# Lipschitz regularity of almost minimizers in a Bernoulli problem with non-
standard growth

by
João Vitor da Silva,111João Vitor da Silva. Universidade Estadual de Campinas
- UNICAMP. Department of Mathematics. Campinas - SP, Brazil. E-mail address:
<EMAIL_ADDRESS>Analía Silva222Analía Silva. Departamento de Matemática,
FCFMyN, Universidad Nacional de San Luis and Instituto de Matemática Aplicada
San Luis (IMASL), UNSL - CONICET. Ejercito de los Andes 950, D5700HHW, San
Luis, Argentina. E-mail address<EMAIL_ADDRESS>
$\&$
Hernán Vivas333Hernán Vivas. Centro Marplatense de Investigaciones Matemáticas
(CEMIM), Universidad Nacional de Mar del Plata, CIC, CONICET. Deán Funes 3350,
7600, Mar del Plata, Argentina. E-mail address<EMAIL_ADDRESS>
###### Abstract

In this work we establish the optimal Lipschitz regularity for non-negative
almost minimizers of the one-phase Bernoulli-type functional

$\mathcal{J}_{\mathrm{G}}(u,\Omega)\mathrel{\mathop{:}}=\int_{\Omega}\left(\mathrm{G}(|\nabla
u|)+\chi_{\\{u>0\\}}\right)\,dx$

where $\Omega\subset\mathbb{R}^{n}$ is a bounded domain and
$\mathrm{G}:[0,\infty)\to[0,\infty)$ is a Young function with
$\mathrm{G}^{\prime}=g$ satisfying the Lieberman’s classical conditions.
Moreover, of independent mathematical interest, we also address a Höder
regularity characterization via Campanato-type estimates in the context of
Orlicz modulars, which is new for such a class of non-standard growth
functionals.

Keywords: Almost minimizers, Lipschitz regularity, Orlicz spaces

MSC 2020: 35B65, 35J70, 35J87, 46E30.

##  1 Introduction and main result

In this manuscript we deal with one-phase problems driven by the
$\mathrm{G}-$energy functionals, with $\mathrm{G}$ a suitable function with
non-polynomial growth. Specifically, we study nonnegative almost minimizers
for one-phase Bernoulli-type functional as follows

$\mathcal{J}_{\mathrm{G}}(u,\Omega)\mathrel{\mathop{:}}=\int_{\Omega}\left(\mathrm{G}(|\nabla
u|)+\chi_{\\{u>0\\}}\right)\,dx$

where $\Omega\subset\mathbb{R}^{n}$ is a bounded Lipschitz domain and
$\mathrm{G}:[0,\infty)\to[0,\infty)$ is a Young function with
$\mathrm{G}^{\prime}=g$ satisfying

(1.1) $\delta\leq\frac{tg^{\prime}(t)}{g(t)}\leq g_{0}\qquad\text{for some
constants}\qquad 1<\delta\leq g_{0}<\infty\quad(\text{Lieberman's
condition}).$

In our research, almost minimizers mean that we will consider functions $0\leq
u\in W^{1,\mathrm{G}}(\Omega)$ such that for some $\beta>0$ and constant
$\kappa\geq 0$ it holds that

(1.2) $\mathcal{J}_{\mathrm{G}}(u,B_{r}(x))\leq(1+\kappa
r^{\beta})\mathcal{J}_{\mathrm{G}}(v,B_{r}(x))$

for any ball $B_{r}(x)$ such that $\overline{B_{r}(x)}\subset\Omega$ and any
$v\in W^{1,\mathrm{G}}(B_{r}(x))$ such that $v=u$ on $\partial B_{r}(x)$ in
the sense of traces (see Definition 2.7 and Subsection 2.2 for more details).

Particularly, in this framework, we address optimal local Lipschitz regularity
for such class of almost minimizers, where the corresponding estimate depends
only on universal parameters

$\|\nabla
u\|_{L^{\infty}(\Omega^{\prime})}\leq\mathrm{C}(\delta,g_{0},\Omega^{\prime},\beta,\kappa)\mathrm{G}^{-1}\left(1+\int_{\Omega}\mathrm{G}\left(|\nabla
u|\right)\>dx\right)\quad\text{for
any}\quad\Omega^{\prime}\subset\subset\Omega.$

We must highlight that the functional $\mathcal{J}_{\mathrm{G}}$ is a natural
generalization of the classical one-phase (Bernoulli type) $p-$energy
functional, which corresponds to the choice of
$\mathrm{G}(t)=\frac{1}{p}t^{p}$ (for each $p\in\left(1,\infty\right)$ fixed).
We will denote by $\mathcal{J}_{p}$ such functional.

Additionally, in the current literature, see e.g. [36], almost minimizers have
been interpreted as a sort of minimizers of “perturbed functionals”. For
instance, one can consider the functional with regional structure (which is
quite similar to certain functionals arising in phase transitions for
nonlinear nonlocal models - see [17])

(1.3)
$\displaystyle\hat{\mathcal{J}}_{\mathrm{G}}(u,\Omega)=\mathcal{J}_{\mathrm{G}}(u,\Omega)+\frac{1}{2}\iint_{\Omega\times\Omega}\Phi(y,z)\Psi(u(y)-u(z))dydz,$

where $\Phi(y,z)=\rho(y)\rho(z)$ with $\rho(x)=\Theta(u(x))$, and
$\Theta:\mathbb{R}\to[0,1]$ fulfils $\Theta\equiv 0$ for $x\leq 0$. Moreover,
the potential $\Psi:\mathbb{R}\to[0,1]$ satisfies $\Psi\equiv 0$ for
$x\in(-\infty,0]$

Now, if we define the $\Psi_{u}-$convolution as

$\displaystyle(\Phi\ast\Psi_{u})(y,z)\mathrel{\mathop{:}}=\iint_{\Omega\times\Omega}\Phi(y,z)\Psi(u(y)-u(z))dydz,$

then (1.3) turns out to be

$\displaystyle\hat{\mathcal{J}}_{\mathrm{G}}(u,\Omega)=\mathcal{J}_{\mathrm{G}}(u,\Omega)+\frac{1}{2}\Phi(y,z)\ast\Psi_{u}(y,z).$

Then, we verify that

$\mathcal{J}_{\mathrm{G}}(u,\Omega)\leq\hat{\mathcal{J}}_{\mathrm{G}}(u,\Omega).$

On the other hand, it is easy to see

$\hat{\mathcal{J}}_{\mathrm{G}}(u,\Omega)\leq\left(1+\frac{1}{2}\omega_{n}r^{n}\right)\mathcal{J}_{\mathrm{G}}(u,\Omega)\leq\left(1+\frac{1}{2}\omega_{n}r^{n}\right)\mathcal{J}_{\mathrm{G}}(v,\Omega),$

for any $v\in W^{1,\mathrm{G}}(B_{r}(x))$ such that $v=u$ on $\partial
B_{r}(x)$. In short, a minimizer for the “perturbed functional”
$\hat{\mathcal{J}}_{\mathrm{G}}(u,\Omega)$ be revealed to be an almost
minimizer for the $\mathrm{G}-$energy functional
$\mathcal{J}_{\mathrm{G}}(u,\Omega)$ for appropriated choices of constants
$\beta$ and $\kappa$. Therefore, it becomes a necessary task to work with
almost minimizers instead of minimizers because of the impossibility of
dealing with weak solutions of certain discontinuous functionals in certain
general frameworks.

Regarding to minimizers in the linear setting, in Alt and Caffarelli’s seminal
paper [3] concerning regularity minimizers to one-phase Bernoulli energy
functional given by

$\displaystyle\mathcal{J}_{2}(u,\Omega)\mathrel{\mathop{:}}=\int_{\Omega}\left(|\nabla
u|^{2}+\chi_{\\{u>0\\}}\right)dx$

it is established the locally Lipschitz regularity estimates. Moreover, it can
be shown that minimizers
$\displaystyle\mathcal{J}_{2}(u_{0},\Omega)=\min_{\mathcal{K}}\mathcal{J}_{2}(u,\Omega)$
are solutions of the following free boundary problem

(1.4) $\left\\{\begin{array}[]{rclcl}\Delta
u_{0}(x)&=&0&\text{in}&\\{u_{0}>0\\}\cap\Omega\\\ |\nabla
u_{0}|&=&\sqrt{2}&\text{on}&\partial\\{u_{0}>0\\}\cap\Omega\\\
u(x)&=&g(x)&\text{on}&\partial\Omega.\end{array}\right.$

in an appropriate distributional sense.

The natural motivations to investigate such a class of free boundary problems
of Bernoulli type (1.4) comes from the analysis of cavities and jets type
problems, see e.g. [12, Section 1.1]. Another relevant models also arise in
combustion theory [6], optimal design problems [50], optimization problems
with constrained volume [2], shape optimization problems [8], [9] and [10] and
phase transitions [47], [48] and [51] just to mention a few scenarios.

Over the past few decades, there has been extensive literature exploring this
research topic, particularly in the context of both single and two-phase
problems, most notably the celebrated viscosity approach to the associated
free boundary developed by Caffarelli in the series of trailblazing works
[13], [14] and [15]. We must refer the reader to [28], [31] and [52] for
comprehensive modern essays on the subject.

Minimizers of the functional $\mathcal{J}_{p}$ have been also considered by
Danielli and Petrosyan in [21], in which the regularity of the free boundary
near flat points was addressed. Recently, Lipschitz estimates of the
minimizers of the functional $\mathcal{J}_{p}$ has been addressed by DiPierro
and Karakhanyan in [37], where the authors also supplied the proof of the
Lipschitz regularity when $p=2$ without making use of monotonicity formulaes.

We must also recall Martinez-Wolanski’s work [46], which consider the
optimization problem of minimizing

$\mathcal{J}_{\mathrm{G},\lambda}(u,\Omega)\mathrel{\mathop{:}}=\int_{\Omega}\left(\mathrm{G}(|\nabla
u|)+\lambda\chi_{\\{u>0\\}}\right)\,dx$

in the class $W^{1,\mathrm{G}}(\Omega)$ with $u-\phi_{0}\in
W_{0}^{1,\mathrm{G}}(\Omega)$, for a bounded function $\phi_{0}\geq 0$ and
$\lambda>0$. The authors prove that solutions to the optimization problem are
locally Lipschitz continuous. Moreover, such solutions satisfy the
corresponding free boundary problem of Bernoulli-type, thereby extending the
Alt-Caffarelli’s results for the scenario of Orlicz-Sobolev framework.
Additionally, they address the Caffarelli’s classification scheme: flat and
Lipschitz free boundaries are locally $C^{1,\alpha}$ for some
$\alpha(\verb"universal")\in(0,1)$. In [7] the authors extend these results to
quasilinear singular/degenerate operators in the nonhomogeneous setting (i.e.
with a nonzero right hand side).

Furthermore, viscosity approaches for one-phase problems driven by the
$p-$Laplace and $p(x)-$Laplace operators has been developed in [44] and [40]
respectively. Additionally, in [20], the authors obtain the Lipschitz
regularity of viscosity solutions of one-phase problems with non-homogeneous
degeneracy (fully nonlinear operators with double phase signature) and some
regularity properties of their free boundaries were addressed.

Now coming back to the setting of almost minimizers, we observe that since
almost minimizers only fulfill a variational inequality, see (1.2), but not a
proper PDE, the main obstacle in facing their regularity properties is the
lack of a monotonicity formula as minimizers do (cf. [3]). In this regard, it
seems to be challenging to derive such techniques to the setting almost
minimizers (cf. [23] and [22] for related topics). In particular, in the quite
recent article [34], De Silva and Savin developed a non-variational approach,
based on Harnack-type inequality, for profiles that do not necessarily satisfy
an infinitesimal equation.

We highlight that almost minimizers of $\mathcal{J}_{2}$ were widely
investigated recently by several authors. In effect, we must cite the David et
al’ work [23], where by developing an original approach combining techniques
from potential theory and geometric measure theory, the authors obtain uniform
rectifiability of the free boundary and, in the one-phase scenario, the
corresponding $C^{1,\alpha}$ almost everywhere regularity. Therefore, the Alt-
Caffarelli’s classical results in [3] were extended to the framework of almost
minimizers. We refer the reader to [22] for generalizations concerning
variable coefficients. Moreover, the analysis of the semilinear scenario with
variable coefficients

$\displaystyle\mathcal{F}_{\gamma}(v;\Omega)=\int_{\Omega}(\langle\mathbb{A}(x)\nabla
v,\nabla
v\rangle+q_{+}(v^{+})^{\gamma}+q_{-}(v^{-})^{\gamma})dx,\quad\text{for}\quad
0\leq\gamma\leq 1\quad\text{and}\quad q_{\pm}\geq 0,$

where $\mathbb{A}$ is a matrix with Hölder continuous coefficients was
explored in [29], thereby proving sharp gradient estimates.

In [33], using non-variational techniques, De Silva and Savin provided a
different approach from that of [23] and [24] to deal with almost minimizers
of $\mathcal{J}_{2}$ and their free boundaries. Precisely, based on ideas
developed by them in [34], they showed that almost minimizers of
$\mathcal{J}_{2}$ are viscosity solutions in a more general sense. Once this
was confirmed, the regularity of the free boundary for almost minimizers
follows by applying the De Silva’s techniques first developed in [30].

Recently, Dipierro et al in [36] obtained a Lipschitz continuity result to
nonnegative almost minimizers for the nonlinear framework, namely for
$\mathcal{J}_{p}$ in $B_{1}$, in the case
$\max\left\\{1,\frac{2n}{n+2}\right\\}<p<\infty$. Precisely, there exists a
universal constant $\mathrm{C}=\mathrm{C}(p,n,\beta,\kappa)>0$ such that

$\|\nabla
u\|_{L^{\infty}(B_{1/2})}\leq\mathrm{C}\left(1+\|u\|_{W^{1,p}(B_{1})}\right)$

Additionally, $u$ is uniformly Lipschitz continuous in a neighborhood of
contact set $\\{u=0\\}$. In effect, the authors’ approach was strongly
inspired by the method introduced by De Silva and Savin in [33]. In their
setting, the main obstacles faced in are concerned the lack of linearity and
the loss of exact descriptions of $p-$harmonic profiles in terms of mean value
properties. To overcome such complication, the authors exploit some regularity
estimates available in the classical literature.

For the vectorial scenarios we must quote the following contributions:

  1. 1.

For a singular system with free boundary, De Silva et al in [32] study the
regularity properties of vector-valued almost minimizers of the functional

$\displaystyle\mathcal{J}_{2}({\textbf{u}};\mathrm{D})=\int_{\mathrm{D}}\left(|\nabla\textbf{u}(x)|^{2}+2|\textbf{u}|\right)dx,$

which is strongly related to a version of the classical obstacle problem (see
[41]).

  2. 2.

For the weakly coupled vectorial $p-$Laplacian, given constant $\lambda>0$ and
a bounded Lipschitz domain $\mathrm{D}\subset\mathbb{R}^{n}$ (for $n\geq 2$),
Shahgholian et al in [5, Theorem 1.1] address local optimal Lipschitz
estimates for almost minimizers of

$\displaystyle\mathcal{J}_{p}({\textbf{v}};\mathrm{D})=\int_{\mathrm{D}}\sum_{i=1}^{m}|\nabla
v_{i}(x)|^{p}+\lambda\chi_{\\{|\bf{v}|>0\\}}(x)dx,\quad(\text{for}\,\,\,1<p<\infty),$

where ${\textbf{v}}=(v_{1},\cdots,v_{m})$, and $m\in\mathbb{N}$.

