ellipticity in the nonlocal
sense of (4.29). This leads for instance to get better bounds and opens the
way to Theorem 1.3, dealing with the intermediate case of double phase
functionals in the sense of Remark 7.

##  5 Approximation of integrands

Here we implement a truncation scheme aimed at approximating the original
integrand $F$ in (1.1) with a family $\\{F_{\varepsilon}\\}$ of integrands
with standard polynomial growth, and converging to $F$ locally uniformly. The
new integrands preserve structure properties as (4.1)-(4.2), with
corresponding control functions $g_{i,\varepsilon},$ still satisfying
relations as those of the original ones. For this we will exploit a few
arguments used in BM  as a starting point. In this section we shall
permanently assume that (4.1)-(4.4) are in force, so that their consequences
(4.5)-(4.12) can be used as well. Additional assumptions as (4.16)-(4.18)
shall also be considered; in case, this will be explicitly mentioned. In the
following we use a parameter $\varepsilon$ that will always be such that
$0<\varepsilon<\min\\{1,T\\}/4$ and use that $\tilde{F}(x,0)\equiv 0$.

###  5.1 General setup

Given $T$ and $\mu$ in (4.2) and (4.4), respecively, we introduce the numbers

$\displaystyle\mu_{\varepsilon}:=\mu+\varepsilon\,,\qquad
T_{\varepsilon}:=T+1/\varepsilon\,.$ (5.1)

With $\tilde{a}\colon\Omega\times(0,\infty)\mapsto[0,\infty)$ that has been
defined in (1.3) as

$\displaystyle\tilde{a}(x,t):=\frac{\tilde{F}^{\prime}(x,t)}{t}$ (5.2)

for $t>0$, which is continuous on $\Omega\times(0,\infty)$, we start
introducing the functions
$\tilde{a}_{\varepsilon}\colon\Omega\times[0,\infty)\to[0,\infty)$ as

$\displaystyle\tilde{a}_{\varepsilon}(x,t):=\begin{cases}\
\frac{\tilde{a}(x,\varepsilon)}{(\varepsilon^{2}+\mu^{2}_{\varepsilon})^{\frac{\gamma-2}{2}}}(t^{2}+\mu^{2}_{\varepsilon})^{\frac{\gamma-2}{2}}\
\ &\mbox{if}\ t\in[0,\varepsilon)\\\ \ \tilde{a}(x,t)\ \ &\mbox{if}\
t\in[\varepsilon,T_{\varepsilon})\\\ \
\frac{\tilde{a}(x,T_{\varepsilon})}{(T_{\varepsilon}^{2}+\mu^{2}_{\varepsilon})^{\frac{\gamma-2}{2}}}(t^{2}+\mu^{2}_{\varepsilon})^{\frac{\gamma-2}{2}}\
\ &\mbox{if}\ t\in[T_{\varepsilon},\infty)\,,\end{cases}$ (5.3)

for every $x\in\Omega$, where $\gamma$ also appears in (4.4); accordingly, we
define

$\begin{cases}\displaystyle
F_{\varepsilon}(x,z):=\tilde{F}_{\varepsilon}(x,\lvert z\rvert),\ \mbox{for}\
\tilde{F}_{\varepsilon}(x,t):=\int_{0}^{t}\tilde{a}_{\varepsilon}(x,s)s\,ds+\varepsilon
L_{\gamma,\varepsilon}(t)\\\ \displaystyle
L_{\gamma,\varepsilon}(t):=\frac{1}{\gamma}\left[(t^{2}+\mu_{\varepsilon}^{2})^{\gamma/2}-\mu_{\varepsilon}^{\gamma}\right]=\int_{0}^{t}(s^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}s\,ds\,,\end{cases}$
(5.4)

and, finally

$\displaystyle\bar{a}_{\varepsilon}(x,t):=\tilde{a}_{\varepsilon}(x,t)+\varepsilon(t^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}$
$\displaystyle\qquad\Longrightarrow
a_{\varepsilon}(x,z):=\bar{a}_{\varepsilon}(x,\lvert
z\rvert)z=\partial_{z}F_{\varepsilon}(x,z)\,.$ (5.5)

Note that $\tilde{a}_{\varepsilon}$ is continuous on $\Omega\times[0,\infty)$
and $\partial_{x}\tilde{a}_{\varepsilon}$ is Carathéodory regular;
$(x,t)\mapsto\tilde{a}^{\prime}_{\varepsilon}(x,t)$ is instead Borel regular.
In view of (4.1) and (4.26), it follows

$\displaystyle\begin{cases}\ t\mapsto\tilde{F}_{\varepsilon}(x,t)\in
C^{1}_{\textnormal{loc}}[0,\infty)\cap
C^{2}_{\textnormal{loc}}\left([0,\infty)\setminus\\{\varepsilon,T_{\varepsilon}\\}\right)\
\ \mbox{for all}\ x\in\Omega\\\ \
x\mapsto\tilde{F}^{\prime}_{\varepsilon}(x,t)\in W^{1,n}(\Omega)\ \ \mbox{for
all}\ \ t\in[0,\infty)\\\ \
t\mapsto\tilde{a}_{\varepsilon}(x,t)\in\textnormal{Lip}_{\textnormal{loc}}[0,\infty)\cap
C^{1}_{\textnormal{loc}}\left([0,\infty)\setminus\\{\varepsilon,T_{\varepsilon}\\}\right)\
\ \mbox{for all}\ x\in\Omega\\\ \ t\mapsto\tilde{F}_{\varepsilon}(x,t)\ \
\mbox{is strictly convex and non-decreasing for all $x\in\Omega$}\\\ \
F_{\varepsilon}\ \mbox{is continuous on $\Omega\times\mathbb{R}^{N\times n}$
and $\tilde{F}_{\varepsilon}(x,0)=0$}\\\ \ F_{\varepsilon}\to
F\quad\mbox{uniformly on $\Omega\times\mathcal{K}$ as $\varepsilon\to 0$,
$\forall\,\mbox{compact}\,\mathcal{K}\subset\mathbb{R}^{N\times n}$ }\\\ \
\mbox{$\partial_{x}\tilde{a}_{\varepsilon}$ is Carath\'{e}odory-regular on
$\Omega\times[0,\infty)$}\,.\end{cases}$ (5.6)

For (5.6)4, see (5.17) below. For the proof of (5.6)5, recall that
$\tilde{F}^{\prime}$ is continuous on $\Omega\times(0,\infty)$. For details
about (5.6)6, see also Lemma 6 below. For (5.6)7 use that
$\partial_{x}\tilde{F}^{\prime}$ is Carathéodory.

Definitions (5.1)-(5.3) lead to introduce new control functions
$g_{i,\varepsilon}\colon\Omega\times[0,\infty)\to(0,\infty)$,
$i\in\\{1,2,3\\}$ as:

$\displaystyle g_{1,\varepsilon}(x,t):=\mathfrak{g}_{1}\begin{cases}\
\frac{g_{1}(x,\varepsilon)}{(\varepsilon^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}}(t^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}\quad&\mbox{if}\
t\in[0,\varepsilon)\\\ \ g_{1}(x,t)\quad&\mbox{if}\
t\in[\varepsilon,T_{\varepsilon})\\\ \
\frac{g_{1}(x,T_{\varepsilon})}{(T_{\varepsilon}^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}}(t^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}\quad&\mbox{if}\
t\in[T_{\varepsilon},\infty)\,,\end{cases}$ (5.7) $\displaystyle
g_{2,\varepsilon}(x,t):=\mathfrak{g}_{2}\begin{cases}\
\left[\frac{g_{2}(x,\varepsilon)}{(\varepsilon^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}}+\varepsilon\right](t^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}\quad&\mbox{if}\
t\in[0,\varepsilon)\\\ \
g_{2}(x,t)+\varepsilon(t^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}\quad&\mbox{if}\
t\in[\varepsilon,T_{\varepsilon})\\\ \
\left[\frac{g_{2}(x,T_{\varepsilon})}{(T_{\varepsilon}^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}}+\varepsilon\right](t^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}\quad&\mbox{if}\
t\in[T_{\varepsilon},\infty)\,,\end{cases}$ (5.8) $\displaystyle
g_{3,\varepsilon}(x,t):=\begin{cases}\
\frac{g_{3}(x,\varepsilon)}{(\varepsilon^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}\varepsilon}(t^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}t\quad&\mbox{if}\
t\in[0,\varepsilon)\\\ \ g_{3}(x,t)\quad&\mbox{if}\
t\in[\varepsilon,T_{\varepsilon})\\\ \
\frac{g_{3}(x,T_{\varepsilon})}{(T_{\varepsilon}^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}T_{\varepsilon}}(t^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}t\quad&\mbox{if}\
t\in[T_{\varepsilon},\infty)\,,\end{cases}$ (5.9)

where the constants $\mathfrak{g}_{1}$ and $\mathfrak{g}_{2}$ are defined by

$\displaystyle\mathfrak{g}_{1}:=\min\\{1,\gamma-1\\}\leq
1\leq\mathfrak{g}_{2}:=20(Nn+\gamma)\,.$ (5.10)

Observe that $g_{1,\varepsilon},g_{2,\varepsilon}$ are still continuous and
$g_{3,\varepsilon}$ is still Carathéodory regular. We next introduce also the
truncated counterparts of the functions defined in (4.13), i.e.,

$\begin{cases}\displaystyle
G_{\varepsilon}(x,t):=\int_{T}^{\max\\{t,T\\}}g_{1,\varepsilon}(x,s)s\,ds\\\
\bar{G}_{\varepsilon}(x,t):=G_{\varepsilon}(x,t)+(T^{2}+1)^{\gamma/2}\,.\end{cases}$
(5.11)

In the following, recalling (5.1), we shall repeatedly use

$1\leq\frac{s^{2}+\mu_{\varepsilon}^{2}}{s^{2}+\mu^{2}}\leq 3,\ \mbox{provided
$s\geq\varepsilon$}$ (5.12)

and that this is a decreasing of $s$.

###  5.2 Four technical lemmas

###### Lemma 3

Under assumptions (4.1)-(4.4).

  * •

There exist constants $\\{\Lambda_{\varepsilon}\\}$ and
$\\{L_{\varepsilon}\\}$ such that the following properties hold:

$\displaystyle\begin{cases}\ \lvert\partial_{zz}F_{\varepsilon}(x,z)\rvert\leq
g_{2,\varepsilon}(x,\lvert z\rvert)\ &\mbox{if}\
(x,z)\in\Omega\times\left\\{\lvert z\rvert\geq T\,,\lvert
z\rvert\not=T_{\varepsilon}\right\\}\\\ \ g_{1,\varepsilon}(x,\lvert
z\rvert)\lvert\xi\rvert^{2}\leq\partial_{zz}F_{\varepsilon}(x,z)\xi\cdot\xi\
&\mbox{if}\ (x,z)\in\Omega\times\left\\{\lvert z\rvert\geq T\,,\lvert
z\rvert\not=T_{\varepsilon}\right\\}\\\ \
\lvert\partial_{xz}F_{\varepsilon}(x,z)\rvert\leq
h(x)g_{3,\varepsilon}(x,\lvert z\rvert)\ &\mbox{for a.e.\,$x\in\Omega$ and
every $z\in\mathbb{R}^{N\times n}$}\\\ \
|\partial_{x}\bar{a}_{\varepsilon}(x,\lvert z\rvert)||z|\leq
h(x)g_{3,\varepsilon}(x,\lvert z\rvert)\ &\mbox{for a.e.\,$x\in\Omega$ and
every $z\in\mathbb{R}^{N\times n}$}\end{cases}$ (5.13)

and

$\displaystyle\begin{cases}\
\lvert\partial_{zz}F_{\varepsilon}(x,z)\rvert\leq\Lambda_{\varepsilon}(\lvert
z\rvert^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}\ &\mbox{if}\
(x,z)\in\Omega\times\left\\{\lvert
z\rvert\not=\varepsilon,T_{\varepsilon}\right\\}\\\ \
\varepsilon\mathfrak{g}_{1}(\lvert
z\rvert^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}\lvert\xi\rvert^{2}\leq\partial_{zz}F_{\varepsilon}(x,z)\xi\cdot\xi\
&\mbox{if}\ (x,z)\in\Omega\times\left\\{\lvert
z\rvert\not=\varepsilon,T_{\varepsilon}\right\\}\\\ \
\lvert\partial_{xz}F_{\varepsilon}(x,z)\rvert\leq L_{\varepsilon}h(x)(\lvert
z\rvert^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}|z|\ &\mbox{for
a.e.\,$x\in\Omega,\ \forall\,z\in\mathbb{R}^{N\times n}$}\end{cases}$ (5.14)

for all $\xi\in\mathbb{R}^{N\times n}$. As a consequence, it follows that

$\displaystyle g_{1,\varepsilon}(x,\lvert
z\rvert)\leq\bar{a}_{\varepsilon}(x,\lvert z\rvert)\leq
g_{2,\varepsilon}(x,\lvert z\rvert)$ (5.15)

holds for all $(x,z)\in\Omega\times\\{\lvert z\rvert\geq T\\}$, and

$\displaystyle\begin{cases}\ \bar{a}_{\varepsilon}(x,\lvert
z\rvert)+\bar{a}_{\varepsilon}^{\prime}(x,\lvert z\rvert)\lvert z\rvert\geq
g_{1,\varepsilon}(x,\lvert z\rvert)\\\ \ \bar{a}_{\varepsilon}(x,\lvert
z\rvert)+\bar{a}_{\varepsilon}^{\prime}(x,\lvert z\rvert)\lvert z\rvert\leq
g_{2,\varepsilon}(x,\lvert z\rvert)\end{cases}$ (5.16)

hold for all $(x,z)\in\Omega\times\\{\lvert z\rvert\geq T,\ \lvert
z\rvert\not=T_{\varepsilon}\\}$.