Taking into account the above results, we will derive uniform Lipschitz
estimates for a general class of almost minimizers of functionals with non-
standard growth. Even if our approach is inspired by the works [33] and [36],
we highlight that the non-homogeneous nature of our functional entails several
difficulties that add to the nonlinear character of the problem already
present in the $p-$Laplace case. The lack of sharp embedding results and the
hard task of handling modulars and norms (for instance when applying Hölder’s
inequality) are two of these challenges which are successfully tackled in this
manuscript, together with the general technical difficulty of dealing with
general behavior different from a power and hence not necessarily homogeneous.

Another gap that we needed to fill was the absence of Campanato-type results
dealing with modulars instead of norms. We consider this point to be of
independent mathematical interest, as the results proved here (see Appendix)
may be useful in other contexts that pertain regularity in Orlicz-Sobolev
settings.

The main regularity result of this proposal is the following (see Section 2
for details on the hypothesis):

######  Theorem 1.1.

Let $\mathrm{G}:[0,\infty)\to[0,\infty)$ be a Young function such that
$g=\mathrm{G}^{\prime}$ is convex and satisfies (1.1) with

$\frac{n(\delta+1)}{n-(\delta+1)}>g_{0}+1.$

Let $u\in W^{1,\mathrm{G}}(\Omega)$ be an almost minimizer of
$\mathcal{J}_{\mathrm{G}}$ with exponent $\beta>0$ and constant $\kappa\geq
0$. Then, $u$ is Lipschitz continuous in the interior of $\Omega$. Moreover,
for any $\Omega^{\prime}\subset\joinrel\subset\Omega$ one has

$\|\nabla
u\|_{L^{\infty}(\Omega^{\prime})}\leq\mathrm{C}\mathrm{G}^{-1}\left(1+\int_{\Omega}\mathrm{G}\left(|\nabla
u|\right)\>dx\right)\quad\text{for
any}\quad\Omega^{\prime}\subset\subset\Omega.$

where $\mathrm{C}=\mathrm{C}(\delta,g_{0},\Omega^{\prime},\beta,\kappa)>0$.

We stress that the nonlinear setting driven by
$\mathcal{J}_{\mathrm{G}}(u,\Omega)$ provides further technical obstacles,
e.g. the sum of two weak solutions is no longer a weak solution. Therefore, it
plays an essential role to develop a theory of $g-$harmonic replacements for
our nonlinear scenario to overcome such complications.

Furthermore, we point out that the regularity results in this work do not
follow from the nowadays classical literature from calculus of variations for
operators with Orlicz-Sobolev structure: Indeed, the integrand of the
$\mathrm{G}-$energy functional has a discontinuous signature, instead, the
non-standard growth scenarios that dealt with continuous integrand or even
Lipschitz continuous one, see e.g. [1] and [45] for such related topic.

Concerning the strategy of the proof of Theorem 1.1: a dichotomy result is the
main step to establishing the optimal Lipschitz continuity. In effect, the
dichotomy property ensures that either the average of the $\mathrm{G}-$energy
of an almost minimizer to $\mathcal{J}_{\mathrm{G}}$ decreases in a suitable
ball or the Luxemburg norm between its gradient and a fixed vector becomes
sufficiently small in a proper sense.

Summarizing, either the average of the $\mathrm{G}-$energy of an almost
minimizer decreases in a smaller domain, or the almost minimizer is close
enough to a linear profile (with good a priori estimates). Once this dichotomy
is proved, the next step is to show that one of such alternatives might be
improved. Precisely, it will hold a kind of geometric improved of flatness,
namely, if an almost minimizer is close to (in the sense of the average of the
$\mathrm{G}-$energy) a suitable linear profile, in a ball, then it will be
closer to a possibly other linear profile in a smaller ball. Therefore, by
iterating such a reasoning, together with the dichotomy property, provides the
desired Lipschitz regularity of almost minimizers to
$\mathcal{J}_{\mathrm{G}}$,

Finally, in our scenario, we need to show that the sum of
$\mathrm{G}-$harmonic replacements solve an equation for which $C^{1,\alpha}$
estimates hold. In effect, this holds true using a linearization strategy,
together with the assumptions of the dichotomy. The main difficulty appears in
the proof of the improvement of dichotomy’s second alternative. The original
strategy to solve such an obstacle strongly relies on De Silva-Savin’s work
[33], where the authors often use harmonic replacements as competitors. This
property holds, for harmonic replacements allowing achieve estimates for the
average of their energy, exploiting Schauder estimates instead of gradient
estimates.

## Some extensions and further comments

We will present a number of interesting building-block models in which our
results work as well.

  1. 1.

Models arising of non-autonomous functionals:

First, by considering the quantitative results (e.g. regularity estimates,
Harnack inequality, see references below) we can to deal with the following
model cases:

     * •

Problems with $(p\&q)-$growth.

$u_{0}+W_{0}^{1,p}(\Omega)\ni
w\mapsto\int_{\Omega}\left(\mathcal{H}_{0}(x,\nabla
w)+\chi_{\\{w>0\\}}\right)dx,$

as long as the integrand $\mathcal{H}_{0}$ enjoys an appropriate double phase
type structure

$L_{1}\cdot\left(\frac{1}{p}|\xi|^{p}+\frac{1}{q}|\xi|^{q}\right)\leq\mathcal{H}_{0}(x,\xi)\leq
L_{2}\cdot\left(\frac{1}{p}|\xi|^{p}+\frac{1}{q}|\xi|^{q}\right),$

where $0<L_{1}\leq L_{2}<\infty$ and $1<p<q<\infty$, which make possible to
access available existence/regularity results for weak solutions. We recommend
to reader see the series of fundamental manuscripts [4], [19], [25] , [26] and
[27] for related topics.

     * •

Problems with Uhlenbeck’s type structure.

We also stress that our approach is particularly refined and quite far-
reaching in order to be employed in other classes of problems. Indeed,

We can also extended our results for Musielak–Orlicz Spaces with non-
homogeneous term as follow (see [35, §2.3])

$\displaystyle u_{0}+W^{1,\varphi}(\Omega)\ni
v\mapsto\int_{\Omega}\left(\varphi(x,|\nabla v|)+\chi_{\\{v>0\\}}\right)dx$

for $(x,\xi)\mapsto\varphi(x,|\xi|)$ with particular structure, including:

       1. ✓

$p-$growth, i.e., $\varphi(x,|t|)=\frac{1}{p}|t|^{p}$;

       2. ✓

Orlicz growth, i.e., $\varphi(x,|t|)=\mathrm{G}(|t|)$;

       3. ✓

Multi phase growth, i.e.,
$\displaystyle\varphi(x,|t|)=\frac{1}{p}|t|^{p}+\sum_{i=1}^{k}\mathfrak{a}_{i}(x)\frac{1}{q_{i}}|t|^{q}$
for $\leq\mathfrak{a}_{i}\in C^{0,1}(\Omega)$;

just to mention a few, (see [18] and [43, Section 2]).

In this case, $\varphi$ must satisfy the Uhlenbeck’s structure: for constants
$1<p\leq q<\infty$

$t\mapsto\varphi(x,t)\quad\text{is in }\quad C^{1}([0,\infty))\cap
C^{2}((0,\infty))\quad\text{and}\quad
p-1\leq\frac{t\varphi^{\prime\prime}(x,t)}{\varphi^{\prime}(x,t)}\leq\max_{1\leq
i\leq k}\\{q_{i}-1\\}.$

  2. 2.

Models with generalized doubly degenerate growth.

Finally, we may consider a wide class of nonlinear degenerate models with non
standard growth properties (cf. [11], and [18] for a survey). An archetypical
example we have in mind concerns models of the form

$\mathcal{J}_{\mathcal{G}_{\mathrm{G},\mathrm{H}}}(u,\Omega)\mathrel{\mathop{:}}=\int_{\Omega}\left(\mathcal{G}_{\mathrm{G},\mathrm{H}}(x,|\nabla
u|)+\chi_{\\{u>0\\}}\right)dx$

where

$\mathcal{G}_{\mathrm{G},\mathrm{H}}(x,\xi)\mathrel{\mathop{:}}=\mathrm{G}(|\xi|)+\mathfrak{a}(x)\mathrm{H}(|\xi|)\quad\text{with}\quad
0\leq\mathfrak{a}\in C^{0,1}(\Omega),$

for Young functions $\mathrm{G},\mathrm{H}:[0,\infty)\to[0,\infty)$ with
$\mathrm{G},\mathrm{H}\in\Delta_{2}\cap\nabla_{2}$,
$\mathrm{G}\prec\mathrm{H}\prec\mathrm{G}^{1+\frac{1}{n}}$, and satisfying the
Lieberman’s condition: there exist constants
$\mathrm{c}_{\mathrm{G}},\mathrm{c}^{\prime}_{\mathrm{G}},\mathrm{c}_{\mathrm{H}},\mathrm{c}^{\prime}_{\mathrm{H}}\geq
1$ such that

$\mathrm{c}^{\prime}_{\mathrm{G}}\leq\frac{t\mathrm{G}^{\prime}(t)}{\mathrm{G}(t)}\leq\mathrm{c}_{\mathrm{G}}\quad\text{and}\quad\mathrm{c}^{\prime}_{\mathrm{H}}\leq\frac{t\mathrm{H}^{\prime}(t)}{\mathrm{H}(t)}\leq\mathrm{c}_{\mathrm{H}}\quad\text{for
all}\quad t>0.$

The rest of the paper is organized as follows: in Section 2 we present the
preliminary definitions and results that will be needed throughout the paper;
Section 3 is devoted to the proof of the most important technical results of
the paper, namely Lemmas 3.1 and 3.3 and, as a consequence, the crucial
interior gradient estimate in Corollary 3.4. In Section 4 we prove the main
result of this paper, i.e. Theorem 1.1. Finally, the Appendix contains the
proof of the Campanato-type regularity result that is used in Corollary 3.4,
summarized in Theorem 4.3.

##  2 Preliminaries

In this section we present some preliminary definitions and results that will
be used in the proofs of the main results of the manuscript. We start with
some basic facts regarding Young functions and Orlicz-Sobolev spaces; these
are rather well known but we include them for the sake of completeness.

###  2.1 Young functions

An application $\mathrm{G}\colon[0,\infty)\longrightarrow[0,\infty)$ is said
to be a _Young function_ if it admits the integral representation
$\displaystyle\mathrm{G}(t)=\int_{0}^{t}g(\tau)\,d\tau$, where $g$ is a right
continuous function defined on $[0,\infty)$ and satisfying:

$\left\\{\begin{array}[]{l}g(0)=0,\quad g(t)>0\text{ for }t>0,\\\ g\text{ is
nondecreasing on }(0,\infty),\\\
\displaystyle\lim_{t\to\infty}g(t)=\infty.\end{array}\right.$

From these properties it is easy to see that a Young function $\mathrm{G}$ is
continuous, nonnegative, strictly increasing and convex on $[0,\infty)$.
Further, we recall that we may extend $g$ to the whole $\mathbb{R}$ in an odd
fashion: for $t<0$ $g(t)=-g(-t)$. And we may (and will) assume without loss of
generality that $\mathrm{G}(1)=\mathrm{G}^{-1}(1)=1$. Note that $\mathrm{G}$
is strictly increasing and hence $\mathrm{G}^{-1}$ is well defined.

In this work we will consider the class of Young functions such that
$g=\mathrm{G}^{\prime}$ is an absolutely continuous function that satisfies
the condition

(2.1) $\delta\leq\frac{tg^{\prime}(t)}{g(t)}\leq g_{0}$

for some constants $1<\delta\leq g_{0}$. This condition was first considered
in the seminal work of G. Lieberman [45] and is the analogous to the
ellipticity condition in the linear theory. For our purposes, we will make the
(rather standard assumption) that

(2.2) $(\delta+1)^{\ast}>g_{0}+1$

where $(\delta+1)^{\ast}$ is the Sobolev conjugate of $(\delta+1)$, i.e.
$(\delta+1)^{\ast}=\frac{n(\delta+1)}{n-(\delta+1)}$. We will also assume that

$g\text{ is convex.}$

This last assumption is analogous to the degenerate setting in the homogeneous
problem, i.e. the case where $p\geq 2$.

Equation (2.1) implies that that $\mathrm{G}$ satisfies

(2.3) $\delta+1\leq\frac{tg(t)}{\mathrm{G}(t)}\leq g_{0}+1;$

this condition is equivalent to ask $\mathrm{G}$ and $\tilde{\mathrm{G}}$ to
satisfy the _$\Delta_{2}$ condition or doubling condition_, i.e.,

(2.4) $\mathrm{G}(2t)\leq
2^{g_{0}+1}\mathrm{G}(t),\qquad\tilde{\mathrm{G}}(2t)\leq
2^{1+\frac{1}{\delta}}\tilde{\mathrm{G}}(t),$

where the _complementary_ function of a Young function $\mathrm{G}$ is the
Young function $\tilde{\mathrm{G}}$ defined as

$\tilde{\mathrm{G}}(t)=\sup\\{ta-\mathrm{G}(a)\colon a>0\\}.$

The doubling condition allow us to split sums as

(2.5) $\mathrm{G}(a+b)\leq 2^{g_{0}}(\mathrm{G}(a)+\mathrm{G}(b)).$

Moreover, integrating (2.3) we have that

(2.6) $\displaystyle t^{g_{0}+1}\leq\mathrm{G}(t)\leq
t^{\delta+1},\quad\text{if }0<t\leq 1$ $\displaystyle
t^{\delta+1}\leq\mathrm{G}(t)\leq t^{g_{0}+1},\quad\text{if }t>1.$

As mentioned in the Introduction, the lack of homogeneity is a serious
technical difficulty in our context. The following lemma gives the
alternatives that hold in this scenario; its proof is elementary so we omit
it.

######  Lemma 2.1.

For $\theta\in[0,1]$ and $t\geq 0$

$\mathrm{G}(\theta t)\leq\theta\mathrm{G}(t),$

and for $\theta\geq 1$ and $t\geq 0$

$\mathrm{G}(\theta t)\geq\theta\mathrm{G}(t).$

More generally, for any, $\theta,t\geq 0$

(2.7)
$\displaystyle\mathrm{G}(t)\min\left\\{\theta^{\delta+1},\theta^{g_{0}+1}\right\\}\leq\mathrm{G}(\theta
t)\leq\mathrm{G}(t)\max\left\\{\theta^{\delta+1},\theta^{g_{0}+1}\right\\},$
(2.8)
$\displaystyle\mathrm{G}^{-1}(t)\min\left\\{\theta^{\frac{1}{\delta+1}},\theta^{\frac{1}{g_{0}+1}}\right\\}\leq\mathrm{G}^{-1}(\theta
t)\leq\mathrm{G}^{-1}(t)\max\left\\{\theta^{\frac{1}{\delta+1}},\theta^{\frac{1}{g_{0}+1}}\right\\}.$

###  2.2 Orlicz-Sobolev spaces

Given a Young function $\mathrm{G}$ and a bounded open set
$\Omega\subset\mathbb{R}^{n}$, we consider the spaces $L^{\mathrm{G}}(\Omega)$
and $W^{1,\mathrm{G}}(\Omega)$ defined as follows:

$\begin{array}[]{rcl}L^{\mathrm{G}}(\Omega)&=&\\{u:\mathbb{R}\to\mathbb{R}\quad\text{measurable
such that}\quad\Phi_{\mathrm{G},\Omega}(u)<\infty\\}\\\
W^{1,\mathrm{G}}(\Omega)&=&\\{u\in L^{\mathrm{G}}(\Omega)\quad\text{such
that}\quad\Phi_{\mathrm{G},\Omega}(|\nabla u|)<\infty\\}\end{array}$

where the modular $\Phi_{\mathrm{G},\Omega}(u)$ stands for

$\displaystyle\Phi_{\mathrm{G},\Omega}(u)=\int_{\Omega}\mathrm{G}(|u|)dx.$

These spaces are endowed with the so-called Luxemburg norm defined as follows:

$\displaystyle\|u\|_{L^{\mathrm{G}}(\Omega)}=\inf\left\\{\lambda>0:\,\,\,\Phi_{\mathrm{G},\Omega}\left(\frac{u}{\lambda}\right)\leq
1\right\\}$

and

$\|u\|_{W^{1,\mathrm{G}}(\Omega)}=\|u\|_{L^{\mathrm{G}}(\Omega)}+\|\nabla
u\|_{L^{\mathrm{G}}(\Omega)}$

In general, it is easy to see that

(2.9)
$\|w\|_{L^{\mathrm{G}}(\Omega)}\leq\max\left\\{\Phi_{\mathrm{G},\Omega}(w)^{\frac{1}{g_{0}+1}},\Phi_{\mathrm{G},\Omega}(w)^{\frac{1}{\delta+1}}\right\\}.$

###  2.3 Technical results concerning $g-$harmonic functions

The notion of $g-$harmonic function will of course be of great importance in
our work; In effect, $g-$harmonic functions are weak solutions of the equation
$\Delta_{g}v=0$ where $\Delta_{g}$ is the $g-$Laplacian operator given by

$\Delta_{g}v=\mathop{\text{div}}\left(g(|\nabla v|)\frac{\nabla v}{|\nabla
v|}\right).$

More precisely, we will be interested in unique weak solutions of the
Dirichlet problem

(2.10) $\left\\{\begin{array}[]{rcrcl}\Delta_{g}v&=&0&\text{in}&\Omega\\\
v&=&h&\text{on}&\partial\Omega\end{array}\right.$

Such weak solutions can be constructed as minimizers of the energy functional
$\displaystyle u\rightarrow\int_{\Omega}\mathrm{G}(|\nabla u|)\>dx$ (with
fixed Dirichlet boundary condition) via the Direct Method of the Calculus of
Variations.

The following proposition will be used, its proof is contained in the
Lieberman’s fundamental work [45, Section 5]. Before stating it, we recall
that a constant will be called universal if it depends only on $n,\delta$ and
$g_{0}$ (and eventually $\beta$ or $\kappa$ in Definition 2.7); multiplicative
constants will be denoted by $C$ and in the proofs they may change from line
to line.

######  Proposition 2.2.

Let $v\in W^{1,\mathrm{G}}(B_{R})$ be a $g-$harmonic function. Then,

  1. 1.

there exists a universal constant $\mathrm{C}>0$ such that for any $x\in
B_{R/2}$

(2.11) $\sup_{B_{R/4}(x)}\mathrm{G}\left(|\nabla
v|\right)\leq\mathrm{C}\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{R/2}(x)}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{R/2}(x)}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{R/2}(x)}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{R/2}(x)}}\mathrm{G}\left(|\nabla
v|\right)\>dx.$

  2. 2.

there exist universal constants $\mathrm{C}>0$ and $\alpha\in(0,1)$ such that

(2.12) $\mathop{\text{osc}}_{B_{\rho}}|\nabla
v|\leq\mathrm{C}\sup_{B_{R}}|\nabla
v|\left(\frac{\rho}{R}\right)^{\alpha},\quad 0<\rho<R.$

######  Remark 2.3.

The estimates in Proposition (2.2) still hold under the more general
assumption that $v$ satisfies $\mathop{\text{div}}(\mathbb{A}(x)\nabla v)=0$
with $x\mapsto\mathbb{A}(x)$ satisfying the structural condition:

$\lambda\frac{g(|z|)}{|z|}|\xi|^{2}\leq\mathbb{A}(x)\xi\cdot\xi\leq\Lambda\frac{g(|z|)}{|z|}|\xi|^{2}.$

See [45] for details.

Another important concept that will be used is that of harmonic replacement in
a ball, which roughly speaking is the $g-$harmonic that coincides with a given
function on the boundary of a ball:

######  Definition 2.4.

The $g-$harmonic replacement of a function $u\in W^{1,G}(\Omega)$ in
$B_{r}(x)$ is the unique $g-$harmonic function $v\in W^{1,G}(B_{r}(x))$ that
coincides with $u$ on $\partial B_{r}(x)$ in the sense of traces, i.e. the
unique weak solution of (2.10) with $\Omega=B_{r}(x)$ and $h=u$.

The following technical lemmas will be used later: the first one is a bound on
the energy of difference between and function and its $g-$harmonic replacement
and the other one is needed to ensure the desired ellipticity (in the sense of
Remark 2.3) so that interior estimates apply:

######  Lemma 2.5.

Let $G$ a Young function satisfying (2.1) and such that
$\mathrm{G}^{\prime}=g$ is a convex function, $u\in W^{1,G}(B_{r})$ and $v$
the $g-$harmonic replacement of $u$ in $B_{r}$. Then the following inequality
holds for a universal constant $\mathrm{C}>0$

$\int_{B_{r}}\mathrm{G}(|\nabla u-\nabla
v|)\,dx\leq\mathrm{C}\int_{B_{r}}\mathrm{G}(|\nabla u|)-\mathrm{G}(|\nabla
v|)\,dx.$

###### Proof.

As $v$ is a weak solution of $\Delta_{g}u=0$, we know that

(2.13) $\int_{B_{r}}g(|\nabla v|)\frac{\nabla v}{|\nabla
v|}\nabla\varphi\,dx=0,$

for all $\varphi\in W_{0}^{1,\mathrm{G}}(B_{r})$.

We consider the following convex combination

$u^{s}(x)=su(x)+(1-s)v(x)\qquad 0\leq s\leq 1$

Observe that $u^{0}=v$ and $u^{1}=u$. Now using that $u-v\in
W^{1,G}_{0}(\Omega)$ and (2.13), we have

$\displaystyle\int_{B_{r}}\mathrm{G}(|\nabla u|)-\mathrm{G}(|\nabla v|)\,dx$
$\displaystyle=\int_{B_{r}}\mathrm{G}(|\nabla u^{1}|)-\mathrm{G}(|\nabla
u^{0}|)\,dx$
$\displaystyle=\int_{B_{r}}\left(\int_{0}^{1}\frac{d}{ds}\mathrm{G}(|\nabla
u^{s}|)\,ds\right)\,dx$ $\displaystyle=\int_{B_{r}}\left(\int_{0}^{1}g(|\nabla
u^{s}|)\frac{\nabla u^{s}}{|\nabla u^{s}|}\cdot\nabla(u-v)\,ds\right)\,dx$
$\displaystyle=\int_{B_{r}}\int_{0}^{1}\left(g(|\nabla u^{s}|)\frac{\nabla
u^{s}}{|\nabla u^{s}|}-g(|\nabla v|)\frac{\nabla v}{|\nabla
v|}\right)\cdot\nabla(u-v)\,ds\,dx.$

Next, note that $u^{s}(x)-v(x)=su(x)-sv(x)=s(u(x)-v(x))$; then using that

$\left(g(|a|)\frac{a}{|a|}-g(|b|)\frac{b}{|b|}\right)\cdot\nabla(a-b)\geq\mathrm{C}\mathrm{G}(|a-b|)$

for a constant $\mathrm{C}=\mathrm{C}(\delta)>0$ (see [16, Lemma 3.1] for a
proof) and (2.7) the last above term is bounded in the following way

$\begin{array}[]{rcl}\displaystyle\int_{B_{r}}\frac{1}{s}\int_{0}^{1}\left(g(|\nabla
u^{s}|)\frac{\nabla u^{s}}{|\nabla u^{s}|}-g(|\nabla v|)\frac{\nabla
v}{|\nabla
v|}\right)\cdot\nabla(u^{s}-v)\,ds\,dx&\geq&\displaystyle\mathrm{C}\int_{0}^{1}\frac{1}{s}\int_{B_{r}}\mathrm{G}(|\nabla
u^{s}-\nabla v|)\,dx\,ds\\\
&\geq&\displaystyle\mathrm{C}\int_{0}^{1}\frac{s^{g_{0}+1}}{s}\int_{B_{r}}\mathrm{G}(|\nabla
u-\nabla v|)\,dx\,ds\\\
&\geq&\displaystyle\frac{\mathrm{C}}{g_{0}+1}\int_{B_{r}}\mathrm{G}(|\nabla
u-\nabla v|)\,dx\end{array}$

and the desired inequality follows. ∎

######  Lemma 2.6.

Let $q\in\mathbb{R}^{n}$ and $|h(x)|<\frac{|q|}{2}$. If
$\overrightarrow{\mathbf{F}}(z)=g(|z|)\frac{z}{|z|}$ for a Young function
satisfying (2.1) and

$\mathfrak{A}(x)\mathrel{\mathop{:}}=\int_{0}^{1}D\overrightarrow{\mathbf{F}}\left(q+th(x)\right)\>dt$

then

(2.14)
$\lambda\frac{g(|q|)}{|q|}|\xi|^{2}\leq\mathfrak{A}(x)\xi\cdot\xi\leq\Lambda\frac{g(|q|)}{|q|}|\xi|^{2}$

for any $\xi\in\mathbb{R}^{n}\setminus\\{0\\}$ and for some constants
$0<\lambda\leq\Lambda$ depending only on $\delta$ and $g_{0}$.

###### Proof.

We start by computing $D\overrightarrow{\mathbf{F}}$: if $i\neq j$

$D_{i}\overrightarrow{\mathbf{F}}_{j}(z)=D_{i}\left(g(|z|)\frac{z_{j}}{|z|}\right)=g^{\prime}(|z|)\frac{z_{i}z_{j}}{|z|^{2}}-g(|z|)\frac{z_{i}z_{j}}{|z|^{3}}$

while

$D_{i}\overrightarrow{\mathbf{F}}_{i}(z)=D_{i}\left(g(|z|)\frac{z_{i}}{|z|}\right)=g^{\prime}(|z|)\frac{z_{i}^{2}}{|z|^{2}}+g(|z|)\left(\frac{1}{|z|}-\frac{z_{i}^{2}}{|z|^{3}}\right).$

Then,

$\displaystyle D\overrightarrow{\mathbf{F}}(z)\xi\cdot\xi$
$\displaystyle=\sum_{i\neq
j}\left(g^{\prime}(|z|)\frac{z_{i}z_{j}}{|z|^{2}}-g(|z|)\frac{z_{i}z_{j}}{|z|^{3}}\right)\xi_{i}\xi_{j}+\sum_{i}\left(g^{\prime}(|z|)\frac{z_{i}^{2}}{|z|^{2}}+g(|z|)\left(\frac{1}{|z|}-\frac{z_{i}^{2}}{|z|^{3}}\right)\right)\xi_{i}^{2}$
$\displaystyle=\sum_{i\neq
j}\left(g^{\prime}(|z|)-\frac{g(|z|)}{|z|}\right)\frac{z_{i}z_{j}\xi_{i}\xi_{j}}{|z|^{2}}+g^{\prime}(|z|)|\xi|^{2}$
$\displaystyle=\left(g^{\prime}(|z|)-\frac{g(|z|)}{|z|}\right)\frac{1}{|z|^{2}}\left(z\cdot\xi\right)^{2}+\frac{g(|z|)}{|z|}|\xi|^{2}$
$\displaystyle\geq\left(\frac{g(|z|)}{|z|}-\left(g^{\prime}(|z|)-\frac{g(|z|)}{|z|}\right)\right)|\xi|^{2}$
$\displaystyle\geq g_{0}\frac{g(|z|)}{|z|}|\xi|^{2}$ $\displaystyle\geq
g^{\prime}(|z|)|\xi|^{2}.$

This implies

$\mathfrak{A}(x)\xi\cdot\xi\geq\int_{0}^{1}g^{\prime}(|q+th(x)|)\>dt|\xi|^{2}\geq
g^{\prime}\left(\frac{|q|}{2}\right)|\xi|^{2}\geq\delta\frac{g\left(\frac{|q|}{2}\right)}{\frac{|q|}{2}}|\xi|^{2}\geq\lambda\frac{g\left(|q|\right)}{|q|}|\xi|^{2}$

where $\lambda\mathrel{\mathop{:}}=2^{1-g_{0}}\delta$, and we have used that
$|q+th(x)|\geq\frac{|q|}{2}$ by hypothesis, that $g^{\prime}$ is increasing
and the doubling condition for $g$. This gives the lower bound on (2.14).

Similarly, we find that

$D\overrightarrow{\mathbf{F}}(z)\xi\cdot\xi\leq\left(\frac{g(|z|)}{|z|}+\left(g^{\prime}(|z|)-\frac{g(|z|)}{|z|}\right)\right)|\xi|^{2}=g^{\prime}(|z|)|\xi|^{2}$

and

$\mathfrak{A}(x)\xi\cdot\xi\leq
g^{\prime}\left(\frac{3|q|}{2}\right)|\xi|^{2}\leq
g_{0}\frac{g\left(\frac{3|q|}{2}\right)}{\frac{3|q|}{2}}|\xi|^{2}\leq\Lambda\frac{g\left(|q|\right)}{|q|}|\xi|^{2}$

now with
$\Lambda\mathrel{\mathop{:}}=\left(\frac{3}{2}\right)^{g_{0}-1}g_{0}$, this
time using that $|q+th(x)|\leq\frac{3|q|}{2}$. ∎

###  2.4 Almost minimizers

In this section we give the rigorous definition of an almost minimizer and
show that they satisfy a nice scaling property:

######  Definition 2.7.

A function $u\in W^{1,\mathrm{G}}(\Omega)$ is an almost minimizer of
$\mathcal{J}_{G}$ with exponent $\beta>0$ and constant $\kappa\geq 0$ if

  1. 1.

$u\geq 0$ a.e. in $\Omega$;

  2. 2.

for any ball $B_{r}(x)$ such that $\overline{B_{r}(x)}\subset\Omega$ and any
$v\in W^{1,G}(B_{r}(x))$ such that $v=u$ on $\partial B_{r}(x)$ in the sense
of traces (i.e. $v-u\in W_{0}^{1,G}(B_{r}(x))$) it holds that

$\mathcal{J}_{\mathrm{G}}(u,B_{r}(x))\leq(1+\kappa
r^{\beta})\mathcal{J}_{\mathrm{G}}(v,B_{r}(x)).$

Almost minimizers satisfy the following scaling property:

######  Lemma 2.8.

If $u\in W^{1,G}(B_{1})$ is an almost minimizer of $\mathcal{J}_{G}$ in
$B_{1}$ with exponent $\beta>0$ and constant $\kappa$ and $0<r<1$ then
$u_{r}(x)\mathrel{\mathop{:}}=\frac{u(rx)}{r}$ is an almost minimizer of
$\mathcal{J}_{G}$ in $B_{1/r}$ with exponent $\beta>0$ and constant
$r^{\beta}\kappa$.

###### Proof.

Assume $u$ is an almost minimizer of $\mathcal{J}_{G}$ in $B_{1}$ with
exponent $\beta>0$ and constant $\kappa$. Let $\bar{x}\in B_{1/r}$ and
$\rho>0$ be such that $\overline{B_{\rho}(\bar{x})}\subset B_{1/r}$ and $v\in
W^{1,G}(B_{\rho}(\bar{x}))$ such that $v=u_{r}$ on $\partial
B_{\rho}(\bar{x})$ in the sense of traces.