  * •

It holds that

$0<s\leq t\ \Longrightarrow\begin{cases}g_{1,\varepsilon}(x,s)s\leq
g_{1,\varepsilon}(x,t)t\\\ \
\tilde{a}_{\varepsilon}(x,s)s\leq\tilde{a}_{\varepsilon}(x,t)t\,.\end{cases}$
(5.17)

In particular, the function $t\mapsto G_{\varepsilon}(\cdot,t)$ in (5.11) is
convex.

  * •

For every $(x,z)\in\Omega\times\\{\lvert z\rvert\geq T\\}$, it holds that:

$\displaystyle\begin{cases}\
c(\nu,\gamma)(|z|^{2}+\mu_{\varepsilon}^{2})^{(\gamma-2)/2}\leq
g_{1,\varepsilon}(x,|z|)\\\ \ G(x,\lvert z\rvert)\leq F(x,z)\\\
G_{\varepsilon}(x,\lvert z\rvert)\leq F_{\varepsilon}(x,z)\,.\end{cases}$
(5.18)

For every $(x,t)\in\Omega\times[T,\infty)$, there holds

$\left\\{\begin{array}[]{c}\displaystyle(t^{2}+\mu_{\varepsilon}^{2})^{\gamma/2}\leq
c(\nu,\gamma)\bar{G}_{\varepsilon}(x,t)\\\
\displaystyle{G}_{\varepsilon}(x,t)\leq
c(\nu,\gamma)[\bar{G}_{\varepsilon}(x,t)]^{2/\gamma}g_{1,\varepsilon}(x,t)\,.\end{array}\right.$
(5.19)

Again for every $(x,z)\in\Omega\times\\{\lvert z\rvert\geq T\\}$, it holds

$\displaystyle\begin{cases}\ c(\nu,\gamma)F(x,z)\geq(\lvert
z\rvert^{2}+\mu^{2})^{\gamma/2}-(T^{2}+\mu^{2})^{\gamma/2}\\\ \
c(\nu,\gamma)F_{\varepsilon}(x,z)\geq(\lvert
z\rvert^{2}+\mu^{2})^{\gamma/2}-(T^{2}+\mu^{2})^{\gamma/2}\,.\end{cases}$
(5.20)

  * •

For every $x\in\Omega$ and for a fixed constant $c\equiv c(\nu,\gamma)$, we
have

$\begin{cases}\tilde{a}_{\varepsilon}(x,t)\leq c\tilde{a}(x,t)\quad\mbox{for
$t\geq\varepsilon$}\\\ (t^{2}+\mu_{\varepsilon}^{2})^{(\gamma-2)/2}\leq
c\tilde{a}(x,t)\quad\mbox{for $t\geq T$}\,.\end{cases}$ (5.21)

  * •

Finally, for another constant $c$ depending on $\nu$ and
$\|\tilde{a}(\cdot,T)\|_{L^{\infty}}(T^{2}+\mu^{2})+T^{\gamma}$, we have that

$F_{\varepsilon}(x,z)\leq cF(x,z)+c\quad\mbox{holds for all
}(x,z)\in\Omega\times\mathbb{R}^{N\times n}$ (5.22)

and that

$\frac{\varepsilon}{\gamma}(t^{2}+\mu_{\varepsilon}^{2})^{\gamma/2}-\frac{\varepsilon\mu_{\varepsilon}^{\gamma}}{\gamma}\leq\tilde{F}_{\varepsilon}(x,t)\leq
c_{\varepsilon}(t^{2}+\mu_{\varepsilon}^{2})^{\gamma/2}$ (5.23)

holds for all $(x,t)\in\Omega\times[0,\infty)$ .

###### Proof

By the very definitions in (5.3)-(5.4), we note that
$\partial_{zz}F_{\varepsilon}(x,z)$ exists for all
$(x,z)\in\Omega\times\left\\{\lvert
z\rvert\not=\varepsilon,T_{\varepsilon}\right\\}$ with

$\displaystyle\partial_{zz}F_{\varepsilon}(x,z)=\begin{cases}\ (\lvert
z\rvert^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}\left[\frac{\tilde{a}(x,\varepsilon)}{(\varepsilon^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}}+\varepsilon\right]\mathcal{C}_{\varepsilon}(z)\\\
\qquad\quad\mbox{if}\ (x,z)\in\Omega\times\\{\lvert z\rvert<\varepsilon\\}\\\
\ \partial_{zz}F(x,z)+\varepsilon(\lvert
z\rvert^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}\mathcal{C}_{\varepsilon}(z)\\\
\qquad\quad\mbox{if}\ (x,z)\ \in\Omega\times\\{\varepsilon<\lvert
z\rvert<T_{\varepsilon}\\}\\\ \ (\lvert
z\rvert^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}\left[\frac{\tilde{a}(x,T_{\varepsilon})}{(T_{\varepsilon}^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}}+\varepsilon\right]\mathcal{C}_{\varepsilon}(z)\\\
\qquad\quad\mbox{if}\ (x,z)\in\Omega\times\\{T_{\varepsilon}<\lvert
z\rvert\\}\,,\end{cases}$

where

$\displaystyle\mathcal{C}_{\varepsilon}(z):=\mathbb{I}_{N\times
n}+(\gamma-2)\frac{z\otimes z}{\lvert
z\rvert^{2}+\mu_{\varepsilon}^{2}}\quad\mbox{for}\ z\in\mathbb{R}^{N\times
n}\,.$

Moreover, recalling (5.3), by the definition of weak derivatives we have

$\displaystyle\partial_{xz}F_{\varepsilon}(x,z)=\begin{cases}\
\frac{\partial_{x}\tilde{a}(x,\varepsilon)}{(\varepsilon^{2}+\mu^{2}_{\varepsilon})^{\frac{\gamma-2}{2}}}(|z|^{2}+\mu^{2}_{\varepsilon})^{\frac{\gamma-2}{2}}z&\mbox{if}\
(x,z)\in\Omega\times\\{\lvert z\rvert<\varepsilon\\}\\\ \
\partial_{x}\tilde{a}(x,|z|)z&\mbox{if}\ (x,z)\
\in\Omega\times\\{\varepsilon\leq\lvert z\rvert<T_{\varepsilon}\\}\\\ \
\frac{\partial_{x}\tilde{a}(x,T_{\varepsilon})}{(T_{\varepsilon}^{2}+\mu^{2}_{\varepsilon})^{\frac{\gamma-2}{2}}}(|z|^{2}+\mu^{2}_{\varepsilon})^{\frac{\gamma-2}{2}}z&\mbox{if}\
(x,z)\in\Omega\times\\{T_{\varepsilon}\leq\lvert z\rvert\\}\,.\end{cases}$

Then (5.13)-(5.14) directly follow by (4.2), (4.8), (5.3), (5.7)-(5.9), and
with the choice

$\displaystyle\Lambda_{\varepsilon}:=\mathfrak{g}_{2}\left[1+\sup_{x\in\Omega}\frac{2g_{2}(x,T_{\varepsilon})}{(T_{\varepsilon}^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}}+\sup_{x\in\Omega,\lvert
z\rvert\in[\varepsilon,T_{\varepsilon}]}\frac{\lvert\partial_{zz}F(x,z)\rvert}{(\lvert
z\rvert^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}}\right],$
$\displaystyle
L_{\varepsilon}:=\sup_{t\in[\varepsilon,T_{\varepsilon}]}\frac{\|g_{3}(\cdot,t)\|_{L^{\infty}(\Omega)}}{(t^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}t}\,.$

Note that we are also using (4.11)2 and (4.4) in order to get upper bounds for
$|\partial_{zz}F_{\varepsilon}(x,z)|$, in (5.13)1 and (5.14)1, respectively;
also (5.12) is used to estimate

$\frac{\tilde{a}(x,\varepsilon)}{(\varepsilon^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}}\leq\frac{2\tilde{a}(x,T_{\varepsilon})}{(T_{\varepsilon}^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}}\leq\frac{2g_{2}(x,T_{\varepsilon})}{(T_{\varepsilon}^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}}\,.$

We also used (4.10)2 to get (5.13)2.

As for (5.15)-(5.16), when $|z|\not=T_{\varepsilon}$, these follow from
definition (5.1) and (5.13), arguing exactly as for (4.10)-(4.11). The case
$|z|=T_{\varepsilon}$ of (5.15) follows by continuity.

The property in (5.17) readily follows from (4.5) and the definitions in
(5.7)-(5.8), also using the fact that
$t\mapsto(t^{2}+\mu_{\varepsilon}^{2})^{(\gamma-2)/2}t$ is increasing.

We now come to (5.18)-(5.20). For $\eqref{z6}_{2}$-$\eqref{z7}_{1}$ we note
that

$\displaystyle F(x,z)$ $\displaystyle\geq$ $\displaystyle\int_{T}^{\lvert
z\rvert}\tilde{a}(x,s)s\,ds\stackrel{{\scriptstyle\eqref{22}_{2}}}{{\geq}}\int_{T}^{\lvert
z\rvert}g_{1}(x,s)s\,ds\stackrel{{\scriptstyle\eqref{GG}}}{{=}}G(x,\lvert
z\rvert)$ (5.24) $\displaystyle\stackrel{{\scriptstyle\eqref{00bis}}}{{\geq}}$
$\displaystyle\nu\int_{T}^{\lvert
z\rvert}(s^{2}+\mu^{2})^{\frac{\gamma-2}{2}}s\,ds\stackrel{{\scriptstyle\eqref{Hnot}}}{{=}}\frac{\nu}{\gamma}\left\\{[E(\lvert
z\rvert)]^{\gamma}-[E(T)]^{\gamma}\right\\}.$

Let us now take care of (5.19)1. For $t\in[T,T_{\varepsilon})$, using (4.6)
and (5.12), we see that (recall that $\varepsilon\leq T/4$)

$\displaystyle\bar{G}_{\varepsilon}(x,t)$
$\displaystyle\geq\frac{\mathfrak{g}_{1}\nu}{\gamma}\left[(t^{2}+\mu^{2})^{\gamma/2}-(T^{2}+\mu^{2})^{\gamma/2}\right]+(T^{2}+1)^{\gamma/2}$
$\displaystyle\geq\frac{(t^{2}+\mu_{\varepsilon}^{2})^{\gamma/2}}{c(\nu,\gamma)}\,,$
(5.25)

while, when $t\geq T_{\varepsilon}$ by analogous means, we have

$\displaystyle\bar{G}_{\varepsilon}(x,t)=\mathfrak{g}_{1}\int_{T_{\varepsilon}}^{t}\frac{g_{1}(x,T_{\varepsilon})}{(T_{\varepsilon}^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}}(s^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}s\,ds$
$\displaystyle\qquad\qquad+\mathfrak{g}_{1}\int_{T}^{T_{\varepsilon}}g_{1}(x,s)s\,ds+(T^{2}+1)^{\gamma/2}$
$\displaystyle\qquad\ \ \
\geq\frac{1}{c(\nu,\gamma)}\int_{T}^{t}(s^{2}+\mu^{2})^{\frac{\gamma-2}{2}}s\,ds+(T^{2}+1)^{\gamma/2}\geq\frac{(t^{2}+\mu_{\varepsilon}^{2})^{\gamma/2}}{c(\nu,\gamma)}\,.$
(5.26)

Implicit in the arguments from (5.25)-(5.26) is also the proof of
$\eqref{z6}_{1}$. Inequalites $\eqref{z6}_{3}$ and $\eqref{z7}_{2}$ follow
using the lower bound in (5.15) and then (5.18)1, as in (5.24). As for
$\eqref{z10a}_{2}$, note that

$\displaystyle G_{\varepsilon}(x,t)=\int_{T}^{t}g_{1,\varepsilon}(x,s)s\,ds$
$\displaystyle\stackrel{{\scriptstyle(\ref{einc})}}{{\leq}}$ $\displaystyle
g_{1,\varepsilon}(x,t)t\int_{T}^{t}\,ds$ (5.27) $\displaystyle=$
$\displaystyle g_{1,\varepsilon}(x,t)t(t-T)\leq g_{1,\varepsilon}(x,t)t^{2}$

holds whenever $t\geq T$. Now $\eqref{z10a}_{2}$ follows using (5.27) with
$\eqref{z10a}_{1}$.