Denoting $\bar{y}=r\bar{x}$ we have

$\overline{B_{\rho}(\bar{x})}\subset
B_{1/r}\>\Rightarrow\>\overline{B_{r\rho}(\bar{y})}\subset B_{1},\quad
x\in\partial
B_{r\rho}(\bar{y})\>\Rightarrow\>\frac{x}{r}\in\partial{B_{\rho}(\bar{x})}.$

Then, by hypothesis

$rv\left(\frac{x}{r}\right)=ru_{r}\left(\frac{x}{r}\right)=u(x)$

when $x\in\partial B_{r\rho}(\bar{y})$ and therefore

$\mathcal{J}_{\mathrm{G}}(u,B_{r\rho}(\bar{y}))\leq(1+\kappa(r\rho)^{\beta})\mathcal{J}_{\mathrm{G}}\left(rv\left(\frac{\cdot}{r}\right),B_{r\rho}(\bar{y})\right).$

But changing variables we have that

$\displaystyle\mathcal{J}_{\mathrm{G}}(u,B_{r\rho}(\bar{y}))$
$\displaystyle=\int_{B_{r\rho}(\bar{y})}\mathrm{G}(|\nabla
u(y)|)+\chi_{\\{u>0\\}}\,dy$
$\displaystyle=r^{n}\int_{B_{\rho}(\bar{x})}\mathrm{G}(|\nabla
u(rx)|)+\chi_{\\{u>0\\}}(rx)\,dx$
$\displaystyle=r^{n}\int_{B_{\rho}(\bar{x})}\mathrm{G}(|\nabla
u_{r}|)+\chi_{\\{u_{r}>0\\}}\,dx$
$\displaystyle=r^{n}\mathcal{J}_{\mathrm{G}}\left(u_{r},B_{\rho}(\bar{x})\right)$

and a similar computation can be done to show that

$\mathcal{J}_{\mathrm{G}}\left(rv\left(\frac{\cdot}{r}\right),B_{r\rho}(\bar{y})\right)=r^{n}\mathcal{J}_{\mathrm{G}}\left(v,B_{\rho}(\bar{x})\right)$

and get the desired inequality. ∎

##  3 Dichotomy lemmas

In this section we prove the main technical results needed in the proof of
Theorem 1.1. We being with the following dichotomy lemma:

######  Lemma 3.1.

There exists $\varepsilon_{0}\in(0,1)$ such that for every
$0<\varepsilon\leq\varepsilon_{0}$ there exist $\eta\in(0,1)$, $\mathrm{M}>1$
and $\sigma_{0}\in(0,1)$ depending on $\varepsilon,n,\delta$ and $g_{0}$ such
that: if $\sigma\leq\sigma_{0},\>a>\mathrm{M},\>u\in W^{1,\mathrm{G}}(B_{1})$
and the following inequality holds for all $v\in W^{1,\mathrm{G}}(B_{1})$ such
that $v=u$ on $\partial B_{1}$:

$\mathcal{J}_{\mathrm{G}}(u,B_{1})\leq(1+\sigma)\mathcal{J}_{\mathrm{G}}(v,B_{1})$

then, if we denote

(3.1) $\mathrm{G}(a)\mathrel{\mathop{:}}=\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}\mathrm{G}(|\nabla
u|)\,dx$

the following dichotomy holds: either

(3.2) $\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}\mathrm{G}(|\nabla
u|)\,dx\leq\mathrm{G}\left(\frac{a}{2}\right)$

or

(3.3) $\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}\mathrm{G}(|\nabla
u-q|)\,dx\leq\mathrm{G}(\varepsilon a)$

where $q\in\mathbb{R}^{n}$ verifies

(3.4)
$\mathrm{G}^{-1}\left(\frac{1}{\mathrm{C}}\mathrm{G}\left(\frac{a}{4}\right)\right)<|q|<\mathrm{C}_{0}a.$

###### Proof.

Let $v$ be the $g-$harmonic replacement of $u$ in $B_{1}$. We start by
computing, with the aid of Lemma 2.5 and the definition of almost minimizer,

$\displaystyle\int_{B_{1}}\mathrm{G}(|\nabla u-\nabla v|)\,dx$
$\displaystyle\leq\mathrm{C}\int_{B_{1}}(\mathrm{G}(|\nabla
u|)-\mathrm{G}(|\nabla v|))\,dx$
$\displaystyle\leq\mathrm{C}\left(\int_{B_{1}}\mathrm{G}(|\nabla
u|)+\chi_{\\{u>0\\}}\,dx-\mathrm{G}(|\nabla v|)\,dx\right)$
$\displaystyle=\mathrm{C}\left(\mathcal{J}_{\mathrm{G}}(u,B_{1})-\int_{B_{1}}\mathrm{G}(|\nabla
v|)\,dx\right)$
$\displaystyle\leq\mathrm{C}((1+\sigma)\mathcal{J}_{\mathrm{G}}(v,B_{1})-\int_{B_{1}}\mathrm{G}(|\nabla
v|)\,dx)$
$\displaystyle=\mathrm{C}\left(\sigma\left(\int_{B_{1}}\mathrm{G}(|\nabla
v|)\,dx+|\\{v>0\\}\cap B_{1}|\right)+|\\{v>0\\}\cap B_{1}|\right)$
$\displaystyle\leq\mathrm{C}\left(\sigma\int_{B_{1}}\mathrm{G}(|\nabla
v|)\,dx+1\right)$
$\displaystyle\leq\mathrm{C}\left(\sigma\int_{B_{1}}\mathrm{G}(|\nabla
u|)\,dx+1\right),$

the last inequality owing to the fact that $v$ minimizes the energy. Taking
average and recalling (3.1) we have

$\displaystyle\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}\mathrm{G}(|\nabla
u-\nabla v|)\,dx\leq\mathrm{C}(\sigma\mathrm{G}(a)+1).$

On the other hand, if we fix $x\in B_{\frac{1}{2}}$, by (2.11) in Proposition
2.2 we have

$\mathrm{G}(|\nabla v(x)|)\leq\sup_{B_{\frac{1}{4}}(x)}\mathrm{G}(|\nabla
v|)\leq\mathrm{C}\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{\frac{1}{2}}(x)}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\frac{1}{2}}(x)}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\frac{1}{2}}(x)}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\frac{1}{2}}(x)}}\mathrm{G}(|\nabla
v|)\,dy\leq\mathrm{C}\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}\mathrm{G}(|\nabla
u|)\,dy\leq\mathrm{C}\mathrm{G}(a)$

so that

(3.5) $\mathrm{G}\left(|\nabla v|\right)\leq C\mathrm{G}(a)\text{ in
}B_{1/2}.$

Further, if we denote $q=\nabla v(0)$, and recall (2.12) in Proposition 2.2 we
have

$\displaystyle\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}\mathrm{G}(|\nabla
v-q|)\,dx$ $\displaystyle\leq\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}\mathrm{G}\left(\mathrm{C}\left(\frac{\eta}{\frac{1}{2}}\right)^{\alpha}\sup_{B_{\frac{1}{2}}}|\nabla
v|\right)\,dx$
$\displaystyle\leq\left(\mathrm{C}\left(\frac{\eta}{\frac{1}{2}}\right)^{\alpha}\right)^{\delta+1}\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}\mathrm{G}\left(\sup_{B_{\frac{1}{2}}}|\nabla
v|\right)\,dx$
$\displaystyle=\mathrm{C}\eta^{\alpha(\delta+1)}\mathrm{G}\left(\sup_{B_{\frac{1}{2}}}|\nabla
v|\right)$

and by (3.5)

$\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}\mathrm{G}(|\nabla
v-q|)\,dx\leq\mathrm{C}\eta^{\alpha(\delta+1)}\mathrm{G}(a)\qquad\mbox{ for
all }\eta\leq\frac{1}{2}.$

Now, using the $\Delta_{2}$ condition (see (2.4)),

$\displaystyle\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}\mathrm{G}(|\nabla
u-q|)\,dx$ $\displaystyle\leq\mathrm{C}\left(\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}\mathrm{G}(|\nabla
u-\nabla v|)\,dx+\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}\mathrm{G}(|\nabla
v-q|)\,dx\right)$ $\displaystyle=\mathfrak{I}+\mathfrak{II}$

These two terms are bounded using the previous estimates:

$\mathfrak{I}\leq\frac{1}{|B_{\eta}|}\int_{B_{1}}\mathrm{G}(|\nabla u-\nabla
v|)\,dx\leq\frac{\mathrm{C}}{\eta^{n}}(\sigma\mathrm{G}(a)+1).$

and

$\mathfrak{II}\leq\mathrm{C}\eta^{\alpha(\delta+1)}\mathrm{G}(a).$

So we get

(3.6) $\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}\mathrm{G}(|\nabla
u-q|)\,dx\leq\mathrm{C}\eta^{-n}\sigma\mathrm{G}(a)+\mathrm{C}\eta^{-n}+\mathrm{C}\mathrm{G}(a)\eta^{\alpha(\delta+1)}$

and

(3.7) $\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}\mathrm{G}(|\nabla
u|)\,dx\leq\mathrm{C}\eta^{-n}\sigma\mathrm{G}(a)+\mathrm{C}\eta^{-n}+\mathrm{C}\mathrm{G}(a)\eta^{\alpha(\delta+1)}+\mathrm{C}\mathrm{G}(|q|).$

Next, we choose $\sigma=\eta^{n+1}$

$\mathrm{C}\eta^{-n}\sigma\mathrm{G}(a)+\mathrm{C}\eta^{-n}+\mathrm{C}\mathrm{G}(a)\eta^{\alpha(\delta+1)}=\mathrm{C}\eta\mathrm{G}(a)+\mathrm{C}\eta^{-n}+\mathrm{C}\mathrm{G}(a)\eta^{\alpha(\delta+1)}$

and $a$ such that

(3.8)
$\mathrm{C}\eta\mathrm{G}(a)+\mathrm{C}\eta^{-n}+\mathrm{C}\mathrm{G}(a)\eta^{\alpha(\delta+1)}\leq\mathrm{G}(\varepsilon
a).$

The inequality (3.8) holds if

$a\geq\mathrm{G}^{-1}\left(\frac{\mathrm{C}\eta^{-n}}{\varepsilon^{g_{0}+1}-\mathrm{C}\eta-\mathrm{C}\eta^{\alpha(\delta+1)}}\right).$

(Note that the denominator is positive if $\eta$ is small enough.) Then,

$\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}\mathrm{G}(|\nabla
u|)\,dx\leq\mathrm{G}(\varepsilon a)+\mathrm{C}\mathrm{G}(|q|).$

Next we split the two possible cases; if

$|q|\leq\mathrm{G}^{-1}\left(\frac{1}{\mathrm{C}}\mathrm{G}\left(\frac{a}{4}\right)\right)$

making $\varepsilon<1/4$ we get

$\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}\mathrm{G}(|\nabla
u|)\,dx\leq
2\mathrm{G}\left(\frac{a}{4}\right)\leq\frac{2}{2^{\delta+1}}\mathrm{G}\left(\frac{a}{2}\right)\leq\mathrm{G}\left(\frac{a}{2}\right)$

which is the first alternative in the Lemma.

On the other hand, if

$|q|>\mathrm{G}^{-1}\left(\frac{1}{\mathrm{C}}\mathrm{G}\left(\frac{a}{4}\right)\right)$

then by (3.6)

$\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}\mathrm{G}(|\nabla
u-q|)\,dx\leq\mathrm{G}(\varepsilon a).$

Noting that (3.5) implies the bound on $|q|$ we finish the proof. ∎

######  Remark 3.2.

Using the $\Delta_{2}$ condition (see in particular (2.7)) we can replace
(3.3) by

(3.9) $\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}}}\mathrm{G}(|\nabla
u-q|)\,dx\leq\varepsilon^{\delta+1}\mathrm{G}(a).$

The following lemma can be seen as an improvement of the previous one in the
case where $\varepsilon$ and $\sigma$ are (universally) small enough.

######  Lemma 3.3.

Let $0<a_{0}<a_{1}$ and $\mathrm{G}$ a Young function satisfying (2.1) such
that $\mathrm{G}^{\prime}=g$ is a convex function. Let $u\in W^{1,G}(B_{1})$
with $u\geq 0$ and:

$\mathcal{J}_{\mathrm{G}}(u,B_{1})\leq(1+\sigma)\mathcal{J}_{\mathrm{G}}(v,B_{1})$

for all $v\in W^{1,\mathrm{G}}(B_{1})$ such that $v=u$ on $\partial B_{1}$.
Define $\mathrm{G}(a)$ be as in (3.1) and assume

$a_{0}\leq a\leq a_{1}.$

Assume also that

$\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}\mathrm{G}(|\nabla
u-q|)\,dx\leq\mathrm{G}\left(a\right)\varepsilon^{\delta+1}$

for $\varepsilon$ small enough with $q\in\mathbb{R}^{n}$ satisfying

(3.10)
$\mathrm{G}^{-1}\left(\frac{1}{\mathrm{C}}\mathrm{G}\left(\frac{a}{8}\right)\right)\leq|q|\leq
2\mathrm{C}_{0}a$

with $C_{0}$ given in (3.4).

Then there exist universal constants $\varepsilon_{0},\alpha,r\in(0,1)$ such
that if $\varepsilon<\varepsilon_{0}$ and $\sigma\leq
c_{0}\mathrm{G}(\varepsilon)$ then

$\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{r}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}}}\mathrm{G}(|\nabla
u-\tilde{q}|)\,dx\leq r^{\alpha(\delta+1)}\mathrm{G}(a)\varepsilon^{\delta+1}$

where $\tilde{q}\in\mathbb{R}^{n}$ verifies

$|q-\tilde{q}|\leq\tilde{C}\varepsilon^{\tau}a\qquad\mbox{ where
}\qquad\tau=\frac{\delta+1}{g_{0}+1}$

###### Proof.

Let $\hat{v}$ be the $g-$harmonic replacement of $u$ in $B_{1/2}$ and define

$\begin{cases}v=\hat{v}&\quad\text{ in }B_{1/2},\\\ v=u&\quad\text{ in
}B_{1}\setminus B_{1/2}.\end{cases}$

Since $v\in W^{1,G}(B_{1})$ and $v=u$ on $\partial B_{1}$ we have

$\displaystyle\mathcal{J}_{\mathrm{G}}(u,B_{1/2})+\mathcal{J}_{\mathrm{G}}(u,B_{1}\setminus
B_{1/2})$ $\displaystyle=\mathcal{J}_{\mathrm{G}}(u,B_{1})$
$\displaystyle\leq\mathcal{J}_{\mathrm{G}}(v,B_{1})+\sigma\mathcal{J}_{\mathrm{G}}(v,B_{1})$
$\displaystyle=\mathcal{J}_{\mathrm{G}}(v,B_{1/2})+\mathcal{J}_{\mathrm{G}}(v,B_{1}\setminus
B_{1/2})+\sigma\mathcal{J}_{\mathrm{G}}(v,B_{1}).$

so

$\mathcal{J}_{\mathrm{G}}(u,B_{1/2})\leq\mathcal{J}_{\mathrm{G}}(v,B_{1/2})+\sigma\mathcal{J}_{\mathrm{G}}(v,B_{1}).$

Now using the definition of $\mathcal{J}_{G}$ on the previous inequality we
have

$\int_{B_{1/2}}\mathrm{G}(|\nabla u|)\,dx+|\\{u>0\\}\cap
B_{1/2}|\leq\int_{B_{1/2}}\mathrm{G}(|\nabla
v|)\,dx+|B_{1/2}|+\sigma\mathcal{J}_{\mathrm{G}}(v,B_{1})$

and hence

$\int_{B_{1/2}}\mathrm{G}(|\nabla u|)-\mathrm{G}(|\nabla
v|)\,dx\leq|B_{1/2}|-|\\{u>0\\}\cap
B_{1/2}|+\sigma\mathcal{J}_{\mathrm{G}}(v,B_{1}).$

Using Lemma 2.5, the fact that

$|B_{1/2}|-|\\{u>0\\}\cap B_{1/2}|=|\\{u=0\\}\cap B_{1/2}|$

and that

$\mathcal{J}_{\mathrm{G}}(v,B_{1})\leq\int_{B_{1}}\mathrm{G}(|\nabla
u|)\,dx+|B_{1}|\leq|B_{1}|\left(\mathrm{G}(a)+1\right)$

we arrive at the following inequality:

(3.11) $\int_{B_{1/2}}\mathrm{G}(|\nabla u-\nabla
v|)\,dx\leq\mathrm{C}\sigma(\mathrm{G}(a)+1)+|\\{u=0\\}\cap B_{1/2}|.$

Next we will prove that

(3.12) $|\\{u=0\\}\cap
B_{1/2}|\leq\mathrm{C}\varepsilon^{\delta+1}\varepsilon^{\gamma}$

for some $\gamma\in(0,1)$. We start out by defining

$\mathfrak{l}(x)\mathrel{\mathop{:}}=q\cdot x+b$

with

$b\mathrel{\mathop{:}}=\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}u(x)\>dx.$

Notice that

$\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}(u(x)-\mathfrak{l}(x))\>dx=-\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}q\cdot
x\>dx=0.$

Denote

$\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}(u(x)-\mathfrak{l}(x))\>dx=(u-\mathfrak{l})_{B_{1}}$

we have by Poincaré inequality

(3.13)
$\|u-\mathfrak{l}-(u-\mathfrak{l})_{B_{1}}\|_{L^{\mathrm{G}}(B_{1})}=\|u-\mathfrak{l}\|_{L^{\mathrm{G}}(B_{1})}\leq\mathrm{C}\|\nabla(u-\mathfrak{l})\|_{L^{\mathrm{G}}(B_{1})}.$

Now, we want to use the fact that

$\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}\mathrm{G}(|\nabla
u-q|)\,dx\leq\mathrm{C}\mathrm{G}\left(a\right)\varepsilon^{\delta+1}$

so we need to compare modulars and norms. Applying (2.9) to $w=\nabla(u-l)$
and using the previous inequality we have

(3.14)
$\|\nabla(u-\mathfrak{l})\|_{L^{\mathrm{G}}(B_{1})}\leq\mathrm{C}\varepsilon^{\tau}\mathrm{G}(a)^{\frac{1}{g_{0}+1}}.$

and by (3.13)

$\|u-\mathfrak{l}\|_{L^{\mathrm{G}}(B_{1})}\leq\mathrm{C}\varepsilon^{\tau}\mathrm{G}(a)^{\frac{1}{g_{0}+1}}.$

Next, we point out that if we denote
$\mathfrak{l}^{-}\mathrel{\mathop{:}}=\min\\{0,l\\}$ then $l^{-}\leq|u-l|$ and
hence

$\mathrm{G}\left(\frac{\mathfrak{l}^{-}}{\lambda}\right)\leq\mathrm{G}\left(\frac{|u-\mathfrak{l}|}{\lambda}\right)$

and we get

(3.15)
$\|\mathfrak{l}^{-}\|_{L^{\mathrm{G}}(B_{1})}\leq\|u-\mathfrak{l}\|_{L^{\mathrm{G}}(B_{1})}\leq\mathrm{C}\varepsilon^{\tau}\mathrm{G}(a)^{\frac{1}{g_{0}+1}}.$

Next, we claim that

(3.16)
$\mathfrak{l}(x)\geq\tilde{\mathrm{c}}\mathrm{G}(a)^{\frac{1}{g_{0}+1}}$

in $B_{1/2}$ for sufficiently small $\varepsilon$ and some constant
$\tilde{c}$. Let us denote

$\sigma\mathrel{\mathop{:}}=\mathrm{G}(a)^{\frac{1}{g_{0}+1}}$

to ease notation and suppose that (3.16) does not hold, i.e. for any $c$ there
exists $x_{0}\in B_{1/2}$ such that

$\mathfrak{l}(x_{0})<\mathrm{c}\sigma.$

Since $\mathfrak{l}(x_{0})=q\cdot x_{0}+b$ and hence

$\mathfrak{l}(x)-b\in\left[\frac{-|q|}{2},\frac{|q|}{2}\right]$

this yields

$\mathrm{c}\sigma>b-\frac{|q|}{2}$

or

(3.17) $\mathrm{c}\sigma+\frac{|q|}{2}>b.$

Let

$\mathcal{B}\mathrel{\mathop{:}}=\left\\{x=-\frac{tq}{|q|}+\eta,t\in\left[\frac{6}{8},\frac{7}{8}\right],\eta\in
B_{1/8}\right\\}$

so that, in particular, $\mathcal{B}\subset B_{1}$.

Now, for $x\in\mathcal{B}$ and using (3.17)

$\mathfrak{l}(x)=-t|q|+\eta\cdot
q+b<\mathrm{c}\sigma+\frac{|q|}{2}-\frac{6}{8}|q|+\frac{1}{8}|q|=\mathrm{c}\sigma-\frac{1}{8}|q|$

but then (3.10) gives

$\mathfrak{l}(x)<\mathrm{c}\sigma-\frac{1}{8}\mathrm{G}^{-1}\left(\frac{1}{\mathrm{C}}\mathrm{G}\left(\frac{\sigma}{8}\right)\right)$

from which we get the bound

$\mathfrak{l}(x)<\mathrm{c}\sigma-\frac{\mathrm{C}}{64}\sigma.$

which taking $\mathrm{c}>0$ small enough gives
$\mathfrak{l}(x)<-\bar{\mathrm{C}}\sigma$ for a positive constant
$\bar{\mathrm{C}}$. Inserting this into (3.15) gives

$\varepsilon^{\tau}\sigma\geq\|\mathfrak{l}^{-}\|_{L^{\mathrm{G}}(B_{1})}>\|\bar{\mathrm{C}}\sigma\|_{L^{\mathrm{G}}(B_{1})}$

which is a contradiction if $\varepsilon>0$ is small enough. Therefore (3.16)
must hold.

The next step of the proof is divided into two cases according to the
embedding properties of Orlicz-Sobolev spaces (see [1, Sections 8.27-8.35]).
First, if

$\int_{1}^{\infty}\frac{\mathrm{G}^{-1}(\tau)}{\tau^{1+\frac{1}{n}}}\>d\tau=\infty$

we consider the the Sobolev conjugate of $\mathrm{G}$

$(\mathrm{G}^{\ast})^{-1}(t)\mathrel{\mathop{:}}=\int_{0}^{t}\frac{\mathrm{G}^{-1}(\tau)}{\tau^{1+\frac{1}{n}}}\>d\tau.$

Then, we have that

$\|u-\mathfrak{l}\|_{L^{\mathrm{G}^{\ast}}(B_{1})}\leq\mathrm{C}\|\nabla(u-\mathfrak{l})\|_{L^{\mathrm{G}}(B_{1})}$

(here we have also used (3.13)) which combined with (3.14) gives

$\|u-\mathfrak{l}\|_{L^{\mathrm{G}^{\ast}}(B_{1})}\leq\mathrm{C}\varepsilon^{\tau}\mathrm{G}(a)^{\frac{1}{g_{0}+1}}.$

On the other hand, (3.16) gives that

$\|\mathfrak{l}^{-}\|_{L^{\mathrm{G}^{\ast}}(B_{1})}\geq\mathrm{c}\mathrm{G}(a)^{\frac{1}{g_{0}+1}}\|\chi_{\\{u=0\\}\cap
B_{1/2}}\|_{\mathrm{G}^{\ast}}\geq\mathrm{c}\mathrm{G}(a)^{\frac{1}{g_{0}+1}}|\\{u=0\\}\cap
B_{1/2}|^{1/(\delta+1)^{\ast}}$

and putting both inequalities together we obtain

$\mathrm{c}\mathrm{G}(a)^{\frac{1}{g_{0}+1}}|\\{u=0\\}\cap
B_{1/2}|^{1/(\delta+1)^{\ast}}\leq\mathrm{C}\varepsilon^{\tau}\mathrm{G}(a)^{\frac{1}{g_{0}+1}}$

or

$|\\{u=0\\}\cap
B_{1/2}|\leq\mathrm{C}\varepsilon^{\tau(\delta+1)^{\ast}}=\mathrm{C}\varepsilon^{\delta+1}\varepsilon^{\gamma}$

which is (3.12) with $\gamma=\tau(\delta+1)^{\ast}-(\delta+1)>0$ (being
positive owing to (2.2)).

The second case is when

$\int_{1}^{\infty}\frac{\mathrm{G}^{-1}(\tau)}{\tau^{1+\frac{1}{n}}}\>d\tau<\infty,$

in which the Morrey-type embedding holds and (again using Poincaré)

$\sup_{B_{1}}|u-\mathfrak{l}|\leq\mathrm{C}\|\nabla(u-\mathfrak{l})\|_{L^{\mathrm{G}}(B_{1})}$

so that

$\sup_{B_{1}}|u-\mathfrak{l}|\leq\mathrm{C}\varepsilon^{\tau}\mathrm{G}(a)^{\frac{1}{g_{0}+1}}.$

But then, using again (3.16), we have that in $B_{1/2}$

$\tilde{\mathrm{c}}\mathrm{G}(a)^{\frac{1}{g_{0}+1}}-u\leq\mathfrak{l}-u\leq\sup_{B_{1}}|u-\mathfrak{l}|\leq\mathrm{C}\varepsilon^{\tau}\mathrm{G}(a)^{\frac{1}{g_{0}+1}}$

so that, for $x\in B_{1/2}$

$\mathrm{G}(a)^{\frac{1}{g_{0}+1}}\left(\tilde{\mathrm{c}}-\mathrm{C}\varepsilon^{\tau}\right)\leq
u(x)$

which means that $u(x)>0$ if $\varepsilon>0$ is small enough. In this case

$|\\{u=0\\}\cap B_{1/2}|=0$

and (3.12) is therefore proved.

With the aid of (3.12), (3.11) reads

(3.18) $\int_{B_{1/2}}\mathrm{G}(|\nabla u-\nabla
v|)\,dx\leq\mathrm{C}\sigma(\mathrm{G}(a)+1)+\mathrm{C}_{0}\varepsilon^{\delta+1}\varepsilon^{\gamma}.$

This readily implies, using (2.5) and the hypothesis, that

$\displaystyle\int_{B_{1/2}}\mathrm{G}(|\nabla v-q|)\,dx$
$\displaystyle\leq\mathrm{C}\left(\int_{B_{1/2}}\mathrm{G}(|\nabla
u-q|)\,dx+\int_{B_{1/2}}\mathrm{G}(|\nabla u-\nabla v|)\,dx\right)$
$\displaystyle\leq\mathrm{C}\varepsilon^{\delta+1}\mathrm{G}(a)+\mathrm{C}\sigma(\mathrm{G}(a)+1)+\mathrm{C}_{0}\varepsilon^{\delta+1}\varepsilon^{\gamma}.$

Now using the fact that $\sigma\leq\mathrm{c}_{0}\varepsilon^{\delta+1}$
($\mathrm{c}_{0}>0$ to be chosen later) and recalling that
$\varepsilon^{\gamma}\leq\mathrm{G}(a)$ and $a\in[a_{0},a_{1}]$ and the we
arrive at

(3.19) $\int_{B_{1/2}}\mathrm{G}(|\nabla
v-q|)\,dx\leq\mathrm{C}\varepsilon^{\delta+1}\mathrm{G}(a).$

The next step is to show that (3.19) implies

(3.20) $|\nabla v(x)-q|\,dx\leq\mathrm{C}\cdot(\varepsilon a)^{\nu},\>\forall
x\in B_{1/2}$

for some $\mathrm{C}>0,\nu\in(0,1)$ and $\varepsilon$ small enough.

Assume that (3.20) is false, then there exists $\hat{x}\in B_{1/2}$ such that

$|\nabla v(\hat{x})-q|\,dx>\mathrm{C}\cdot(\varepsilon a)^{\nu}.$

By interior regularity, for any $x\in B_{1/8}(\hat{x})$ we have

$|\nabla v(x)-\nabla v(\hat{x})|\,dx\leq\mathrm{C}|x-\hat{x}|^{\alpha}$

with a constant $\mathrm{C}>0$ that, thanks to (2.11), depends only on
$n,g_{0}$ and $a_{1}$. Then,

$\displaystyle|\nabla v(x)-q|$ $\displaystyle\geq|\nabla v(\hat{x})-q|-|\nabla
v(x)-\nabla v(\hat{x})|$ $\displaystyle>\mathrm{C}\cdot(\varepsilon
a)^{\nu}-\mathrm{C}|x-\hat{x}|^{\alpha}$
$\displaystyle\geq\frac{\mathrm{C}\cdot(\varepsilon a)^{\nu}}{2}$

as long as $x\in B_{\left(\frac{\mathrm{C}\cdot(\varepsilon
a)^{\nu}}{2}\right)^{1/\alpha}}(\hat{x})$ which is contained in $B_{1/2}$ for
$\varepsilon$ small enough. Then

$\displaystyle\int_{B_{1/2}}\mathrm{G}\left(|\nabla v(x)-q|\right)\>dx$
$\displaystyle\geq\int_{B_{\left(\frac{\mathrm{C}\cdot(\varepsilon
a)^{\nu}}{2\mathrm{C}}\right)^{1/\alpha}}(\hat{x})}\mathrm{G}\left(|\nabla
v(x)-q|\right)\,dx$
$\displaystyle\geq\mathrm{G}\left(\frac{\mathrm{C}\cdot(\varepsilon
a)^{\nu}}{2\mathrm{C}}\right)\left|\frac{\mathrm{C}\cdot(\varepsilon
a)^{\nu}}{2\mathrm{C}}\right|^{n/\alpha}|B_{1}|.$

This, together with (3.19) gives

$\mathrm{G}\left(\frac{\mathrm{C}\cdot(\varepsilon
a)^{\nu}}{2}\right)\left(\frac{\mathrm{C}\cdot(\varepsilon
a)^{\nu}}{2}\right)^{n/\alpha}|B_{1}|\leq\mathrm{C}\varepsilon^{\delta+1}\mathrm{G}(a)$

or, using (2.6),

$\left(\frac{\mathrm{C}\cdot(\varepsilon
a)^{\nu}}{2}\right)^{g_{0}+1+\frac{n}{\alpha}}|B_{1}|\leq\varepsilon^{\delta+1}\mathrm{G}(a)$

which gives a contradiction choosing $\nu$ appropriately and for $\varepsilon$
small enough thus proving (3.20).