The proof of (5.21)1 follows straightaway from the definition of
$\tilde{a}_{\varepsilon}$ in (5.3), the first property in (4.4) and (5.12);
the relation in (5.21)2 instead follows from (4.6), (4.10)2 and again (5.12).

For the proof of (5.22) we use (4.6) and (5.12), to get, in the case $t\leq T$

$\displaystyle\tilde{F}_{\varepsilon}(x,t)$ $\displaystyle\leq
c\left[\frac{\tilde{a}(x,T)}{(T^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}}+\varepsilon\right]\int_{0}^{t}(s^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}s\,ds$
$\displaystyle\leq
c\|\tilde{a}(\cdot,T)\|_{L^{\infty}}(T^{2}+\mu^{2})+c(T^{\gamma}+1)\,.$ (5.28)

On the other hand, when $t>T$, using (5.21) gives that
$\tilde{F}_{\varepsilon}(x,t)-\tilde{F}_{\varepsilon}(x,T)\leq
c\tilde{F}(x,t)$ and this, together with the content of the last display,
gives (5.21) again. Finally, the proof of (5.23) follows straightaway from the
definitions in (5.3)-(5.4).

###### Lemma 4

Under assumptions (4.1)-(4.4)

  * •

If (4.18) is also in force for some $\sigma\geq 0$, then for every
$(x,t)\in\Omega\times[T,\infty)$ it holds that

$\displaystyle\frac{g_{2,\varepsilon}(x,t)}{g_{1,\varepsilon}(x,t)}\leq
c[\bar{G}_{\varepsilon}(x,t)]^{\sigma}\,,\quad\mbox{where $c\equiv
c(n,N,\nu,\gamma,c_{\textnormal{b}})$}\,.$ (5.29)

In particular, if $g_{2}/g_{2}\leq c_{\textnormal{b}}$ holds as in (4.25),
then it also holds that

$\displaystyle\frac{g_{2,\varepsilon}(x,t)}{g_{1,\varepsilon}(x,t)}\leq
c(n,N,\nu,\gamma,c_{\textnormal{b}})\,.$ (5.30)

  * •

If (4.16) is also in force, then, with $\mathfrak{g}_{1}$ as in (5.10), for
a.e. $x\in\Omega$ and every $z\in\mathbb{R}^{N\times n}$ with $|z|\geq T$, it
holds that

$\displaystyle\lvert\partial_{x}g_{1,\varepsilon}(x,t)\rvert
t\leq\frac{h(x)}{\mathfrak{g}_{1}^{\hat{\sigma}}(1+\hat{\sigma})}\partial_{t}[\bar{G}_{\varepsilon}(x,t)]^{1+\hat{\sigma}}\,.$
(5.31)

  * •

For all $(x,t)\in\Omega\times[T,\infty)$, the following holds:

$\varepsilon_{1}<\varepsilon_{2}<\min\\{1,T\\}/4\Longrightarrow
g_{1,\varepsilon_{2}}(x,t)\leq c(\gamma)g_{1,\varepsilon_{1}}(x,t)$ (5.32)

so that, this time for all $(x,t)\in\Omega\times[0,\infty)$, it follows

$G_{\varepsilon_{2}}(x,t)\leq
c(\gamma)G_{\varepsilon_{1}}(x,t)\quad\mbox{and}\quad\lim_{\varepsilon\to
0}\,G_{\varepsilon}(x,t)=\mathfrak{g}_{1}G(x,t)\,.$ (5.33)

The last convergence occurs uniformly on compact subsets of $[0,\infty)$.

###### Proof

For (5.29), note that (4.6), (4.12)1 and (5.12) imply

$T\leq t\leq
T_{\varepsilon}\quad\Longrightarrow\quad\frac{1}{\mathfrak{g}_{1}}\frac{g_{2}(x,t)}{g_{1}(x,t)}\leq\frac{g_{2,\varepsilon}(x,t)}{g_{1,\varepsilon}(x,t)}\leq
c(\nu,\gamma)\,\frac{g_{2}(x,t)}{g_{1}(x,t)}$ (5.34)

and that the definitions in (5.7)-(5.8) imply

$t>T_{\varepsilon}\Longrightarrow\frac{g_{2,\varepsilon}(x,t)}{g_{1,\varepsilon}(x,t)}=\frac{g_{2,\varepsilon}(x,T_{\varepsilon})}{g_{1,\varepsilon}(x,T_{\varepsilon})}\,.$
(5.35)

Then, for $T\leq t\leq T_{\varepsilon}$ we have

$\displaystyle\frac{g_{2,\varepsilon}(x,t)}{g_{1,\varepsilon}(x,t)}\stackrel{{\scriptstyle\eqref{ccrrss}}}{{\leq}}c\frac{g_{2}(x,t)}{g_{1}(x,t)}\stackrel{{\scriptstyle\eqref{9}}}{{\leq}}c[\bar{G}(x,t)]^{\sigma}\stackrel{{\scriptstyle(\ref{g1e})}}{{\leq}}c[\bar{G}_{\varepsilon}(x,t)]^{\sigma}$
(5.36)

while, for $T_{\varepsilon}<t$, we have

$\displaystyle\frac{g_{2,\varepsilon}(x,t)}{g_{1,\varepsilon}(x,t)}\stackrel{{\scriptstyle(\ref{ccrrss1})}}{{=}}\frac{g_{2,\varepsilon}(x,T_{\varepsilon})}{g_{1,\varepsilon}(x,T_{\varepsilon})}\stackrel{{\scriptstyle\eqref{banal}}}{{\leq}}[\bar{G}_{\varepsilon}(x,T_{\varepsilon})]^{\sigma}\leq
c[\bar{G}_{\varepsilon}(x,t)]^{\sigma}\,,$

and (5.29) is completely proved. The proof of (5.30) follows as in the last
two displays by formally taking $\sigma=0$. The proof of (5.31) is a
straightforward consequence of the definition in (5.7) and (4.16) when $t\leq
T_{\varepsilon}$. In the case $t>T_{\varepsilon}$, we instead have, also using
(4.16)

$\displaystyle\lvert\partial_{x}g_{1,\varepsilon}(x,t)\rvert t$
$\displaystyle\stackrel{{\scriptstyle(\ref{g1e})}}{{=}}$
$\displaystyle\frac{\mathfrak{g}_{1}|\partial_{x}g_{1}(x,T_{\varepsilon})|}{(T_{\varepsilon}^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}}(t^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}t$
$\displaystyle\leq$
$\displaystyle\frac{\mathfrak{g}_{1}h(x)[\bar{G}(x,T_{\varepsilon})]^{\hat{\sigma}}g_{1}(x,T_{\varepsilon})}{(T_{\varepsilon}^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}}(t^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}t$
$\displaystyle\leq$
$\displaystyle\frac{h(x)}{\mathfrak{g}_{1}^{\hat{\sigma}}}[\bar{G}_{\varepsilon}(x,t)]^{\hat{\sigma}}g_{1,\varepsilon}(x,t)t$
$\displaystyle=$
$\displaystyle\frac{h(x)}{\mathfrak{g}_{1}^{\hat{\sigma}}(1+\hat{\sigma})}\partial_{t}[\bar{G}_{\varepsilon}(x,t)]^{1+\hat{\sigma}}$

that is, (5.31). Finally, the properties in (5.32)-(5.33), follow directly
from the definitions in (5.7) and (5.11), using (4.4) and (5.12).

###### Lemma 5

Under assumptions (4.1)-(4.4) and (4.17), the inequalities

$\left\\{\begin{array}[]{c}\displaystyle g_{3,\varepsilon}(x,t)t\leq
c[\bar{G}_{\varepsilon}(x,t)]^{1+\sigma}\\\ \\\
\displaystyle\frac{[g_{3,\varepsilon}(x,t)]^{2}}{g_{1,\varepsilon}(x,t)}\leq
c[\bar{G}_{\varepsilon}(x,t)]^{1+2\sigma}\end{array}\right.$ (5.37)

hold for all $t\geq T$ and for a.e. $x\in\Omega$, where $c\equiv
c(n,N,\nu,\gamma,c_{\textnormal{b}})$.

###### Proof

When $t\in[T,T_{\varepsilon})$, the proof of (5.37) follows directly by the
definitions (5.7), (5.9), (5.11) and from assumption (4.17). Therefore we
restrict ourselves to the case $t\geq T_{\varepsilon}$. For this, we set
$t_{\varepsilon}:=t/T_{\varepsilon}$, and

$\mathcal{Q}_{\varepsilon}(t):=\frac{\bar{G}_{\varepsilon}(x,T_{\varepsilon})}{\bar{G}_{\varepsilon}(x,t)}\leq
1$ (5.38)

and bound via $\eqref{8}_{1}$ and (5.12) as follows:

$\displaystyle g_{3,\varepsilon}(x,t)t$ $\displaystyle\leq
cg_{3}(x,T_{\varepsilon})T_{\varepsilon}\left(\frac{t^{2}+\mu_{\varepsilon}^{2}}{T_{\varepsilon}^{2}+\mu_{\varepsilon}^{2}}\right)^{\frac{\gamma-2}{2}}t_{\varepsilon}^{2}$
$\displaystyle\leq
c[\bar{G}_{\varepsilon}(x,T_{\varepsilon})]^{1+\sigma}t_{\varepsilon}^{\gamma}\leq
c(n,\gamma,c_{\textnormal{b}})[\mathcal{Q}_{\varepsilon}(t)]^{1+\sigma}t_{\varepsilon}^{\gamma}[\bar{G}_{\varepsilon}(x,t)]^{1+\sigma}\,.$
(5.39)

In order to bound $\mathcal{Q}_{\varepsilon}(t)$ we start observing that if
$t_{\varepsilon}\leq 1000$, then (5.37)1 follows using (5.38). When
$t_{\varepsilon}>1000$, we instead estimate

$\displaystyle\mathcal{Q}_{\varepsilon}(t)$
$\displaystyle\stackrel{{\scriptstyle(\ref{gte})}}{{\leq}}$
$\displaystyle\frac{\bar{G}(x,T_{\varepsilon})}{\displaystyle\int_{T_{\varepsilon}}^{t}g_{1,\varepsilon}(x,s)s\,ds+(T^{2}+1)^{\gamma/2}}$
$\displaystyle\stackrel{{\scriptstyle(\ref{g1e})}}{{\leq}}$
$\displaystyle\frac{\bar{G}(x,T_{\varepsilon})}{\frac{\mathfrak{g}_{1}g_{1}(x,T_{\varepsilon})}{\gamma(T_{\varepsilon}^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}}T_{\varepsilon}^{\gamma}\left[\left(t_{\varepsilon}^{2}+(\mu_{\varepsilon}/T_{\varepsilon})^{2}\right)^{\gamma/2}-\left(1+(\mu_{\varepsilon}/T_{\varepsilon})^{2}\right)^{\gamma/2}\right]}$
$\displaystyle\stackrel{{\scriptstyle t_{\varepsilon}>1000}}{{\leq}}$
$\displaystyle\frac{c(\gamma)\bar{G}(x,T_{\varepsilon})(T_{\varepsilon}^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}}{g_{1}(x,T_{\varepsilon})T_{\varepsilon}^{\gamma}t_{\varepsilon}^{\gamma}}$
$\displaystyle\stackrel{{\scriptstyle(\ref{wewewe})}}{{\leq}}$ $\displaystyle
c(\gamma)\left[\frac{g_{1}(x,T_{\varepsilon})T_{\varepsilon}^{2}(T_{\varepsilon}^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}}{g_{1}(x,T_{\varepsilon})T_{\varepsilon}^{\gamma}t_{\varepsilon}^{\gamma}}+\frac{(T^{\gamma}+1)(T_{\varepsilon}^{2}+\mu_{\varepsilon}^{2})^{\frac{\gamma-2}{2}}}{g_{1}(x,T_{\varepsilon})T_{\varepsilon}^{\gamma}t_{\varepsilon}^{\gamma}}\right]$
$\displaystyle\stackrel{{\scriptstyle\eqref{00bis}}}{{\leq}}$
$\displaystyle\frac{c(\gamma)}{t_{\varepsilon}^{\gamma}}\left[1+\frac{T^{\gamma}+1}{\nu
T_{\varepsilon}^{\gamma}}\right]\,.$

It follows that

$\mathcal{Q}_{\varepsilon}(t)\leq\frac{c(\nu,\gamma)}{t_{\varepsilon}^{\gamma}}$
(5.40)

Inserting this last estimate in (5.39), we get

$g_{3,\varepsilon}(x,t)t\leq
ct_{\varepsilon}^{-\sigma\gamma}[\bar{G}_{\varepsilon}(x,t)]^{1+\sigma}\leq
c[\bar{G}_{\varepsilon}(x,t)]^{1+\sigma},$ (5.41)

where $c\equiv c(n,\nu,\gamma,c_{\textnormal{b}})$ and the proof of (5.37)1
follows in the case $t_{\varepsilon}>1000$ too. As for (5.37)2, similarly to
(5.37)1, when $t\in[T,T_{\varepsilon})$ the proof follows from (5.7), (5.9)
and assumption $\eqref{8}_{2}$, while for $t\geq T_{\varepsilon}$, using
$\eqref{8}_{2}$ we have

$\displaystyle\frac{[g_{3,\varepsilon}(x,t)]^{2}}{g_{1,\varepsilon}(x,t)}$
$\displaystyle\leq
c\frac{[g_{3}(x,T_{\varepsilon})]^{2}}{g_{1}(x,T_{\varepsilon})}\left(\frac{t^{2}+\mu_{\varepsilon}^{2}}{T_{\varepsilon}^{2}+\mu_{\varepsilon}^{2}}\right)^{\frac{\gamma-2}{2}}t_{\varepsilon}^{2}$
$\displaystyle\leq
c(n,N,\gamma)[\mathcal{Q}_{\varepsilon}(t)]^{1+2\sigma}t_{\varepsilon}^{\gamma}[\bar{G}_{\varepsilon}(x,t)]^{1+2\sigma}$

and (5.37)2 follows using (5.40) in the last estimate and again arguing as for
(5.41).