Next we consider the vectorial function
$\overrightarrow{\mathbf{F}}:\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n}$
given by

$\overrightarrow{\mathbf{F}}(z)\mathrel{\mathop{:}}=g(|z|)\frac{z}{|z|}$

and compute

$\displaystyle\overrightarrow{\mathbf{F}}(\nabla
v)-\overrightarrow{\mathbf{F}}(q)$
$\displaystyle=\int_{0}^{1}\frac{d}{dt}\overrightarrow{\mathbf{F}}\left(q+t(\nabla
v-q)\right)\>dt$
$\displaystyle=\int_{0}^{1}D\overrightarrow{\mathbf{F}}\left(q+t(\nabla
v-q)\right)\cdot(\nabla v-q)\>dt$ $\displaystyle=\mathfrak{A}(x)(\nabla v-q)$

with
$\displaystyle\mathfrak{A}(x)\mathrel{\mathop{:}}=\int_{0}^{1}D\overrightarrow{\mathbf{F}}\left(q+t(\nabla
v-q)\right)\>dt$. Nonetheless, we have that

$\mathop{\text{div}}\overrightarrow{\mathbf{F}}(\nabla
v)=\mathop{\text{div}}\overrightarrow{\mathbf{F}}(q)=0$

and hence

$\mathop{\text{div}}\left(\mathfrak{A}(x)(\nabla v-q)\right)=0.$

Applying interior estimates (i.e. using Proposition 2.2 with Remark 2.3 and
Lemma 2.6) and using (3.19) we have that

$\mathrm{G}(|\nabla v(x)-q)|)\leq\sup_{B_{R/4}(x)}\mathrm{G}\left(|\nabla
v-q|\right)\leq\mathrm{C}\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{R/2}(x)}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{R/2}(x)}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{R/2}(x)}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{R/2}(x)}}\mathrm{G}\left(|\nabla
v-q|\right)\>dx\leq\mathrm{C}\mathrm{G}(a)\varepsilon^{\delta+1}.$

or, since $\mathrm{G}$ is increasing,

$|\nabla
v(x)-q|\leq\mathrm{C}\mathrm{G}^{-1}\left(\mathrm{G}(a)\varepsilon^{\delta+1}\right).$

In particular,

$|\bar{q}|\mathrel{\mathop{:}}=|\nabla
v(0)-q|\leq\mathrm{C}\mathrm{G}^{-1}\left(\mathrm{G}(a)\varepsilon^{\delta+1}\right)$

so that

$|\nabla v(x)-q-\bar{q}|\leq
2\mathrm{C}\mathrm{G}^{-1}\left(\mathrm{G}(a)\varepsilon^{\delta+1}\right),\>x\in
B_{1/2}.$

This, using again (2.12) in Proposition 2.2 we have

$\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{r}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}}}\mathrm{G}(|\nabla
v-q-\bar{q}|)\,dx\leq\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{r}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}}}\mathrm{G}\left(\mathrm{C}\left(\frac{r}{\frac{1}{2}}\right)^{\mu}\sup_{B_{\frac{1}{2}}}|\nabla
v-q-\bar{q}|\right)\leq\mathrm{C}r^{\mu(\delta+1)}\mathrm{G}(a)\varepsilon^{\delta+1}.$

This and (3.18), together with the doubling condition, give

$\displaystyle\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{r}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}}}\mathrm{G}(|\nabla
u-q-\bar{q}|)\,dx$
$\displaystyle\leq\mathrm{C}\left(\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{r}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}}}\mathrm{G}\left(|\nabla
u-\nabla v|\right)\>dx+\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{r}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}}}\mathrm{G}(|\nabla
v-q-\bar{q}|)\,dx\right)$
$\displaystyle\leq\frac{\mathrm{C}}{r^{n}}\left(\sigma(\mathrm{G}(a)+1)+\mathrm{C}_{0}\varepsilon^{\delta+1}\varepsilon^{\gamma}\right)+\mathrm{C}r^{\mu(\delta+1)}\mathrm{G}(a)\varepsilon^{\delta+1}$
$\displaystyle\leq\tilde{\mathrm{C}}r^{-n}\sigma(\mathrm{G}(a)+1)+\tilde{\mathrm{C}}r^{-n}\varepsilon^{\delta+1}\varepsilon^{\gamma}+\tilde{\mathrm{C}}r^{\mu(\delta+1)}\mathrm{G}(a)\varepsilon^{\delta+1}.$

Now defining $\alpha_{0}\mathrel{\mathop{:}}=\mu$ and for $\alpha<\alpha_{0}$,
we set

$r^{\delta+1}=(3\tilde{\mathrm{C}})^{\frac{1}{\alpha-\alpha_{0}}},\quad\varepsilon_{0}\mathrel{\mathop{:}}=\left(\frac{r^{\alpha(\delta+1)+n}\mathrm{G}(a_{0})}{3\tilde{\mathrm{C}}}\right)^{\frac{1}{\gamma}},\quad\mathrm{c}_{0}\mathrel{\mathop{:}}=\frac{r^{\alpha(\delta+1)+n}\mathrm{G}(a_{0})}{3\tilde{\mathrm{C}}(\mathrm{G}(a_{1})+1)}$

so that

$\displaystyle\tilde{\mathrm{C}}r^{\mu(\delta+1)}\mathrm{G}(a)\varepsilon^{\delta+1}=\tilde{\mathrm{C}}r^{(\alpha_{0}-\alpha)(\delta+1)}r^{\alpha(\delta+1)}\mathrm{G}(a)\varepsilon^{\delta+1}=\frac{1}{3}r^{\alpha(\delta+1)}\mathrm{G}(a)\varepsilon^{\delta+1}$
$\displaystyle\tilde{\mathrm{C}}r^{-n}\varepsilon^{\delta+1}\varepsilon^{\gamma}\leq\tilde{\mathrm{C}}r^{-n}\frac{r^{\alpha(\delta+1)+n}\mathrm{G}(a_{0})}{4\tilde{\mathrm{C}}}\varepsilon^{\delta+1}\leq\frac{1}{3}r^{\alpha(\delta+1)}\mathrm{G}(a)\varepsilon^{\delta+1}$
$\displaystyle\tilde{\mathrm{C}}r^{-n}\sigma(\mathrm{G}(a)+1)\leq\tilde{\mathrm{C}}r^{-n}\frac{r^{\alpha(\delta+1)+n}\mathrm{G}(a_{0})}{4\tilde{\mathrm{C}}(\mathrm{G}(a_{1})+1)}\varepsilon^{\delta+1}(\mathrm{G}(a)+1)\leq\frac{1}{3}r^{\alpha(\delta+1)}\mathrm{G}(a)\varepsilon^{\delta+1}.$

All this gives

$\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{r}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}}}\mathrm{G}(|\nabla
u-q-\bar{q}|)\,dx\leq r^{\alpha(\delta+1)}\mathrm{G}(a)\varepsilon^{\delta+1}$

and hence the desired result with $\tilde{q}\mathrel{\mathop{:}}=q+\bar{q}$. ∎

As a consequence of the previous lemma we obtain the crucial $L^{\infty}$
estimate for the gradient of an almost minimizer for sufficiently small
$\kappa$.

######  Corollary 3.4.

Let $a_{1}>a_{0}>0$ and let $u$ be an almost minimizer of $\mathcal{J}_{G}$ in
$B_{1}$ with constant $\kappa$ and exponent $\beta$. Set

$\mathrm{G}(a)\mathrel{\mathop{:}}=\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}\mathrm{G}(|\nabla
u|)\,dx$

and assume there exist $q\in\mathbb{R}^{n}$ and constants
$\mathrm{C}_{0},\mathrm{C}_{1}>0$ such that

(3.21) $\mathrm{G}(a)\in[a_{0},a_{1}],\quad\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}\mathrm{G}(|\nabla
u-q|)\,dx\leq\mathrm{C}_{1}\varepsilon^{\delta+1}\mathrm{G}(a),\quad\mathrm{G}^{-1}\left(\frac{1}{C}\mathrm{G}\left(\frac{a}{4}\right)\right)<|q|<2\mathrm{C}_{0}a.$

Then there exist $\varepsilon_{0},\kappa_{0}>0$ and $\gamma\in(0,1)$ depending
on $n,\delta,g_{0},\beta,a_{0}$ and $a_{1}$ such that for
$0<\varepsilon\leq\varepsilon_{0}$ and
$0<\kappa\leq\kappa_{0}\varepsilon^{\delta+1}$ there exists a linear function

$\mathfrak{l}(x)\mathrel{\mathop{:}}=q\cdot x+b$

such that

$\|u-\mathfrak{l}\|_{C^{1,\gamma}(B_{1/2})}\leq\mathrm{C}\mathrm{G}^{-1}\left(\mathrm{C}\rho^{-n-\alpha(\delta+1)}\varepsilon^{\delta+1}\mathrm{G}(a)\right)$

where $\mathrm{C}=\mathrm{C}(n,\delta,g_{0})>0$.

Moreover

(3.22) $\|\nabla u\|_{L^{\infty}(B_{1/2})}\leq\hat{\mathrm{C}}a$

where $\hat{\mathrm{C}}>0$ is a universal constant.

###### Proof.

We may assume that $u$ is an almost minimizer in $B_{2}$ with the same
constant and exponent. The goal is to iterate Lemma 3.3 with
$\alpha\mathrel{\mathop{:}}=\min\left\\{\frac{\alpha_{0}}{2},\frac{\beta}{g_{0}+1}\right\\}$,
$\alpha_{0}$ from Lemma 3.3.

We are going to show by induction that for some
$\rho\in(0,1),\>\tilde{\mathrm{C}}>0$ universal and all $k\geq 0$ there exists
$q_{k}\in\mathbb{R}^{n}$ such that

(3.23) $\displaystyle\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{\rho^{k}}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\rho^{k}}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\rho^{k}}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\rho^{k}}}}\mathrm{G}\left(|\nabla
u-q_{k}|\right)\>dx\leq\mathrm{C}\rho^{k\alpha(\delta+1)}\varepsilon^{\delta+1}\mathrm{G}(a),$
(3.24)
$\displaystyle\mathrm{G}^{-1}\left(\frac{1}{\mathrm{C}}\mathrm{G}\left(\frac{a}{8}\right)\right)-\tilde{\mathrm{C}}\varepsilon^{\tau}a\left(\frac{1-\rho^{k\alpha\tau}}{1-\rho^{\alpha\tau}}\right)\leq|q_{k}|\leq
2\mathrm{C}_{0}a+\tilde{\mathrm{C}}\varepsilon^{\tau}a\left(\frac{1-\rho^{k\alpha\tau}}{1-\rho^{\alpha\tau}}\right),$
(3.25)
$\displaystyle\mathrm{G}\left(\frac{|q|}{2}\right)\leq\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{\rho^{k}}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\rho^{k}}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\rho^{k}}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\rho^{k}}}}\mathrm{G}\left(|\nabla
u|\right)\>dx\leq 2\mathrm{G}\left(|q|\right)$

with

$\tau\mathrel{\mathop{:}}=\left(\frac{\delta+1}{g_{0}+1}\right).$

For $k=0$, pick $q_{0}\mathrel{\mathop{:}}=q$; then (3.23) and (3.24) follow
just from the corresponding inequalities in (3.21).

Now for the inductive step let $r\mathrel{\mathop{:}}=\rho^{k}$ and
$u_{r}(x)\mathrel{\mathop{:}}=\frac{1}{r}u(rx)$ and get, by inductive
hypothesis, that

$\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}\mathrm{G}\left(|\nabla
u_{r}-q_{k}|\right)\>dx\leq\mathrm{C}_{1}\rho^{k\alpha(\delta+1)}\varepsilon^{\delta+1}\mathrm{G}(a)=\mathrm{C}_{1}\varepsilon_{k}^{\delta+1}\mathrm{G}(a)$

if we set $\varepsilon_{k}\mathrel{\mathop{:}}=\rho^{k\alpha}\varepsilon$.
Then, Lemma 3.3 implies that there exists $q_{k+1}\in\mathbb{R}^{n}$ such that

$\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{\rho}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\rho}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\rho}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\rho}}}\mathrm{G}\left(|\nabla
u_{r}-q_{k+1}|\right)\>dx\leq\mathrm{C}_{1}\varepsilon_{k}^{\delta+1}\mathrm{G}(a)\rho^{(\delta+1)\alpha}=\mathrm{C}_{1}\rho^{(k+1)\alpha(\delta+1)}\varepsilon^{\delta+1}\mathrm{G}(a)$

and

(3.26)
$|q_{k}-q_{k+1}|<\tilde{\mathrm{C}}a\varepsilon_{k}^{\tau}=\mathrm{C}a\varepsilon^{\tau}\rho^{k\alpha\tau}.$

But, rescaling the integral back (and recalling the definition of $r$), we
have that

$\displaystyle\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{\rho}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\rho}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\rho}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\rho}}}\mathrm{G}\left(|\nabla
u_{r}-q_{k+1}|\right)\>dx$
$\displaystyle=\frac{1}{\rho^{n}|B_{1}|}\int_{B_{\rho}}\mathrm{G}\left(|\nabla
u_{r}-q_{k+1}|\right)\>dx$
$\displaystyle=\frac{1}{(r\rho)^{n}|B_{1}|}\int_{B_{r}\rho}\mathrm{G}\left(|\nabla
u-q_{k+1}|\right)\>dx$ $\displaystyle=\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{\rho^{k+1}}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\rho^{k+1}}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\rho^{k+1}}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\rho^{k+1}}}}\mathrm{G}\left(|\nabla
u-q_{k+1}|\right)\>dx$

so (3.23) is proved. Incidentally, we point out that the application of Lemma
3.3 is where the smallness restriction on $\kappa$ appears.

Regarding (3.24), we can compute, using the inductive hypothesis and (3.26),

$\displaystyle|q_{k+1}|$ $\displaystyle\leq|q_{k+1}-q_{k}|+|q_{k}|$
$\displaystyle\leq\mathrm{C}a\varepsilon^{\tau}\rho^{k\alpha\tau}+2\mathrm{C}_{0}a+\tilde{\mathrm{C}}\varepsilon^{\tau}a\left(\frac{1-\rho^{k\alpha\tau}}{1-\rho^{\alpha\tau}}\right).$

Then, we have that

(3.27) $|q_{k+1}|\leq
2\mathrm{C}_{0}a+\tilde{\mathrm{C}}\varepsilon^{\tau}a\left(\rho^{k\alpha\tau}+\frac{1-\rho^{k\alpha\tau}}{1-\rho^{\alpha\tau}}\right)=2\mathrm{C}_{0}a+\tilde{\mathrm{C}}\varepsilon^{\tau}a\left(\frac{1-\rho^{(k+1)\alpha\tau}}{1-\rho^{\alpha\tau}}\right)$

which is the upper bound on (3.24). For the lower bound, recall again that the
inductive hypothesis and (3.26) give

$\displaystyle|q_{k+1}|$ $\displaystyle\geq|q_{k}|-|q_{k+1}-q_{k}|$
$\displaystyle\geq\mathrm{G}^{-1}\left(\frac{1}{\mathrm{C}}\mathrm{G}\left(\frac{a}{8}\right)\right)-\tilde{\mathrm{C}}\varepsilon^{\tau}a\left(\frac{1-\rho^{k\alpha\tau}}{1-\rho^{\alpha\tau}}\right)-\tilde{\mathrm{C}}a\varepsilon^{\tau}\rho^{k\alpha\tau}a.$

It remains to prove (3.25).

$\displaystyle\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{\rho^{k+1}}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\rho^{k+1}}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\rho^{k+1}}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\rho^{k+1}}}}\mathrm{G}\left(|\nabla
u|\right)\>dx-\mathrm{G}(|q|)$
$\displaystyle\leq\frac{\mathrm{C}}{2}\left(\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{\rho^{k+1}}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\rho^{k+1}}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\rho^{k+1}}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\rho^{k+1}}}}\mathrm{G}\left(|\nabla
u-q_{k+1}|\right)\>dx+\mathrm{G}(|q_{k+1}|)-\mathrm{G}(|q|)\right)$
$\displaystyle\leq\frac{\mathrm{C}}{2}\left(\mathrm{C}\rho^{(k+1)\alpha(\delta+1)}\varepsilon^{\delta+1}\mathrm{G}(a)+\sum_{j=0}^{k}|\mathrm{G}(|q_{j+1}|)-\mathrm{G}(|q_{j}|)|\right).$

Now, for some $q_{j}^{\ast}$ between $q_{j+1}$ and $q_{j}$ we have

$|\mathrm{G}(|q_{j+1}|)-\mathrm{G}(|q_{j}|)|=g\left(|q_{j}^{\ast}|\right)|q_{j+1}-q_{j}|$

and recalling that $g$ is increasing we have

$\displaystyle\sum_{j=0}^{k}|\mathrm{G}(|q_{j+1}|)-\mathrm{G}(|q_{j}|)|$
$\displaystyle\leq\sum_{j=0}^{k}g\left(|q_{j+1}|\right)|q_{j+1}-q_{j}|$
$\displaystyle\leq\mathrm{C}\sum_{j=0}^{k}(\mathrm{c}g(a)+\mathrm{c}\varepsilon^{\delta\tau}g(a))a(\varepsilon\rho^{j\alpha})^{\tau}$
$\displaystyle=\mathrm{C}(\mathrm{c}g(a)a\varepsilon^{\tau}+\mathrm{c}\varepsilon^{\delta\tau}g(a)a\varepsilon^{\tau})\sum_{j=0}^{k}\rho^{j\alpha\tau}$
$\displaystyle\leq\mathrm{C}\left(\mathrm{G}(a)\varepsilon^{\tau}+\varepsilon^{(\delta+1)\tau}\mathrm{G}(a)\right)\frac{1}{1-\rho^{\alpha\tau}}$
$\displaystyle=\mathrm{C}\left(1+\frac{1}{1-\rho^{\alpha\tau}}\right)\varepsilon^{\tau}\mathrm{G}(a)$

Putting this inequality together with the previous one, we wind up with

$\left|\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{\rho^{k+1}}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\rho^{k+1}}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\rho^{k+1}}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\rho^{k+1}}}}\mathrm{G}\left(|\nabla
u|\right)\>dx-\mathrm{G}(|q|)\right|\leq\mathrm{C}(\varepsilon^{\tau}+\varepsilon^{\delta})\mathrm{G}(a)\leq\mathrm{G}\left(\frac{|q|}{2}\right)$

as desired.

Now we want to apply the Campanato-type estimate in the Appendix (Theorem 4.3)
with $\Omega=B_{1/2}$ and $\lambda=n+\alpha(\delta+1)>n$; for that, we must
first show that

(3.28) $\varrho^{-\lambda}\inf_{\xi\in\mathbb{R}^{n}}\int_{B_{1/2}\cap
B_{\varrho}(x_{0})}\mathrm{G}(|\nabla
u-q-\xi|)\>dx\leq\mathrm{C}\varepsilon^{\delta+1}\mathrm{G}(a).$

For that we split the cases $\varrho\in(0,1)$ and $\varrho\geq 1$.

In the latter case the bound is obtained by

$\varrho^{-\lambda}\inf_{\xi\in\mathbb{R}^{n}}\int_{B_{1/2}\cap
B_{\varrho}(x_{0})}\mathrm{G}(|\nabla
u-q-\xi|)\>dx\leq\int_{B_{1/2}}\mathrm{G}(|\nabla
u-q|)\>dx\leq|B_{1}|\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{1}}}\mathrm{G}(|\nabla
u-q|)\>dx$

and the hypothesis.

If $\varrho\in(0,1)$ then we use a $\rho-$adyc argument: let
$\rho^{k+1}\leq\varrho\leq\rho^{k}$ and use the (3.23) to bound

$\displaystyle\varrho^{-\lambda}\inf_{\xi\in\mathbb{R}^{n}}\int_{B_{1/2}\cap
B_{\varrho}(x_{0})}\mathrm{G}(|\nabla u-q-\xi|)\>dx$
$\displaystyle\leq\rho^{-\lambda(k+1)}\int_{B_{1/2}\cap
B_{\rho^{k}}(x_{0})}\mathrm{G}(|\nabla u-q_{k}|)\>dx$
$\displaystyle\leq\rho^{-\lambda(k+1)+kn}|B_{1}|\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{\rho^{k}}(x_{0})}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\rho^{k}}(x_{0})}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\rho^{k}}(x_{0})}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\rho^{k}}(x_{0})}}\mathrm{G}(|\nabla
u-q_{k}|)\>dx$
$\displaystyle\leq\mathrm{C}\rho^{-(n+\alpha(\delta+1))(k+1)+kn+k\alpha(\delta+1)}|B_{1}|\varepsilon^{\delta+1}\mathrm{G}(a)$
$\displaystyle=\mathrm{C}\rho^{-n-\alpha(\delta+1)}|B_{1}|\varepsilon^{\delta+1}\mathrm{G}(a)$

and we get (3.28).

Since we already have a bound on
$\displaystyle\int_{B_{1/2}}\mathrm{G}(|\nabla u-q|)\>dx$, Theorem 4.3 says
that $\nabla u-q$ belongs to $C^{0,\gamma}(B_{1/2})$ and

(3.29) $[\nabla
u-q]_{C^{0,\gamma}(B_{1/2})}\leq\mathrm{C}\mathrm{G}^{-1}\left(\rho^{-n-\alpha(\delta+1)}\varepsilon^{\delta+1}\mathrm{G}(a)\right).$

Now if we define $\mathfrak{l}(x)=u(0)+x\cdot q$ then in $B_{1/2}$ it holds

$|u(x)-\mathfrak{l}(x)|=\left|\frac{d}{dt}\int_{0}^{1}\left(\nabla
u(tx)-q\right)\cdot
x\>dt\right|\leq\mathrm{C}\mathrm{G}^{-1}\left(\rho^{-n-\alpha(\delta+1)}\varepsilon^{\delta+1}\mathrm{G}(a)\right)$

and therefore

$\|u-\mathfrak{l}\|_{L^{\infty}(B_{1/2})}\leq\mathrm{C}\mathrm{G}^{-1}\left(\rho^{-n-\alpha(\delta+1)}\varepsilon^{\delta+1}\mathrm{G}(a)\right)$

which together with (3.29) and a standard interpolation inequality (see for
instance [39, page 9]) gives the first part of the result.

As for (3.22), simply note that

$\|\nabla
u\|_{L^{\infty}(B_{1/2})}\leq\|\nabla(u-q)\|_{L^{\infty}(B_{1/2})}+|q|$

But taking limit in (3.27) we have

$|q|\leq
2\mathrm{C}_{0}a+\tilde{\mathrm{C}}\varepsilon^{\tau}a\left(\frac{1}{1-\rho^{\alpha\tau}}\right)$

and therefore

$\|\nabla
u\|_{L^{\infty}(B_{1/2})}\leq\mathrm{C}\mathrm{G}^{-1}\left(\mathrm{C}\rho^{-n-\alpha(\delta+1)}\varepsilon^{\delta+1}\mathrm{G}(a)\right)+2\mathrm{C}_{0}a+\tilde{\mathrm{C}}\varepsilon^{\tau}a\left(\frac{1}{1-\rho^{\alpha\tau}}\right)\leq\hat{\mathrm{C}}a$

as desired. ∎

##  4 Proof of the Theorem 1.1

In this section we give the proof of Theorem 1.1. As is customary we reduce
the proof to the case where $\Omega$ is a ball, the general case following
from a covering procedure.

###### Proof of Theorem 1.1.

Let us define

$\mathrm{G}\left(a(\tau)\right)\mathrel{\mathop{:}}=\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{\tau}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\tau}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\tau}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\tau}}}\mathrm{G}\left(|\nabla
u|\right)\>dx$

and for $r\in(0,\eta]$ we want to show that

(4.1)
$\mathrm{G}(a(r))\leq\mathrm{C}(\eta,\mathrm{M})\left(1+\mathrm{G}(a(1))\right),$

Let consider the set $\mathcal{K}\subset\mathbb{N}=\\{0,1,2,\cdots\\}$
containing all the $k$ such that

(4.2) $\mathrm{G}(a(\eta^{k}))\leq\eta^{-n}\mathrm{M}+2^{-k}\mathrm{G}(a(1)).$

It is easy to check that $0\in\mathcal{K}$, then $\mathcal{K}\neq\emptyset$.
So, we have two different possibilities: $\mathcal{K}=\mathbb{N}$ or
$\mathcal{K}\subset\mathbb{N}$ and $\mathcal{K}\neq\mathbb{N}$.

In the case $\mathcal{K}=\mathbb{N}$ we have that for $r\in(0,1)$ there exists
a (minimum) value of $k\geq 0$ such that $\eta^{k+1}\leq r<\eta^{k}$. Then

$\displaystyle\mathrm{G}(a(r))$ $\displaystyle=\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{r}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}}}\mathrm{G}\left(|\nabla
u|\right)\>dx$
$\displaystyle\leq\frac{|B_{\eta^{k}}|}{|B_{r}|}\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{\eta^{k}}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta^{k}}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta^{k}}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta^{k}}}}\mathrm{G}\left(|\nabla
u|\right)\>dx$ $\displaystyle\leq\eta^{-n}\mathrm{G}(a(\eta^{k}))$
$\displaystyle\leq\eta^{-n}\left(\eta^{-n}\mathrm{M}+2^{-k}\mathrm{G}(a(1))\right)$
$\displaystyle\leq\mathrm{C}(\eta,\mathrm{M})(1+\mathrm{G}(a(1)))$

so (4.1) holds.

In the other case, there exists a minimum integer $k+1$ for which the
inequality fails (4.2). Further, it must happen that

(4.3) $\mathrm{G}(a(\eta^{k}))>\mathrm{M};$

if not, we have that

$\mathrm{G}(a(\eta^{k+1}))\leq\eta^{-n}\mathrm{G}(a(\eta^{k}))\leq\eta^{-n}\mathrm{M}\leq\eta^{-n}\mathrm{M}+2^{-{(k+1)}}\mathrm{G}(a(1))$

and (4.2) is also satisfied for $k+1$.

Now, $\mathrm{G}(a(\eta^{k}))>\mathrm{M}$ gives, owing to Lemma 3.1 in
$B_{\eta}$ we have that either

$\mathrm{G}(a(\eta^{k+1}))\leq\mathrm{G}\left(\frac{a(\eta^{k})}{2}\right)$

or

$\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{\eta^{k+1}}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta^{k+1}}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta^{k+1}}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta^{k+1}}}}\mathrm{G}(|\nabla
u-q|)\,dx\leq\varepsilon^{\delta+1}\mathrm{G}(a(\eta^{k}))$

where $q\in\mathbb{R}^{n}$ verifies

$\frac{1}{\mathrm{C}}\mathrm{G}\left(\frac{a(\eta^{k})}{4}\right)<\mathrm{G}(|q|)<\mathrm{C}\mathrm{G}(a((\eta^{k})).$

Suppose that
$\mathrm{G}(a(\eta^{k+1}))\leq\mathrm{G}(\frac{a(\eta^{k})}{2})\leq\frac{\mathrm{G}(a(\eta^{k}))}{2}$.
Since

(4.4)
$\mathrm{G}(a(\eta^{k+1}))>\eta^{-n}\mathrm{M}+2^{-{(k+1)}}\mathrm{G}(a(1))\geq\frac{\eta^{-n}\mathrm{M}+2^{-{k}}\mathrm{G}(a(1))}{2}\geq\frac{\mathrm{G}(a(\eta^{k}))}{2}$

this leads to a contradiction.

If the second alternative holds true, then we are in situation to apply
Corollary 3.4. Observe that by (4.3) and (4.4) we have that

$\mathrm{G}(a(\eta^{k+1}))\geq\frac{\mathrm{M}}{2}.$

On the other hand,

$\mathrm{G}(a(\eta^{k+1}))\leq\eta^{-n}\mathrm{G}(a(\eta^{k}))\leq\eta^{-n}(\eta^{-n}\mathrm{M}+2^{-k}\mathrm{G}(a(1)))\leq\eta^{-n}(\eta^{-n}\mathrm{M}+\mathrm{G}(a(1))).$

Now, we apply Corollary 3.4, with $a_{0}=\frac{\mathrm{M}}{2}$ and
$a_{1}=\eta^{-n}(\eta^{-n}\mathrm{M}+\mathrm{G}(a(1)))$ and taking
$2\varepsilon$ instead $\varepsilon$. Then

$\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{\eta}^{k+1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}^{k+1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}^{k+1}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{\eta}^{k+1}}}\mathrm{G}\left(|\nabla
u-q|\right)\>dx\leq(2\varepsilon)^{\delta+1}\mathrm{G}(a(\eta^{k}))$

and

(4.5) $\|\nabla
u\|_{L^{\infty}\left(B_{\frac{\eta^{k+1}}{2}}\right)}\leq\mathrm{C}a(\eta^{k}).$

By (4.5), for all $r\in\left(0,\frac{\eta^{k+1}}{2}\right)$.

$\displaystyle\mathrm{G}(a(r))$ $\displaystyle=\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{r}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}}}\mathrm{G}\left(|\nabla
u|\right)\>dx\leq\mathrm{C}\mathrm{G}(a(\eta^{k}))\leq\mathrm{C}(\eta^{-n}\mathrm{M}+2^{-k}\mathrm{G}(a(1)))$
$\displaystyle\leq\mathrm{C}(\eta^{-n}\mathrm{M}+\mathrm{G}(a(1)))\leq\mathrm{C}(\mathrm{M},\eta)(1+\mathrm{G}(a(1))).$

If $r\in\left[\frac{\eta^{k+1}}{2},\eta\right)$ then there exists $k_{r}$ such
that $\eta^{k_{r}+1}<r\leq\eta^{k_{r}}$. Then

$\frac{1}{\eta^{k_{r}}}\leq\frac{1}{r}\leq\frac{2}{\eta^{k_{r}+1}}$

and

$k_{r}\leq k+\mathrm{C}_{\ast}$

So, there are two cases, first suppose that $k_{r}\in\\{0,\cdots,k\\}$ then

$\displaystyle\mathrm{G}(a(r))$
$\displaystyle=\frac{1}{|B_{\eta^{k_{r}+1}}|}\int_{B_{\eta}^{k_{r}}}\mathrm{G}\left(|\nabla
u|\right)\>dx\leq\eta^{-n}\mathrm{G}(\eta^{k_{r}})\leq\eta^{-n}(\eta^{-n}\mathrm{M}+2^{-k_{r}}\mathrm{G}(a(1)))$
$\displaystyle\leq\eta^{-n}(\eta^{-n}\mathrm{M}+\mathrm{G}(a(1)))\leq\mathrm{C}(\mathrm{M},\eta)(1+\mathrm{G}(a(1))).$

If $k_{r}>k$ then

$\displaystyle\mathrm{G}(a(r))$
$\displaystyle=\frac{1}{|B_{\eta^{k+\mathrm{C}_{\ast}+1}}|}\int_{B_{\eta}^{k}}\mathrm{G}\left(|\nabla
u|\right)\>dx\leq\eta^{-n(\mathrm{C}_{\ast}+1)}\mathrm{G}(\eta^{k})\leq\eta^{-n(\mathrm{C}_{\ast}+1)}(\eta^{-n}\mathrm{M}+2^{-k}\mathrm{G}(a(1)))$
$\displaystyle\leq\eta^{-n(\mathrm{C}_{\ast}+1)}(\eta^{-n}\mathrm{M}+\mathrm{G}(a(1)))\leq\mathrm{C}(\mathrm{M},\eta)(1+\mathrm{G}(a(1))).$

so, the proof of (4.1) is completed.

Now, we extend (4.1) for balls with center $x_{0}$ and radius small enough.
So, for all $r\in[0,\eta]$ we have

(4.6)
$\mathrm{G}(a(r,x_{0}))\leq\mathrm{C}(\eta,\mathrm{M})\left(1+\mathrm{G}(a(1))\right),$

where

$\mathrm{G}\left(a(r,x_{0})\right)\mathrel{\mathop{:}}=\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{r}(x_{0})}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}(x_{0})}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}(x_{0})}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}(x_{0})}}\mathrm{G}\left(|\nabla
u|\right)\>dx$

For $x_{0}\in B_{\frac{1}{2}}$

$\mathrm{G}(|\nabla u(x_{0})|)=\lim_{r\to
0}\mathrm{G}(a(r,x_{0}))\leq\mathrm{C}(\eta,\mathrm{M})(1+\mathrm{G}(a(1))=\mathrm{C}\cdot\left(1+\int_{B_{1}}\mathrm{G}\left(|\nabla
u|\right)\>dx\right)$

Finally, we obtain

$\|\nabla
u\|_{L^{\infty}\left(B_{\frac{1}{2}}\right)}\leq\mathrm{G}^{-1}\left(1+\int_{B_{1}}\mathrm{G}\left(|\nabla
u|\right)\>dx\right)$

∎

## Appendix: a Campanato-type estimate

In this section we prove a Campanato-type result in the context of Orlicz
modulars, i.e. that functions with appropriate average decay are Hölder
continuous. As far as the authors’ are concerned, this result is new in the
specialized literature and hence has independent interest. The proof follows
Fernández Bonder’s personal notes [38] for the homogeneous setting.