###### Lemma 6

Under assumptions (4.1)-(4.4) and (4.18) for some $\sigma\geq 0$, consider a
ball $B\Subset\Omega$ with $\texttt{r}(B)\leq 1$, numbers
$0<\varepsilon_{1}<\varepsilon_{2}\leq\min\\{1,T\\}/4$ and let $w\in
W^{1,\gamma}(B;\mathbb{R}^{N})$. If $\bar{G}(\cdot,\lvert Dw\rvert)\in
L^{p}(B)$ for some $p>1+\sigma$, then

$\int_{B}\left|F_{\varepsilon_{1}}(x,Dw)-F_{\varepsilon_{2}}(x,Dw)\right|\,dx\leq\texttt{o}(\varepsilon_{2})\int_{B}[\bar{G}_{\varepsilon_{1}}(x,\lvert
Dw\rvert)]^{p}\,dx+\texttt{o}(\varepsilon_{2})\,,$ (5.42)

holds, where $\texttt{o}(\varepsilon_{2})$ denotes a quantity such that
$\texttt{o}(\varepsilon_{2})\to 0$ as $\varepsilon_{2}\to 0$, and

$\int_{B}\left|F(x,Dw)-F_{\varepsilon_{2}}(x,Dw)\right|\,dx\leq\texttt{o}(\varepsilon_{2})\int_{B}[\bar{G}(x,\lvert
Dw\rvert)]^{p}\,dx+\texttt{o}(\varepsilon_{2})\,.$ (5.43)

###### Proof

Note that (5.33) implies $G_{\varepsilon_{1}}(x,t)\leq c(\gamma)G(x,t)$, and
therefore we also have that $\bar{G}_{\varepsilon_{1}}(\cdot,\lvert
Dw\rvert)\in L^{p}(B)$. Next, we denote

$\bar{F}_{\varepsilon}(x,t)=\tilde{F}_{\varepsilon}(x,t)-\varepsilon
L_{\gamma,\varepsilon}(t):=\int_{0}^{t}\tilde{a}_{\varepsilon}(x,s)s\,ds$
(5.44)

and in the following we always take $x\in B$. In the case it is
$|z|\leq\varepsilon_{2}$, by (4.4) and (5.12) we easily have

$\displaystyle\left|\bar{F}_{\varepsilon_{1}}(x,|z|)-\bar{F}_{\varepsilon_{2}}(x,|z|)\right|\leq\left|\bar{F}_{\varepsilon_{1}}(x,\varepsilon_{2})\right|+\left|\bar{F}_{\varepsilon_{2}}(x,\varepsilon_{2})\right|$
$\displaystyle\ \leq
c\|\tilde{a}(\cdot,1)\|_{L^{\infty}(B)}\int_{0}^{\varepsilon_{2}}(s^{2}+\mu_{\varepsilon_{1}}^{2})^{\frac{\gamma-2}{2}}s\,ds$
$\displaystyle\qquad\quad+c\|\tilde{a}(\cdot,1)\|_{L^{\infty}(B)}\int_{0}^{\varepsilon_{2}}(s^{2}+\mu_{\varepsilon_{2}}^{2})^{\frac{\gamma-2}{2}}s\,ds\leq\texttt{o}(\varepsilon_{2})\,.$

When $\varepsilon_{2}\leq|z|\leq T_{\varepsilon_{2}}$, recalling that
$\tilde{a}_{\varepsilon_{1}}(x,|z|)\equiv\tilde{a}_{\varepsilon_{2}}(x,|z|)$,
by also using the information in the last display we find

$\left|\bar{F}_{\varepsilon_{1}}(x,|z|)-\bar{F}_{\varepsilon_{2}}(x,|z|)\right|\leq\left|\bar{F}_{\varepsilon_{1}}(x,\varepsilon_{2})\right|+\left|\bar{F}_{\varepsilon_{2}}(x,\varepsilon_{2})\right|\leq\texttt{o}(\varepsilon_{2})\,.$
(5.45)

Finally, note that (4.4) and (5.12) imply

$\tilde{a}_{\varepsilon_{2}}(x,s)\leq
c(\gamma)\tilde{a}_{\varepsilon_{1}}(x,s)\leq
c(\gamma)\bar{a}_{\varepsilon_{1}}(x,s)\quad\mbox{for
$s>T_{\varepsilon_{2}}$}$ (5.46)

therefore, when $|z|>T_{\varepsilon_{2}}$, we get

$\displaystyle\left|\bar{F}_{\varepsilon_{1}}(x,|z|)-\bar{F}_{\varepsilon_{2}}(x,|z|)\right|$
$\displaystyle\stackrel{{\scriptstyle(\ref{triine})}}{{\leq}}$
$\displaystyle\left|\bar{F}_{\varepsilon_{1}}(x,T_{\varepsilon_{2}})-\bar{F}_{\varepsilon_{2}}(x,T_{\varepsilon_{2}})\right|$
$\displaystyle\qquad+c\int_{T_{\varepsilon_{2}}}^{|z|}\tilde{a}_{\varepsilon_{1}}(x,s)s\,ds$
$\displaystyle\stackrel{{\scriptstyle(\ref{dec2})}}{{\leq}}$ $\displaystyle
c\int_{T_{\varepsilon_{2}}}^{|z|}\bar{a}_{\varepsilon_{1}}(x,s)s\,ds+\texttt{o}(\varepsilon_{2})$
$\displaystyle\stackrel{{\scriptstyle\eqref{a1}}}{{\leq}}$ $\displaystyle
c\int_{T_{\varepsilon_{2}}}^{|z|}g_{2,\varepsilon_{1}}(x,s)s\,ds+\texttt{o}(\varepsilon_{2})$
$\displaystyle\stackrel{{\scriptstyle(\ref{z9})}}{{\leq}}$ $\displaystyle
c\int_{T_{\varepsilon_{2}}}^{|z|}[\bar{G}_{\varepsilon_{1}}(x,s)]^{\sigma}g_{1,\varepsilon_{1}}(x,s)s\,ds+\texttt{o}(\varepsilon_{2})$
$\displaystyle\leq$ $\displaystyle
c[\bar{G}_{\varepsilon_{1}}(x,|z|)]^{\sigma}\int_{T_{\varepsilon_{2}}}^{|z|}g_{1,\varepsilon_{1}}(x,s)s\,ds+\texttt{o}(\varepsilon_{2})$
$\displaystyle\leq$ $\displaystyle
c[\bar{G}_{\varepsilon_{1}}(x,|z|)]^{1+\sigma}+\texttt{o}(\varepsilon_{2})\,,$

and we have also used that $s\mapsto\bar{G}_{\varepsilon_{1}}(x,s)$ is non-
decreasing. Using the content of the last four displays, and using also Hölder
inequality, we get

$\displaystyle\int_{B}\left|\bar{F}_{\varepsilon_{1}}(x,|Dw|)-\bar{F}_{\varepsilon_{2}}(x,|Dw|)\right|\,dx$
$\displaystyle\quad\leq\texttt{o}(\varepsilon_{2})|B|+c\int_{B\cap\\{|Dw|>T_{\varepsilon_{2}}\\}}[\bar{G}_{\varepsilon_{1}}(x,\lvert
Dw\rvert)]^{1+\sigma}\,dx$
$\displaystyle\quad\leq\texttt{o}(\varepsilon_{2})|B|+c|B\cap\\{|Dw|>T_{\varepsilon_{2}}\\}|^{\frac{p-1-\sigma}{p}}\left(\int_{B}[\bar{G}_{\varepsilon_{1}}(x,\lvert
Dw\rvert)]^{p}\,dx\right)^{\frac{1+\sigma}{p}}$

for $c\equiv c(\nu,\gamma,p)$. On the other hand, observe that

$\displaystyle|B\cap\\{|Dw|>T_{\varepsilon_{2}}\\}|$ $\displaystyle\leq$
$\displaystyle T_{\varepsilon_{2}}^{-\gamma
p}\int_{B\cap\\{|Dw|>T_{\varepsilon_{2}}\\}}|Dw|^{\gamma p}\,dx$
$\displaystyle\stackrel{{\scriptstyle\eqref{z10a}_{1}}}{{\leq}}$
$\displaystyle c\varepsilon_{2}^{\gamma
p}\int_{B}[\bar{G}_{\varepsilon_{1}}(x,\lvert Dw\rvert)]^{p}\,dx\,.$

Last two inequalities yield

$\displaystyle\int_{B}\left|\bar{F}_{\varepsilon_{1}}(x,|Dw|)-\bar{F}_{\varepsilon_{2}}(x,|Dw|)\right|\,dx$
$\displaystyle\leq|B|\texttt{o}(\varepsilon_{2})$
$\displaystyle\quad+c\varepsilon_{2}^{\gamma(p-1-\sigma)}\int_{B}[\bar{G}_{\varepsilon_{1}}(x,\lvert
Dw\rvert)]^{p}\,dx,$

where $c\equiv c(n,N,\nu,\gamma,c_{\textnormal{a}},c_{\textnormal{b}})$. By
using again (5.19)1, we get

$\displaystyle\int_{B}\left|\varepsilon_{1}L_{\gamma,\varepsilon_{1}}(Dw)-\varepsilon_{2}L_{\gamma,\varepsilon_{2}}(Dw)\right|\,dx$
$\displaystyle\leq c\varepsilon_{2}\int_{B}(|Dw|^{2}+1)^{\gamma/2}\,dx$
$\displaystyle\leq c\varepsilon_{2}\int_{B}[\bar{G}_{\varepsilon_{1}}(x,\lvert
Dw\rvert)]^{p}\,dx\,.$

Combining the content of the last two displays and recalling (5.4) and (5.44)
we arrive at (5.42). As for (5.43), this follows from (5.6)6 and (5.42)
letting $\varepsilon_{1}\to 0$. Indeed, note that Fatou’s lemma works for the
left-hand one; as for the right-hand side, we use again that
$G_{\varepsilon}(x,t)\lesssim G(x,t)$ and the second information in (5.33),
and finally Lebesgue dominated convergence.

##  6 A priori estimates

In this section we develop basic a priori estimates. These are obtained for
solutions to certain elliptic systems associated to the integrands defined in
Section 5, (5.4). Unless differently specified, we shall permanently assume
that set in (4.20) is in force; all properties (4.1)-(4.18) will be therefore
available as well. With $0<\varepsilon<\min\\{1,T\\}/4$ and
$\mathcal{B}\Subset\Omega$ being a ball such that $\texttt{r}(\mathcal{B})\leq
1$, we consider a weak solution $u\in
W^{1,\gamma}(\mathcal{B};\mathbb{R}^{N})$ to the system

$\ \begin{cases}-\,{\rm div}\,a_{\varepsilon}(x,Du)=f\ \ \mbox{in}\
\mathcal{B}\\\ f\in L^{\infty}(\mathcal{B};\mathbb{R}^{N}),\
|f|\leq|\mathfrak{f}|,\ \mbox{where}\ \mathfrak{f}\in
L^{n}(\mathcal{B};\mathbb{R}^{N})\,,\end{cases}$ (6.1)

with $a_{\varepsilon}\colon\Omega\times\mathbb{R}^{N\times
n}\to\mathbb{R}^{N\times n}$ being defined as in (5.1). This setting will be
kept for the rest of Section 6. In particular, all the balls considered in the
following will have radius $\leq 1$. Eventually, we shall consider additional
restrictions of the type $\texttt{r}(\mathcal{B})\leq R_{*}$, where $R_{*}$
will be a (small) threshold radius to be determined as a function of the fixed
parameters of the problem, but independently of the solution $u$ considered.
From (6.1), and taking (5.14) and (5.23) into account, it follows that $u$ is
a minimizer of the functional

$\displaystyle w\mapsto\int_{\mathcal{B}}\left[F_{\varepsilon}(x,Dw)-f\cdot
w\right]\,dx=\int_{\mathcal{B}}\left[\tilde{F}_{\varepsilon}(x,\lvert
Dw\rvert)-f\cdot w\right]\,dx\,.$ (6.2)

We recall that $\bar{a}_{\varepsilon}(\cdot)$, defined in (5.1), is such that
$x\mapsto\bar{a}_{\varepsilon}(x,t)\in W^{1,d}(\Omega)$ for all $t\geq 0$,
$t\mapsto\bar{a}_{\varepsilon}(x,t)\in
W_{\textnormal{loc}}^{1,\infty}[0,\infty)\cap
C^{1}_{\textnormal{loc}}([0,\infty)\setminus\\{\varepsilon,T_{\varepsilon}\\})$
i.e., it is locally $C^{1}$-regular outside
$\\{\varepsilon,T_{\varepsilon}\\}$ and it is such that
$\bar{a}_{\varepsilon}^{\prime}(x,0)=0$ for all $x\in\Omega$. This implies
that $x\mapsto a_{\varepsilon}(x,z)\in W^{1,d}(\Omega;\mathbb{R}^{N\times n})$
for all $z\in\mathbb{R}^{N\times n}$ and $z\mapsto a_{\varepsilon}(x,z)\in
W^{1,\infty}_{\textnormal{loc}}(\mathbb{R}^{N\times n};\mathbb{R}^{N\times
n})$ for all $x\in\Omega$. Finally, the functions
$t\mapsto\bar{a}_{\varepsilon}(\cdot,t)$ and $g_{1,\varepsilon}(\cdot,t)$ are
non-decreasing when $\gamma\geq 2$; this is indeed an easy consequence of
assumption (4.4)1 (see display after (4.5)). A direct consequence of
(5.15)-(5.16) is

$\displaystyle\lvert\bar{a}_{\varepsilon}^{\prime}(x,\lvert
z\rvert)\rvert\lvert z\rvert\leq g_{2,\varepsilon}(x,\lvert
z\rvert)\quad\mbox{for all}\ \ (x,z)\in\Omega\times\\{\lvert z\rvert\geq T,\
\lvert z\rvert\not=T_{\varepsilon}\\}\,.$ (6.3)

Indeed, if $\bar{a}_{\varepsilon}^{\prime}(x,\lvert z\rvert)\geq 0$, then
(6.3) trivially follows from (5.16)2; otherwise (5.16)1, and then (5.15), give
$|\bar{a}_{\varepsilon}^{\prime}(x,\lvert z\rvert)|\lvert
z\rvert=-\bar{a}_{\varepsilon}^{\prime}(x,\lvert z\rvert)\lvert z\rvert\leq-
g_{1,\varepsilon}(x,\lvert z\rvert)+\bar{a}_{\varepsilon}(x,\lvert
z\rvert)\leq\bar{a}_{\varepsilon}(x,\lvert z\rvert)\leq
g_{2,\varepsilon}(x,\lvert z\rvert)$. Similarly to (5.15), by (5.3) and
(5.14), we have that

$\begin{cases}\ \varepsilon\mathfrak{g}_{1}(\lvert
z\rvert^{2}+\mu_{\varepsilon}^{2})^{(\gamma-2)/2}\leq\bar{a}_{\varepsilon}(x,\lvert
z\rvert)\leq\Lambda_{\varepsilon}(\lvert
z\rvert^{2}+\mu_{\varepsilon}^{2})^{(\gamma-2)/2}\\\ \
|\bar{a}_{\varepsilon}^{\prime}(x,\lvert z\rvert)|\leq c_{\varepsilon}(\lvert
z\rvert^{2}+\mu_{\varepsilon}^{2})^{(\gamma-3)/2}\end{cases}$ (6.4)

holds this time for all $(x,z)\in\Omega\times\mathbb{R}^{N\times n}$ (must be
$|z|\not=\varepsilon,T_{\varepsilon}$ for (6.4)2). From (5.13)-(5.14) we can
apply Proposition 6 from Section 12, and this yields

$\displaystyle Du\in
L^{\infty}_{\textnormal{loc}}(\mathcal{B};\mathbb{R}^{N\times
n})\quad\mbox{and}\quad u\in
W^{2,2}_{\textnormal{loc}}(\mathcal{B};\mathbb{R}^{N})\,.$ (6.5)

In turn, this implies that

$a_{\varepsilon}(\cdot,Du)=\bar{a}_{\varepsilon}(\cdot,|Du|)Du\in
W^{1,2}_{\textnormal{loc}}(\mathcal{B};\mathbb{R}^{N\times n})\,.$ (6.6)

This follows for instance from the results in (dele, , Theorem 1.5), together
with the expression (chain rule) of $Da_{\varepsilon}(\cdot,Du)$. It is
sufficient to check that $\bar{a}_{\varepsilon}(\cdot,|Du|)\in
W^{1,2}_{\textnormal{loc}}(\mathcal{B})$ and that the corresponding chain rule
applies to $D_{s}(\bar{a}_{\varepsilon}(\cdot,|Du|))$, for every
$s\in\\{1,\ldots,n\\}$. After extending $t\mapsto\bar{a}_{\varepsilon}(x,t)$
to $\mathbb{R}$ by even reflection, we apply (dele, , Theorem 1.5) to the
vector field $B\colon\mathcal{B}\times\mathbb{R}\mapsto\mathbb{R}^{n}$ defined
by $(B(\cdot))_{j}\equiv(\bar{a}_{\varepsilon}(\cdot)\delta_{sj})$. For this,
note that for every $t\in\mathbb{R}$, $x\mapsto\tilde{a}_{\varepsilon}(x,t)\in
W^{1,n}(\mathcal{B})$ (this is (dele, , Theorem 1.5, (i))), and that
$\partial_{x}\bar{a}_{\varepsilon}(\cdot)=\partial_{x}\tilde{a}_{\varepsilon}(\cdot)$
is still Carathéodory regular; this follows from the definitions (5.2)-(5.3)
and by the fact that $\partial_{x}\tilde{F}^{\prime}$ is Carathéodory regular
by assumptions (this is (dele, , Theorem 1.5, (ii))). The crucial point to
apply the results from dele  is that, as described in (5.6), the set of non-
differentiable points of the (extended) partial function
$t\mapsto\bar{a}_{\varepsilon}(x,t)$ is contained in
$\\{-T_{\varepsilon},-\varepsilon,\varepsilon,T_{\varepsilon}\\}$ for every
$x\in\mathcal{B}$, and it is therefore a null set which is independent of $x$
(this is (dele, , Theorem 1.5, (iii))). Finally, (dele, , Theorem 1.5, (iv))
is verified thanks to (5.13)1 and (6.4)2. Therefore, by (6.5), (dele, ,
Theorem 1.5) applies and (6.6) follows with the corresponding chain rule (see
(6.8) and the related discussion a few lines below). Note that we have used
(6.5), that, together with (5.14)4, also implies
$\partial_{x_{s}}a_{\varepsilon}(\cdot,Du)\in L^{n}_{{\rm
loc}}(\Omega;\mathbb{R}^{N\times n})$.

Let us write, with abuse of notation (keep in mind
$\partial_{z}a_{\varepsilon}(x,z)\equiv\partial_{zz}F_{\varepsilon}(x,z)$ is
not defined when $|z|=\varepsilon,T_{\varepsilon}$)

$\displaystyle(\partial_{z}a_{\varepsilon}(x,z))^{\alpha\beta}_{ij}=\bar{a}_{\varepsilon}(x,\lvert
z\rvert)\delta_{ij}\delta_{\alpha\beta}+\mathds{1}_{\mathcal{D}}(|z|)\bar{a}_{\varepsilon}^{\prime}(x,\lvert
z\rvert)\lvert z\rvert\frac{z^{\alpha}_{i}z^{\beta}_{j}}{\lvert z\rvert^{2}}$
(6.7)

for $z\in\mathbb{R}^{N\times n}\setminus\\{0\\}$, and here we are denoting by
$\mathds{1}_{\mathcal{D}}(\cdot)$ the indicator function of the set
$\mathcal{D}:=\mathbb{R}\setminus\\{\varepsilon,T_{\varepsilon}\\}$. We
explain (6.7) as follows. Recalling that $|Du|\in W^{1,2}_{{\rm loc}}(\Omega)$
by (6.5)-(6.6), we have that

$\displaystyle D_{s}[a_{\varepsilon}(\cdot,Du)]$ $\displaystyle=$
$\displaystyle\partial_{x_{s}}a_{\varepsilon}(x,Du)+\partial_{z}a_{\varepsilon}(x,Du)DD_{s}u$
(6.8) $\displaystyle=$
$\displaystyle\partial_{x_{s}}\bar{a}_{\varepsilon}(x,|Du|)Du+\bar{a}_{\varepsilon}(x,|Du|)\mathds{I}_{N\times
n}DD_{s}u$
$\displaystyle+\mathds{1}_{\mathcal{D}}(|Du|)\bar{a}_{\varepsilon}^{\prime}(x,|Du|)D_{s}|Du|$

holds a.e. in $\mathcal{B}$, for every $s\in\\{1,\ldots,n\\}$. Here we are
using Einstein’s convention on repeated indexes and $\mathds{I}_{N\times
n}\equiv(\delta_{ij}\delta_{\alpha\beta})$. Exactly as in the autonomous case,
the presence of $\mathds{1}_{\mathcal{D}}(|Du|)$ in (6.7)-(6.8) then accounts
for the fact that the term $\bar{a}_{\varepsilon}^{\prime}(x,|Du|)D_{s}|Du|$
is interpreted as zero at those points where $D|Du|=0$, and in particular, for
a.e. $x$ such that $|Du(x)|\in\\{\varepsilon,T_{\varepsilon}\\}$, i.e., where
$\bar{a}_{\varepsilon}^{\prime}(x,|Du(x)|)$ alone does not make sense; see
(dele, , Theorem 1.5). Note that, in particular, from (6.7), (5.13)1,2 and
(5.15), it follows that

$g_{1,\varepsilon}(x,|z|)|\xi|^{2}\leq\partial_{z}a_{\varepsilon}(x,z)\xi\cdot\xi\,,\quad|\partial_{z}a_{\varepsilon}(x,z)|\leq
g_{2,\varepsilon}(x,|z|)\,,$ (6.9)

whenever $|z|\geq T$ and $\xi\in\mathbb{R}^{N\times n}$.

Similar arguments apply to $G_{\varepsilon}(\cdot,|Du|)$, this time using
directly (dele, , Corollary 1.7). Indeed, change/extend
$g_{1,\varepsilon}(\cdot)$ on $\Omega\times(\infty,T)$ as
$g_{1,\varepsilon}(x,t):=g_{1,\varepsilon}(x,T)$ for $t\leq T$ (this is
continuous as $g_{1,\varepsilon}$ is continuous), and, in dele , take
$b\colon\mathcal{B}\times\mathbb{R}\to\mathbb{R}^{n}$ defined as
$(b)_{j}:=(g_{1,\varepsilon}(x,t)t\delta_{ij})_{j}$ for every
$i\in\\{1,\ldots,n\\}$, with $w(\cdot)\equiv T$,
$u(\cdot)\equiv\max\\{|Du|,T\\}\in W^{1,2}_{{\rm loc}}(\mathcal{B})\cap
L^{\infty}_{{\rm loc}}(\mathcal{B})$ (here $u(\cdot)$ is the one from dele ,
and not the solution considered in (6.1)). Finally, note that
$\partial_{x_{i}}g_{1,\varepsilon}\in L^{n}_{{\rm
loc}}(\mathcal{B}\times\mathbb{R})$ by (5.31). Then (dele, , Corollary 1.7)
implies

$G_{\varepsilon}(\cdot,|Du|)\in W^{1,2}_{\textnormal{loc}}(\mathcal{B})$
(6.10)

and

$\displaystyle D_{i}G_{\varepsilon}(x,\lvert Du\rvert)$
$\displaystyle=\int_{T}^{\max\\{\lvert
Du\rvert,T\\}}\partial_{x_{i}}g_{1,\varepsilon}(x,t)t\,dt$
$\displaystyle\qquad+g_{1,\varepsilon}(x,\lvert
Du\rvert)D_{i}\max\\{|Du|,T\\}$

holds a.e. in $\mathcal{B}$. In particular, on $\\{\lvert Du\rvert>T\\}$, we
have

$\displaystyle D_{i}G_{\varepsilon}(x,\lvert
Du\rvert)=g_{1,\varepsilon}(x,\lvert Du\rvert)\lvert Du\rvert D_{i}\lvert
Du\rvert+\int_{T}^{\lvert
Du\rvert}\partial_{x_{i}}g_{1,\varepsilon}(x,t)t\,dt$
$\displaystyle\quad=g_{1,\varepsilon}(x,\lvert
Du\rvert)\sum_{\alpha=1}^{N}\sum_{s=1}^{n}D_{i}D_{s}u^{\alpha}D_{s}u^{\alpha}+\int_{T}^{\lvert
Du\rvert}\partial_{x_{i}}g_{1,\varepsilon}(x,t)t\,dt$ (6.11)

for every $i\in\\{1,\ldots,n\\}$, where we have also used that

$\displaystyle D_{i}\lvert Du\rvert=\frac{1}{\lvert
Du\rvert}\sum_{\alpha=1}^{N}\sum_{s=1}^{n}D_{i}D_{s}u^{\alpha}D_{s}u^{\alpha}\,.$
(6.12)

Note also that $D_{i}G_{\varepsilon}(x,\lvert Du\rvert)\equiv 0$ a.e. on the
complement of $\\{\lvert Du\rvert>T\\}$.

###### Lemma 7

For $\lambda\equiv(\lambda_{i})\in\mathbb{R}^{n}$ and
$z=(z^{\alpha}_{i})\in\mathbb{R}^{N\times n}$, $1\leq i\leq n$ and
$1\leq\alpha\leq N$, with $|z|\geq T$

$\displaystyle\bar{a}_{\varepsilon}(x,\lvert
z\rvert)\lambda\cdot\lambda+\mathds{1}_{\mathcal{D}}(|z|)\bar{a}_{\varepsilon}^{\prime}(x,\lvert
z\rvert)\lvert z\rvert\sum_{\alpha=1}^{N}\frac{\lvert\lambda\cdot
z^{\alpha}\rvert^{2}}{\lvert z\rvert^{2}}$
$\displaystyle\qquad\geq\min\left\\{\bar{a}_{\varepsilon}(x,\lvert
z\rvert),\bar{a}_{\varepsilon}(x,\lvert
z\rvert)+\mathds{1}_{\mathcal{D}}(|z|)\bar{a}_{\varepsilon}^{\prime}(x,\lvert
z\rvert)\lvert z\rvert\right\\}\lvert\lambda\rvert^{2}$
$\displaystyle\qquad\geq g_{1,\varepsilon}(x,\lvert
z\rvert)\lvert\lambda\rvert^{2}$ (6.13)

holds for every $x\in\Omega$.

###### Proof

Indeed, (6.13) is trivial by (5.15) if
$\mathds{1}_{\mathcal{D}}(|z|)\bar{a}_{\varepsilon}^{\prime}(x,\lvert
z\rvert)\geq 0$ (note that (5.15) holds when $|z|=T_{\varepsilon}$ too).
Otherwise, we can estimate simply $\bar{a}_{\varepsilon}^{\prime}(x,\lvert
z\rvert)\lvert\lambda\cdot
z^{\alpha}\rvert^{2}\geq\bar{a}_{\varepsilon}^{\prime}(x,\lvert
z\rvert)|z^{\alpha}|^{2}|\lambda|^{2}$ for every $\alpha$ and then use
$\eqref{a2}_{1}$.

###  6.1 Caccioppoli inequality for powers, when $n>2$

Up to Section 6.3 included, we concentrate on the case $n>2$, and for the
following we set

$m:=\frac{d}{d-2}>1\quad\mbox{and}\quad 1\stackrel{{\scriptstyle
n<d}}{{<}}\chi:=\frac{2^{*}}{2m}=\frac{n}{n-2}\frac{d-2}{d}\,.$ (6.14)

The main result here is

###### Proposition 1

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

$\begin{cases}\sigma+\hat{\sigma}<\frac{s_{*}}{n}-\frac{s_{*}}{d}\\\ 1\leq
s_{*}<\min\left\\{2m(1+\sigma+\hat{\sigma}),\frac{2n}{n-2}\right\\}\,.\end{cases}$
(6.15)

Then, for every $p\in[1,\infty)$, there exists a positive radius $R_{*}\equiv
R_{*}(\texttt{data},\mathfrak{f}(\cdot),p)\linebreak\leq 1$ such that if
$\texttt{r}(\mathcal{B})\leq R_{*}$ and $B_{\varsigma}\Subset B_{\varrho}$ are
concentric balls contained in $\mathcal{B}$, then

$\displaystyle\|G_{\varepsilon}(\cdot,\lvert
Du\rvert)\|_{L^{p}(B_{\varsigma})}\leq\frac{c}{(\varrho-\varsigma)^{\beta_{p}}}\left[\|F_{\varepsilon}(\cdot,Du)\|_{L^{s_{*}}(B_{\varrho})}^{\theta_{p}}+1\right]$
(6.16)

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

Needless to say, (6.15) is implied by (4.15)1 for $s_{*}=1$. The proof of
Proposition 1 will take this and the subsequent Sections 6.2-6.3; in the
following, all the balls considered but $\mathcal{B}$, will be concentric with
$B_{\varsigma}\Subset B_{\varrho}$ from the statement. We recall it is
$\texttt{r}(\mathcal{B})\leq 1$; the size of $R_{*}$ will be chosen in due
course of the proof. Observe that all the foregoing computations, except those
involving $f$, still work in the case $n=2$; this case will be treated in
Section 6.7 below. To start with the proof of Proposition 1, by (6.5)-(6.7) we
pass to the differentiated form of system (6.1), that is

$\displaystyle\sum_{s=1}^{n}\int_{\mathcal{B}}\left[\partial_{z}a_{\varepsilon}(x,Du)DD_{s}u\cdot
D\varphi+\partial_{x_{s}}a_{\varepsilon}(x,Du)\cdot D\varphi+f\cdot
D_{s}\varphi\right]\,dx=0\,,$ (6.17)

which holds for all $\varphi\in W^{1,2}_{0}(\mathcal{B};\mathbb{R}^{N})$ with
compact support in $\mathcal{B}$, by density. Indeed, note that the terms
multiplying $D\varphi$ belong to $L^{2}_{{\rm loc}}(\mathcal{B})$ and
$L^{n}_{{\rm loc}}(\mathcal{B})$; this follows from (5.13)3 and (6.5). We now
consider concentric balls $B_{\varsigma}\subset B_{\tau_{1}}\Subset
B_{\tau_{2}}\subset B_{\varrho}$; in particular it is
$\varsigma\leq\tau_{1}<\tau_{2}\leq\varrho\leq 1$. In (6.17) we take
$\varphi\equiv\varphi_{s}:=\eta^{2}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa+1}D_{s}u$ for $s\in\\{1,\ldots,n\\}$, where $\kappa\geq 0$
and $\eta\in C^{1}_{{\rm c}}(B_{\varrho})$ satisfies
$\mathds{1}_{B_{\tau_{1}}}\leq\eta\leq\mathds{1}_{B_{\tau_{2}}}$ and $\lvert
D\eta\rvert\lesssim 1/(\tau_{2}-\tau_{1})$. Note that (6.5) and (6.10) imply
that $\varphi_{s}\in W^{1,2}_{0}(\mathcal{B};\mathbb{R}^{N})$, has support
contained in $B_{\tau_{2}}$ and, as such, it is admissible in (6.17). It
follows

$\displaystyle D\varphi_{s}=$
$\displaystyle\,\eta^{2}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa+1}DD_{s}u$
$\displaystyle+(\kappa+1)\eta^{2}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa}D_{s}u\otimes DG_{\varepsilon}(x,\lvert Du\rvert)$
$\displaystyle+2\eta[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa+1}D_{s}u\otimes D\eta\,.$ (6.18)

By (6.5) and (6.10) it follows that $\varphi\in
W^{1,2}_{0}(\mathcal{B};\mathbb{R}^{N})$ and it is therefore admissible in
(6.17). By the last identity we have

$\displaystyle\sum_{s=1}^{n}\int_{\mathcal{B}}\partial_{z}a_{\varepsilon}(x,Du)DD_{s}u\cdot
D\varphi_{s}\,dx$
$\displaystyle\quad=\sum_{s=1}^{n}\int_{\mathcal{B}}\eta^{2}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa+1}\partial_{z}a_{\varepsilon}(x,Du)DD_{s}u\cdot
DD_{s}u\,dx$
$\displaystyle\qquad+(\kappa+1)\sum_{s=1}^{n}\int_{\mathcal{B}}\eta^{2}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa}\partial_{z}a_{\varepsilon}(x,Du)DD_{s}u$
$\displaystyle\hskip 147.95433pt\cdot\left(D_{s}u\otimes
DG_{\varepsilon}(x,\lvert Du\rvert)\right)\,dx$
$\displaystyle\qquad+2\sum_{s=1}^{n}\int_{\mathcal{B}}\eta[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa+1}\partial_{z}a_{\varepsilon}(x,Du)DD_{s}u\cdot(D_{s}u\otimes
D\eta)\,dx$
$\displaystyle\quad=:\mbox{(I)}_{z}+\mbox{(II)}_{z}+\mbox{(III)}_{z}\,.$
(6.19)

Note that in the display above and in the following ones until (6.25), as
$G_{\varepsilon}(x,t)\equiv 0$ for $t\leq T$, all the integrals above actually
extend only over $B\cap\\{\lvert Du\rvert>T\\}$, therefore we can always use
(6) when computing the derivatives of $G_{\varepsilon}$. To proceed, we have

$\displaystyle\mbox{(I)}_{z}\stackrel{{\scriptstyle(\ref{inaggiunta})}}{{\geq}}$
$\displaystyle\int_{\mathcal{B}}\eta^{2}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa+1}g_{1,\varepsilon}(x,\lvert Du\rvert)\lvert
D^{2}u\rvert^{2}\,dx=:\mathcal{S}_{1}\,.$

We temporarily shorten the notation as follows (recall (4.12)1):

$\displaystyle\mathcal{H}_{\kappa}(Du):=(\kappa+1)[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa}\frac{\bar{a}_{\varepsilon}(x,\lvert
Du\rvert)}{g_{1,\varepsilon}(x,\lvert Du\rvert)}\,,$
$\displaystyle\mathcal{H}_{\kappa}^{\prime}(Du):=(\kappa+1)\mathds{1}_{\mathcal{D}}(|Du|)[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa}\frac{\bar{a}_{\varepsilon}^{\prime}(x,\lvert
Du\rvert)\lvert Du\rvert}{g_{1,\varepsilon}(x,\lvert Du\rvert)}\,.$

As all the integrals extend over $\\{|Du|>T\\}$ by (4.6), it is
$g_{1,\varepsilon}(x,\lvert Du\rvert)>0$. Recalling (6.7)-(6.12), we then re-
write

$\displaystyle\mbox{(II)}_{z}$
$\displaystyle=\int_{\mathcal{B}}\eta^{2}\mathcal{H}_{\kappa}(Du)DG_{\varepsilon}(x,\lvert
Du\rvert)\cdot DG_{\varepsilon}(x,\lvert Du\rvert)\,dx$ $\displaystyle\hskip
17.07164pt+\sum_{\alpha=1}^{N}\int_{\mathcal{B}}\eta^{2}\mathcal{H}_{\kappa}^{\prime}(Du)\frac{\left(DG_{\varepsilon}(x,\lvert
Du\rvert)\cdot Du^{\alpha}\right)^{2}}{\lvert Du\rvert^{2}}\,dx$
$\displaystyle\hskip
17.07164pt-\int_{\mathcal{B}}\eta^{2}\mathcal{H}_{\kappa}(Du)\int_{T}^{\lvert
Du\rvert}\partial_{x}g_{1,\varepsilon}(x,t)t\,dt\cdot
DG_{\varepsilon}(x,\lvert Du\rvert)\,dx$ $\displaystyle\hskip
17.07164pt-\sum_{\alpha=1}^{N}\int_{\mathcal{B}}\eta^{2}\mathcal{H}_{\kappa}^{\prime}(Du)\left(\int_{T}^{\lvert
Du\rvert}\partial_{x}g_{1,\varepsilon}(x,t)t\,dt\cdot Du^{\alpha}\right)$
$\displaystyle\hskip 122.34685pt\times\frac{(DG_{\varepsilon}(x,\lvert
Du\rvert)\cdot Du^{\alpha})}{\lvert Du\rvert^{2}}\,dx$
$\displaystyle=:\mbox{(II)}_{z,1}+\mbox{(II)}_{z,2}+\mbox{(II)}_{z,3}+\mbox{(II)}_{z,4}\,.$
(6.20)

We now observe that

$\mbox{(II)}_{z,1}+\mbox{(II)}_{z,2}\geq(\kappa+1)\int_{\mathcal{B}}\eta^{2}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa}\lvert DG_{\varepsilon}(x,\lvert
Du\rvert)\rvert^{2}\,dx=:\mathcal{S}_{2}\,.$ (6.21)

Indeed, this follows from (6.13) with $\lambda\equiv DG_{\varepsilon}$ and
$z\equiv Du$. Then we have, by using (5.15), (5.29), (5.31) and (6.3)

$\displaystyle\ \ |\mbox{(II)}_{z,3}|+|\mbox{(II)}_{z,4}|\leq
c(\kappa+1)\int_{\mathcal{B}}\eta^{2}h(x)[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa}\frac{g_{2,\varepsilon}(x,\lvert
Du\rvert)}{g_{1,\varepsilon}(x,\lvert Du\rvert)}$ $\displaystyle\hskip
99.58464pt\times\left(\int_{T}^{\lvert
Du\rvert}\partial_{t}[\bar{G}_{\varepsilon}(x,t)]^{1+\hat{\sigma}}\,dt\right)\lvert
DG_{\varepsilon}(x,\lvert Du\rvert)\rvert\,dx$ $\displaystyle\quad\leq
c(\kappa+1)\int_{\mathcal{B}}\eta^{2}h(x)[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa}[\bar{G}_{\varepsilon}(x,\lvert
Du\rvert)]^{1+\sigma+\hat{\sigma}}\lvert DG_{\varepsilon}(x,\lvert
Du\rvert)\rvert\,dx$
$\displaystyle\quad\leq\bar{\varepsilon}\mathcal{S}_{2}+\frac{c(\kappa+1)}{\bar{\varepsilon}}\int_{\mathcal{B}}\eta^{2}[h(x)]^{2}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa}[\bar{G}_{\varepsilon}(x,\lvert
Du\rvert)]^{2(1+\sigma+\hat{\sigma})}\,dx$
$\displaystyle\quad\leq\bar{\varepsilon}\mathcal{S}_{2}+\frac{c(\kappa+1)\lVert
h\rVert_{L^{d}(\mathcal{B})}^{2}}{\bar{\varepsilon}}$ $\displaystyle\hskip
56.9055pt\times\left(\int_{\mathcal{B}}\eta^{2m}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{m\kappa}[\bar{G}_{\varepsilon}(x,\lvert
Du\rvert)]^{2m(1+\sigma+\hat{\sigma})}\,dx\right)^{1/m}\,,$

where $c\equiv c(n,N,\nu,\gamma,c_{\textnormal{b}})$, and arbitrary
$\bar{\varepsilon}\in(0,1)$. Note that in last two lines we have used Young
and Hölder’s inequalities. Similarly to (6.20), we also have

$\displaystyle\mbox{(III)}_{z}$ $\displaystyle=2\int_{\mathcal{B}}\frac{\eta
G_{\varepsilon}(x,\lvert
Du\rvert)\mathcal{H}_{\kappa}(Du)}{\kappa+1}DG_{\varepsilon}(x,\lvert
Du\rvert)\cdot D\eta\,dx$ $\displaystyle\ \
+2\sum_{\alpha=1}^{N}\int_{\mathcal{B}}\frac{\eta G_{\varepsilon}(x,\lvert
Du\rvert)\mathcal{H}_{\kappa}^{\prime}(Du)}{(\kappa+1)|Du|^{2}}(DG_{\varepsilon}(x,\lvert
Du\rvert)\cdot Du^{\alpha})(Du^{\alpha}\cdot D\eta)\,dx$ $\displaystyle\ \
-2\int_{\mathcal{B}}\frac{\eta G_{\varepsilon}(x,\lvert
Du\rvert)\mathcal{H}_{\kappa}(Du)}{\kappa+1}\left(\int_{T}^{\lvert
Du\rvert}\partial_{x}g_{1,\varepsilon}(x,t)t\,dt\cdot D\eta\right)\,dx$
$\displaystyle\ \ -2\sum_{\alpha=1}^{N}\int_{\mathcal{B}}\frac{\eta
G_{\varepsilon}(x,\lvert
Du\rvert)\mathcal{H}_{\kappa}^{\prime}(Du)}{(\kappa+1)|Du|^{2}}$
$\displaystyle\ \ \qquad\quad\times\left(\int_{T}^{\lvert
Du\rvert}\partial_{x}g_{1,\varepsilon}(x,t)t\,dt\cdot
Du^{\alpha}\right)(Du^{\alpha}\cdot D\eta)\,dx$
$\displaystyle=:\mbox{(III)}_{z,1}+\mbox{(III)}_{z,2}+\mbox{(III)}_{z,3}+\mbox{(III)}_{z,4}\,.$

Using again (5.15), (5.29), (6.3) and Young inequality, we have, for every
$\bar{\varepsilon}\in(0,1)$

$\displaystyle|\mbox{(III)}_{z,1}|+|\mbox{(III)}_{z,2}|$ $\displaystyle\ \leq
c\int_{\mathcal{B}}\eta[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa+1}\frac{g_{2,\varepsilon}(x,\lvert
Du\rvert)}{g_{1,\varepsilon}(x,\lvert Du\rvert)}\lvert
DG_{\varepsilon}(x,\lvert Du\rvert)\rvert|D\eta|\,dx$ $\displaystyle\ \leq
c\int_{\mathcal{B}}\eta[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa+1}[\bar{G}_{\varepsilon}(x,\lvert Du\rvert)]^{\sigma}\lvert
DG_{\varepsilon}(x,\lvert Du\rvert)\rvert|D\eta|\,dx$ $\displaystyle\
\leq\bar{\varepsilon}\mathcal{S}_{2}+\frac{c}{\bar{\varepsilon}(\kappa+1)}\int_{\mathcal{B}}\lvert
D\eta\rvert^{2}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa}[\bar{G}_{\varepsilon}(x,\lvert
Du\rvert)]^{2(1+\sigma)}\,dx$ $\displaystyle\
\leq\bar{\varepsilon}\mathcal{S}_{2}+\frac{c}{\bar{\varepsilon}(\kappa+1)}\left(\int_{\mathcal{B}}\lvert
D\eta\rvert^{2m}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{m\kappa}[\bar{G}_{\varepsilon}(x,\lvert
Du\rvert)]^{2m(1+\sigma)}\,dx\right)^{1/m}$

(recall that $\bar{G}_{\varepsilon}\geq 1$) and, using now also (5.31), we get

$\displaystyle|\mbox{(III)}_{z,3}|+|\mbox{(III)}_{z,4}|$ $\displaystyle\ \leq
c\int_{\mathcal{B}}\eta h(x)[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa+1}[\bar{G}_{\varepsilon}(x,|Du|)]^{1+\sigma+\hat{\sigma}}\lvert
D\eta\rvert\,dx$ $\displaystyle\ \leq
c\int_{\mathcal{B}}\eta^{2}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa+2}\,dx$ $\displaystyle\quad\ +c\int_{\mathcal{B}}\lvert
h(x)\rvert^{2}\lvert D\eta\rvert^{2}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa}[\bar{G}_{\varepsilon}(x,\lvert
Du\rvert)]^{2(1+\sigma+\hat{\sigma})}\,dx$ $\displaystyle\ \leq
c\int_{\mathcal{B}}\eta^{2}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa}[\bar{G}_{\varepsilon}(x,\lvert
Du\rvert)]^{2(1+\sigma+\hat{\sigma})}\,dx$ $\displaystyle\quad\ +c\lVert
h\rVert^{2}_{L^{d}(\mathcal{B})}\left(\int_{\mathcal{B}}\lvert
D\eta\rvert^{2m}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{m\kappa}[\bar{G}_{\varepsilon}(x,\lvert
Du\rvert)]^{2m(1+\sigma+\hat{\sigma})}\,dx\right)^{1/m}$ $\displaystyle\ \leq
c\left(\lVert h\rVert^{2}_{L^{d}(\mathcal{B})}+1\right)$ (6.22)
$\displaystyle\ \quad\times\left(\int_{\mathcal{B}}(\eta^{2m}+\lvert
D\eta\rvert^{2m})[G_{\varepsilon}(x,\lvert
Du\rvert)]^{m\kappa}[\bar{G}_{\varepsilon}(x,\lvert
Du\rvert)]^{2m(1+\sigma+\hat{\sigma})}\,dx\right)^{1/m}$

with $c\equiv c(n,N,\nu,\gamma,c_{\textnormal{b}})$. Now we look at the second
group of terms stemming from (6.17)

$\displaystyle\sum_{s=1}^{n}\int_{\mathcal{B}}\partial_{x_{s}}a_{\varepsilon}(x,Du)\cdot
D\varphi_{s}\,dx$
$\displaystyle\stackrel{{\scriptstyle(\ref{latesta})}}{{=}}\sum_{s=1}^{n}\int_{\mathcal{B}}\eta^{2}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa+1}\partial_{x_{s}}a_{\varepsilon}(x,Du)\cdot DD_{s}u\,dx$
$\displaystyle\
+(\kappa+1)\sum_{s=1}^{n}\int_{\mathcal{B}}\eta^{2}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa}\partial_{x_{s}}a_{\varepsilon}(x,Du)\cdot(D_{s}u\otimes
DG_{\varepsilon}(x,\lvert Du\rvert))\,dx$ $\displaystyle\
+2\sum_{s=1}^{n}\int_{\mathcal{B}}\eta[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa+1}\partial_{x_{s}}a_{\varepsilon}(x,Du)\cdot(D_{s}u\otimes
D\eta)\,dx$
$\displaystyle\quad=:\mbox{(I)}_{x}+\mbox{(II)}_{x}+\mbox{(III)}_{x}\,.$

From $\eqref{growthd}_{3}$,$\eqref{z11}_{2}$, Hölder and Young inequalities,
we obtain, for arbitrary $\bar{\varepsilon}\in(0,1)$

$\displaystyle\lvert\mbox{(I)}_{x}\rvert\leq\,c\int_{\mathcal{B}}\eta^{2}h(x)[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa+1}g_{3,\varepsilon}(x,\lvert Du\rvert)\lvert
D^{2}u\rvert\,dx$ $\displaystyle\
\leq\bar{\varepsilon}\mathcal{S}_{1}+\frac{c}{\bar{\varepsilon}}\int_{\mathcal{B}}\eta^{2}[h(x)]^{2}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa+1}\frac{[g_{3,\varepsilon}(x,\lvert
Du\rvert)]^{2}}{g_{1,\varepsilon}(x,\lvert Du\rvert)}\,dx$ $\displaystyle\
\leq\bar{\varepsilon}\mathcal{S}_{1}+\frac{c}{\bar{\varepsilon}}\lVert
h\rVert^{2}_{L^{d}(\mathcal{B})}\left(\int_{\mathcal{B}}\eta^{2m}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{m\kappa}[\bar{G}_{\varepsilon}(x,\lvert
Du\rvert)]^{2m(1+\sigma)}\,dx\right)^{1/m},$

where $c\equiv c(n,N,\nu,\gamma,c_{\textnormal{b}})$. Using this time
$\eqref{z11}_{1}$ we get

$\displaystyle\lvert\mbox{(II)}_{x}\rvert\leq$
$\displaystyle\,c(\kappa+1)\int_{\mathcal{B}}\eta^{2}h(x)[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa}g_{3,\varepsilon}(x,\lvert Du\rvert)\lvert Du\rvert\lvert
DG_{\varepsilon}(x,\lvert Du\rvert)\rvert\,dx$ $\displaystyle\leq$
$\displaystyle\,\bar{\varepsilon}\mathcal{S}_{2}+\frac{c(\kappa+1)}{\bar{\varepsilon}}\int_{\mathcal{B}}\eta^{2}[h(x)]^{2}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa}[g_{3,\varepsilon}(x,\lvert Du\rvert)]^{2}\lvert
Du\rvert^{2}\,dx$ $\displaystyle\leq$
$\displaystyle\,\bar{\varepsilon}\mathcal{S}_{2}+\frac{c(\kappa+1)}{\bar{\varepsilon}}\lVert
h\rVert^{2}_{L^{d}(\mathcal{B})}$ $\displaystyle\hskip
28.45274pt\times\left(\int_{\mathcal{B}}\eta^{2m}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{m\kappa}[\bar{G}_{\varepsilon}(x,\lvert
Du\rvert)]^{2m(1+\sigma)}\,dx\right)^{1/m}$

with $c\equiv c(n,N,\nu,\gamma,c_{\textnormal{b}})$. Again, $\eqref{z11}_{1}$,
Young and Hölder inequalities, as in (6.22), give

$\displaystyle\lvert\mbox{(III)}_{x}\rvert\leq$
$\displaystyle\,c\int_{\mathcal{B}}\eta h(x)[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa+1}g_{3,\varepsilon}(x,\lvert Du\rvert)\lvert Du\rvert\lvert
D\eta\rvert\,dx$ $\displaystyle\leq$
$\displaystyle\,c\int_{\mathcal{B}}\eta^{2}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa+2}\,dx$ $\displaystyle+c\int_{\mathcal{B}}\lvert
D\eta\rvert^{2}[h(x)]^{2}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa}[g_{3,\varepsilon}(x,\lvert Du\rvert)]^{2}\lvert
Du\rvert^{2}\,dx$ $\displaystyle\leq$ $\displaystyle\,c\left(\lVert
h\rVert^{2}_{L^{d}(\mathcal{B})}+1\right)$ (6.23)
$\displaystyle\times\left(\int_{\mathcal{B}}(\eta^{2m}+\lvert
D\eta\rvert^{2m})[G_{\varepsilon}(x,\lvert
Du\rvert)]^{m\kappa}[\bar{G}_{\varepsilon}(x,\lvert
Du\rvert)]^{2m(1+\sigma)}\,dx\right)^{1/m}$

where $c\equiv c(n,N,\nu,\gamma,c_{\textnormal{b}})$. Finally, we examine the
contributions to (6.17) coming from the terms featuring $f$:

$\displaystyle\sum_{s=1}^{n}\int_{\mathcal{B}}f\cdot
D_{s}\varphi_{s}\,dx\stackrel{{\scriptstyle(\ref{latesta})}}{{=}}\sum_{s=1}^{n}\int_{\mathcal{B}}\eta^{2}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa+1}(f\cdot D_{s}D_{s}u)\,dx$
$\displaystyle\qquad+(\kappa+1)\sum_{s=1}^{n}\int_{\mathcal{B}}\eta^{2}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa}D_{s}G_{\varepsilon}(x,\lvert Du\rvert)(f\cdot D_{s}u)\,dx$
$\displaystyle\qquad+2\sum_{s=1}^{n}\int_{\mathcal{B}}\eta[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa+1}D_{s}\eta(f\cdot D_{s}u)\,dx$
$\displaystyle=:\,\mbox{(I)}_{f}+\mbox{(II)}_{f}+\mbox{(III)}_{f}\,.$

Here recall that $n>2$. Using $\eqref{z10a}_{2}$, Hölder and Young
inequalities we get

$\displaystyle\lvert\mbox{(I)}_{f}\rvert\leq$
$\displaystyle\,c\int_{\mathcal{B}}\eta^{2}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa+1}\lvert f\rvert\lvert D^{2}u\rvert\,dx$
$\displaystyle\leq$
$\displaystyle\,\bar{\varepsilon}\mathcal{S}_{1}+\frac{c}{\bar{\varepsilon}}\int_{\mathcal{B}}\eta^{2}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa}\lvert f\rvert^{2}[g_{1,\varepsilon}(x,\lvert
Du\rvert)]^{-1}G_{\varepsilon}(x,\lvert Du\rvert)\,dx$ $\displaystyle\leq$
$\displaystyle\,\bar{\varepsilon}\mathcal{S}_{1}+\frac{c}{\bar{\varepsilon}}\int_{\mathcal{B}}\eta^{2}\lvert
f\rvert^{2}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa}[\bar{G}_{\varepsilon}(x,\lvert Du\rvert)]^{2/\gamma}\,dx$
$\displaystyle\leq$
$\displaystyle\,\bar{\varepsilon}\mathcal{S}_{1}+\frac{c}{\bar{\varepsilon}}\lVert
f\rVert_{L^{n}(\mathcal{B})}^{2}\left(\int_{\mathcal{B}}\eta^{2^{*}}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{2^{*}\kappa/2}[\bar{G}_{\varepsilon}(x,\lvert
Du\rvert)]^{2^{*}}\,dx\right)^{2/2^{*}}\,,$

with $c\equiv c(n,N,\nu,\gamma)$ and $2^{*}$ is as in (2.2). Moreover, by
$\eqref{z10a}_{1}$, we get

$\displaystyle\lvert\mbox{(II)}_{f}\rvert\leq$
$\displaystyle\,c(\kappa+1)\int_{\mathcal{B}}\eta^{2}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa}\lvert f\rvert\lvert DG_{\varepsilon}(x,\lvert
Du\rvert)\rvert\lvert Du\rvert\,dx$ $\displaystyle\leq$
$\displaystyle\,\bar{\varepsilon}\mathcal{S}_{2}+\frac{c(\kappa+1)}{\bar{\varepsilon}}\int_{\mathcal{B}}\eta^{2}\lvert
f\rvert^{2}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa}[\bar{G}_{\varepsilon}(x,\lvert Du\rvert)]^{2/\gamma}\,dx$
$\displaystyle\leq$
$\displaystyle\,\bar{\varepsilon}\mathcal{S}_{2}+\frac{c(\kappa+1)\lVert
f\rVert^{2}_{L^{n}(\mathcal{B})}}{\bar{\varepsilon}}$
$\displaystyle\qquad\quad\times\left(\int_{\mathcal{B}}\eta^{2^{*}}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{2^{*}\kappa/2}[\bar{G}_{\varepsilon}(x,\lvert
Du\rvert)]^{2^{*}}\,dx\right)^{2/2^{*}}\,,$

for $c\equiv c(n,N,\nu,\gamma)$. Finally, again from $\eqref{z10a}_{1}$ we
deduce that

$\displaystyle\lvert\mbox{(III)}_{f}\rvert\leq
c\int_{\mathcal{B}}\eta[G_{\varepsilon}(x,\lvert Du\rvert)]^{\kappa+1}\lvert
f\rvert\lvert D\eta\rvert\lvert Du\rvert\,dx$ $\displaystyle\quad\leq
c\int_{\mathcal{B}}\lvert D\eta\rvert^{2}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa+2}\,dx$
$\displaystyle\qquad\qquad+c\int_{\mathcal{B}}\eta^{2}\lvert
f\rvert^{2}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\kappa}[\bar{G}_{\varepsilon}(x,\lvert Du\rvert)]^{2/\gamma}\,dx$
$\displaystyle\quad\leq c\left(\int_{\mathcal{B}}\lvert
D\eta\rvert^{2m}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{m\kappa}[\bar{G}_{\varepsilon}(x,\lvert
Du\rvert)]^{2m}\,dx\right)^{1/m}$ $\displaystyle\qquad\ +c\lVert
f\rVert_{L^{n}(\mathcal{B})}^{2}\left(\int_{\mathcal{B}}\eta^{2^{*}}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{2^{*}\kappa/2}[\bar{G}_{\varepsilon}(x,\lvert
Du\rvert)]^{2^{*}}\,dx\right)^{2/2^{*}}$ (6.24)

where $c\equiv c(n,N,\nu,\gamma)$. In the previous three displays, we also
used that, by $\eqref{gte}_{2}$ it is $\bar{G}_{\varepsilon}(\cdot)\geq 1$ for
all $(x,t)\in\Omega\times[0,\infty)$, thus, for instance
$[\bar{G}_{\varepsilon}(\cdot)]^{1/\gamma}\leq\bar{G}_{\varepsilon}(\cdot)$.

Merging the content of displays from (6.19) to (6.24), with (6.17), choosing
$\bar{\varepsilon}>0$ small enough (in order to reabsorb terms in the usual
way), we get, after a few standard manipulations, and again using that
$\bar{G}_{\varepsilon}(\cdot)\geq 1$,

$\displaystyle\mathcal{S}_{1}+\mathcal{S}_{2}\leq c(\kappa+1)\left(\lVert
h\rVert_{L^{d}(\mathcal{B})}^{2}+1\right)$ (6.25)
$\displaystyle\quad\times\left(\int_{\mathcal{B}}(\eta^{2m}+\lvert
D\eta\rvert^{2m})[G_{\varepsilon}(x,\lvert
Du\rvert)]^{m\kappa}[\bar{G}_{\varepsilon}(x,\lvert
Du\rvert)]^{2m(1+\sigma+\hat{\sigma})}\,dx\right)^{1/m}$
$\displaystyle\qquad+c(\kappa+1)\lVert
f\rVert_{L^{n}(\mathcal{B})}^{2}\left(\int_{\mathcal{B}}\eta^{2^{*}}[G_{\varepsilon}(x,\lvert
Du\rvert)]^{2^{*}\kappa/2}[\bar{G}_{\varepsilon}(x,\lvert
Du\rvert)]^{2^{*}}\,dx\right)^{2/2^{*}}\,,$

with $c\equiv c(n,N,\nu,\gamma,c_{\textnormal{b}})$. Now we note that, by
$\eqref{gte}_{2}$ it follows

$\displaystyle
G_{\varepsilon}(x,t)\leq\bar{G}_{\varepsilon}(x,t)\leq(T^{2}+1)^{\gamma/2}\left[G_{\varepsilon}(x,t)+1\right]\,,$
(6.26)

for all $(x,t)\in\Omega\times[0,\infty)$, therefore, recalling (6.21),
estimate (6.25) can be rearranged as

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

where $c\equiv c(n,N,\nu,\gamma,T,c_{\textnormal{b}})$ and every number
$\kappa\geq 0$. Note that this is the first time a dependence of the constants
on $T$ appears, via (6.26); no blow-up occurs when $T\to 0$, compare with
Remark 5. Sobolev-Poincaré inequality and the previous display give

$\displaystyle\left(\int_{\mathcal{B}}\eta^{2^{*}}\left[[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\frac{2^{*}}{2}(\kappa+2)}+1\right]\,dx\right)^{2/2^{*}}$
$\displaystyle\leq\left(\int_{\mathcal{B}}\eta^{2^{*}}\left[[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\frac{\kappa}{2}+1}+1\right]^{2^{*}}\,dx\right)^{2/2^{*}}$
$\displaystyle\leq c\int_{\mathcal{B}}\lvert D(\eta([G_{\varepsilon}(x,\lvert
Du\rvert)]^{\frac{\kappa}{2}+1}+1))\rvert^{2}\,dx$ $\displaystyle\leq
c(\kappa+1)^{2}\left(\lVert h\rVert_{L^{d}(\mathcal{B})}^{2}+1\right)$ (6.27)
$\displaystyle\quad\times\left(\int_{\mathcal{B}}(\eta^{2m}+\lvert
D\eta\rvert^{2m})\left[[G_{\varepsilon}(x,\lvert
Du\rvert)]^{m(\kappa+2+2\sigma+2\hat{\sigma})}+1\right]\,dx\right)^{1/m}$
$\displaystyle\qquad\ \ +c_{\textnormal{ab}}(\kappa+1)^{2}\lVert
f\rVert_{L^{n}(\mathcal{B})}^{2}\left(\int_{\mathcal{B}}\eta^{2^{*}}\left[[G_{\varepsilon}(x,\lvert
Du\rvert)]^{\frac{2^{*}}{2}(\kappa+2)}+1\right]\,dx\right)^{2/2^{*}},$

with $c,c_{\textnormal{ab}}\equiv
c,c_{\textnormal{ab}}(n,N,\nu,\gamma,T,c_{\textnormal{b}})$. We now determine
the radius $R_{*}$ such that $\texttt{r}(\mathcal{B})\leq R_{*}$ as in the
statement of Proposition 1. Specifically, we fix $\bar{\kappa}\geq 0$ and,
using the absolute continuity of the integral, determine $R_{*}\equiv
R_{*}(n,N,\nu,\gamma,\linebreak
T,c_{\textnormal{b}},\mathfrak{f}(\cdot),\bar{\kappa})\in(0,1)$, such that

$\displaystyle c_{\textnormal{ab}}(\bar{\kappa}+1)^{2}\lVert
f\rVert_{L^{n}(\mathcal{B})}^{2}\stackrel{{\scriptstyle(\ref{a0})}}{{\leq}}c_{\textnormal{ab}}(\bar{\kappa}+1)^{2}\lVert\mathfrak{f}\rVert_{L^{n}(\mathcal{B})}^{2}\leq
1/2\,.$ (6.28)

With (6.28) being now in force, for all $\kappa\leq\bar{\kappa}$ estimate
(6.27) becomes

$\displaystyle\left(\int_{\mathcal{B}}\eta^{2^{*}}\left[[G_{\varepsilon}(x,\lvert