Let us denote

$\Omega_{x_{0},\varrho}\mathrel{\mathop{:}}=\Omega\cap
B_{\varrho}(x_{0})\quad\text{and}\quad
u_{x_{0},\varrho}\mathrel{\mathop{:}}=\frac{1}{|\Omega_{x_{0},\varrho}|}\int_{\Omega_{x_{0},\varrho}}u(x)\>dx=\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0pt\Omega_{x_{0},\varrho}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{\Omega_{x_{0},\varrho}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{\Omega_{x_{0},\varrho}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{\Omega_{x_{0},\varrho}}}u(x)dx.$

For our purposes, the analogous of the $L^{p}$ norm and Campanato seminorm
(see for instance [42]) will be given, for $\lambda\geq 0$, by

(4.7)
$\Phi(u)\mathrel{\mathop{:}}=\int_{\Omega}\mathrm{G}(|u|)\>dx,\quad\Phi_{\lambda}(u)\mathrel{\mathop{:}}=\sup_{\begin{subarray}{c}x_{0}\in\Omega\\\
\varrho>0\end{subarray}}\varrho^{-\lambda}\int_{\Omega_{x_{0},\varrho}}\mathrm{G}(|u-u_{x_{0},\varrho}|)\>dx$

These quantities are not homogeneous and hence not a norm/seminorm, but they
capture the essential properties that will be used to prove the desired
continuity.

We will work with functions satisfying

(4.8) $\Phi(u)+\Phi_{\lambda}(u)\leq\mathrm{C}_{\Phi}$

for some constant $\mathrm{C}_{\Phi}$. The first step is to prove that every
point in $\Omega$ is a Lebesgue point of $u$:

######  Proposition 4.1.

If $u$ satisfies (4.8) and $\lambda>n$ then there exists $\tilde{u}=u$ a.e.
such that

$\lim_{r\rightarrow 0^{+}}u_{x_{0},r}=\tilde{u}(x_{0})$

uniformly in $\Omega$.

###### Proof.

Step 1. Let $0<r_{1}<r_{2}$. Then for $y\in B_{r_{1}}(x_{0})$ and using (2.5)
we have that

$\mathrm{G}(|u_{x_{0},r_{1}}-u_{x_{0},r_{2}}|)\leq
2^{g_{0}}\left(\mathrm{G}(|u_{x_{0},r_{1}}-u(y)|)+\mathrm{G}(|u_{x_{0},r_{2}}-u(y)|)\right).$

Integrating over $B_{r_{1}}(x_{0})$ we get that

$|B_{1}|r_{1}^{n}\mathrm{G}(|u_{x_{0},r_{1}}-u_{x_{0},r_{2}}|)\leq
2^{g_{0}}\Phi_{\lambda}(u)(r_{1}^{\lambda}+r_{2}^{\lambda})$

and we get

$\mathrm{G}(|u_{x_{0},r_{1}}-u_{x_{0},r_{2}}|)\leq
2^{g_{0}}\frac{r_{1}^{\lambda}+r_{2}^{\lambda}}{|B_{1}|r_{1}^{n}}\Phi_{\lambda}(u).$

or

(4.9)
$|u_{x_{0},r_{1}}-u_{x_{0},r_{2}}|\leq\mathrm{G}^{-1}\left(2^{g_{0}}\frac{r_{1}^{\lambda}+r_{2}^{\lambda}}{|B_{1}|r_{1}^{n}}\Phi_{\lambda}(u)\right).$

Step 2. We will iterate (4.9) dyadically; for $j\in\mathbb{N}_{0}$ and fixed
$r>0$ we have

$\mathrm{G}(|u_{x_{0},2^{-(j+1)}r}-u_{x_{0},2^{-j}r}|)\leq
2^{g_{0}}\frac{2^{-(j+1)\lambda}r^{\lambda}+2^{-j\lambda}r^{\lambda}}{|B_{1}|2^{(-j-1)n}r^{n}}\Phi_{\lambda}(u).$

This means that

$|u_{x_{0},2^{-(j+1)}r}-u_{x_{0},2^{-j}r}|\leq\mathrm{G}^{-1}\left(\frac{2^{g_{0}}}{|B_{1}|}\Phi_{\lambda}(u)\right)r^{\frac{\lambda-n}{g_{0}+1}}2^{\frac{j(n-\lambda)}{g_{0}+1}}$

and since

$|u_{x_{0},r}-u_{x_{0},2^{-k}r}|\leq\sum_{j=0}^{k}|u_{x_{0},2^{-(j+1)}r}-u_{x_{0},2^{-j}r}|$

we get

(4.10)
$|u_{x_{0},r}-u_{x_{0},2^{-k}r}|\leq\mathrm{G}^{-1}\left(\frac{2^{g_{0}}}{|B_{1}|}\Phi_{\lambda}(u)\right)r^{\frac{\lambda-n}{g_{0}+1}}\sum_{j=0}^{k}2^{\frac{j(n-\lambda)}{g_{0}+1}}$

Step 3. Now we can use (4.10) in the following way

$\displaystyle|u_{x_{0},2^{-k}r}-u_{x_{0},2^{-(k+i)}r}|$
$\displaystyle\leq\mathrm{G}^{-1}\left(\frac{2^{g_{0}}}{|B_{1}|}\Phi_{\lambda}(u)\right)\left(\frac{r}{2^{k}}\right)^{\frac{\lambda-n}{g_{0}+1}}\sum_{j=0}^{i-1}2^{\frac{j(n-\lambda)}{g_{0}+1}}$
$\displaystyle\leq\mathrm{C}\mathrm{G}^{-1}\left(\frac{2^{g_{0}}}{|B_{1}|}\Phi_{\lambda}(u)\right)\left(\frac{r}{2^{k}}\right)^{\frac{\lambda-n}{g_{0}+1}}$

where $\mathrm{C}>0$ depends only on $g_{0},n$ and $\lambda$. In particular
$\left\\{u_{x_{0},2^{-k}r}\right\\}_{k}$ is a Cauchy sequence (uniformly in
$x_{0}$) and hence has a limit which we call $\tilde{u}(x_{0})$:

$\lim_{k\rightarrow\infty}u_{x_{0},2^{-k}r}=:\tilde{u}(x_{0}).$

Notice that: a) for almost every $x_{0}$ Lebesgue’s Differentiation Theorem
says that $\tilde{u}(x_{0})=u(x_{0})$; and b) $\tilde{u}$ does not depend on
the specific choice of $r$. Indeed, passing to the limit as
$i\rightarrow\infty$ above we obtain that

$|u_{x_{0},2^{-k}r}-\tilde{u}(x_{0})|\leq\mathrm{C}\mathrm{G}^{-1}\left(\frac{2^{g_{0}}}{|B_{1}|}\Phi_{\lambda}(u)\right)\left(\frac{r}{2^{k}}\right)^{\frac{\lambda-n}{g_{0}+1}}$

and then, if we consider a $r^{\prime}>0$, we can bound, using (4.9) and the
previous inequality

$\displaystyle|u_{x_{0},2^{-k}r^{\prime}}-\tilde{u}(x_{0})|$
$\displaystyle\leq|u_{x_{0},2^{-k}r^{\prime}}-u_{x_{0},2^{-k}r}|+|u_{x_{0},2^{-k}r}-\tilde{u}(x_{0})|$
$\displaystyle\leq\mathrm{G}^{-1}\left(2^{g_{0}}\frac{{r^{\prime}}^{\lambda}+r^{\lambda}}{|B_{1}|{r^{\prime}}^{n}}\Phi_{\lambda}(u)2^{-k(\lambda-n)}\right)+\mathrm{C}\mathrm{G}^{-1}\left(\frac{2^{g_{0}}}{|B_{1}|}\Phi_{\lambda}(u)\right)\left(\frac{r}{2^{k}}\right)^{\frac{\lambda-n}{g_{0}+1}}$
$\displaystyle\leq
2^{-k\frac{\lambda-n}{g_{0}+1}}\left(\mathrm{G}^{-1}\left(2^{g_{0}}\frac{{r^{\prime}}^{\lambda}+r^{\lambda}}{|B_{1}|{r^{\prime}}^{n}}\Phi_{\lambda}(u)\right)+\mathrm{C}\mathrm{G}^{-1}\left(\frac{2^{g_{0}}}{|B_{1}|}\Phi_{\lambda}(u)\right)r^{\frac{\lambda-n}{g_{0}+1}}\right)$

which goes to $0$ as $k\rightarrow\infty$.

Finally, we go from discrete to continuous: we pass to the limit on $k$ to
obtain in (4.10) to get

(4.11)
$|u_{x_{0},r}-\tilde{u}(x_{0})|\leq\mathrm{G}^{-1}\left(\frac{2^{g_{0}}}{|B_{1}|}\Phi_{\lambda}(u)\right)r^{\frac{\lambda-n}{g_{0}+1}}\sum_{j=0}^{\infty}2^{\frac{j(n-\lambda)}{g_{0}+1}}$

and taking the limit $r\rightarrow 0^{+}$ we get the desired uniform
convergence. ∎

Next we want to show that $u$ is indeed Hölder continuous in $\Omega$. We
identify it with $\tilde{u}$ and drop the $\>\tilde{}\>$ for simplicity.

######  Proposition 4.2.

If $u$ satisfies (4.8) and $\lambda>n$ then there exists
$\mathrm{C}=\mathrm{C}(n,g_{0},\lambda)>0$ such that

$\sup_{\begin{subarray}{c}x,y\in\Omega\\\ x\neq
y\end{subarray}}\frac{|u(x)-u(y)|}{|x-y|^{\gamma}}\leq\mathrm{C}\mathrm{G}^{-1}\left(\frac{2^{g_{0}}}{|B_{1}|}\Phi_{\lambda}(u)\right)$

with $\gamma=\frac{\lambda-n}{g_{0}+1}$.

###### Proof.

We start with the following claim: if $x,y\in\Omega$ and $r=2|x-y|$ then

$|u_{x,r}-u_{y,r}|\leq\mathrm{C}\mathrm{G}^{-1}\left(\frac{2^{g_{0}}}{|B_{1}|}\Phi_{\lambda}(u)\right)|x-y|^{\gamma}$

with $\gamma=\frac{\lambda-n}{g_{0}+1}$.

To prove the claim we note that for $z\in B_{r}(x)\cap B_{r}(y)$ it holds that

$\mathrm{G}(|u_{x,r}-u_{y,r}|)\leq
2^{g_{0}}\left(\mathrm{G}(|u_{x,r}-u(z)|)+\mathrm{G}(|u_{y,r}-u(z)|)\right)$

and so integrating over $B_{r}(x)\cap B_{r}(y)$ we get that

$\displaystyle|B_{r}(x)\cap B_{r}(y)|\mathrm{G}(|u_{x,r}-u_{y,r}|)$
$\displaystyle\leq
2^{g_{0}}\left(\int_{B_{r}(x)}\mathrm{G}(|u_{x,r}-u(z)|)\>dz+\int_{B_{r}(y)}\mathrm{G}(|u_{y,r}-u(z)|)\>dz\right)$
$\displaystyle\leq 2^{g_{0}+1}r^{\lambda}\Phi_{\lambda}(u)$

which leads to

$|u_{x,r}-u_{y,r}|\leq\mathrm{G}^{-1}\left(\frac{1}{|B_{r}(x)\cap
B_{r}(y)|}2^{g_{0}+1}r^{\lambda}\Phi_{\lambda}(u)\right).$

The claim follows simply by noting that $B_{\frac{r}{2}}(x)\subset
B_{r}(x)\cap B_{r}(y)$.

Since

$|u(x)-u(y)|\leq|u(x)-u_{x,r}|+|u_{x,r}-u_{y,r}|+|u_{y,r}-u(y)|$

the proposition follows using the claim and (4.11). ∎

We remark that $\Phi_{\lambda}(u)$ is equivalent to

$\sup_{\begin{subarray}{c}x_{0}\in\Omega\\\
\varrho>0\end{subarray}}\varrho^{-\lambda}\inf_{\xi\in\mathbb{R}}\int_{\Omega_{x_{0},\varrho}}\mathrm{G}(|u(x)-\xi|)\>dx;$

indeed, one inequality being trivial, the other one follows by noting that
Jensen’s inequality gives

$\mathrm{G}(|u_{x_{0},\varrho}-\xi|)=\mathrm{G}\left(\left|\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0pt\Omega_{x_{0},\varrho}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{\Omega_{x_{0},\varrho}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{\Omega_{x_{0},\varrho}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{\Omega_{x_{0},\varrho}}}(u(x)-\xi)\>dx\right|\right)\leq\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0pt\Omega_{x_{0},\varrho}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{\Omega_{x_{0},\varrho}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{\Omega_{x_{0},\varrho}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{\Omega_{x_{0},\varrho}}}\mathrm{G}\left(\left|u(x)-\xi\right|\right)\>dx$

from where

$\displaystyle\int_{\Omega_{x_{0},\varrho}}\mathrm{G}\left(\left|u(x)-u_{x_{0},\varrho}\right|\right)\>dx$
$\displaystyle\leq
2^{g_{0}}\left(\int_{\Omega_{x_{0},\varrho}}\mathrm{G}\left(\left|u(x)-\xi\right|\right)\>dx+\int_{\Omega_{x_{0},\varrho}}\mathrm{G}\left(\left|\xi-
u_{x_{0},\varrho}\right|\right)\>dx\right)$ $\displaystyle\leq
2^{g_{0}+1}\int_{\Omega_{x_{0},\varrho}}\mathrm{G}\left(\left|u(x)-\xi\right|\right)\>dx.$

In summary, we have the following theorem:

######  Theorem 4.3.

Let $u$ be a measurable function satisfying

$\int_{\Omega}\mathrm{G}(|u|)\>dx+\sup_{\begin{subarray}{c}x_{0}\in\Omega\\\
\varrho>0\end{subarray}}\varrho^{-\lambda}\inf_{\xi\in\mathbb{R}}\int_{\Omega_{x_{0},\varrho}}\mathrm{G}(|u(x)-\xi|)\>dx\leq\mathrm{C}_{0}$

for some positive constant $\mathrm{C}_{0}$ and $\lambda>n$. Then, $u\in
C^{0,\gamma}(\Omega)$ with $\gamma=\frac{\lambda-n}{g_{0}+1}$ and

$\sup_{\begin{subarray}{c}x,y\in\Omega\\\ x\neq
y\end{subarray}}\frac{|u(x)-u(y)|}{|x-y|^{\gamma}}\leq\mathrm{C}\mathrm{G}^{-1}\left(\sup_{\begin{subarray}{c}x_{0}\in\Omega\\\
\varrho>0\end{subarray}}\varrho^{-\lambda}\inf_{\xi\in\mathbb{R}}\int_{\Omega_{x_{0},\varrho}}\mathrm{G}(|u(x)-\xi|)dx\right)$

######  Remark 4.4.

One could consider $\lambda=n+g_{0}+1$ in the previous theorem and obtain
$u\in C^{0,1}(\Omega)$, i.e. $u$ is Lipschitz continuous in $\Omega$.

######  Definition 4.5 ([49, Definition 1.2]).

Let $p\in[1,\infty)$ and a $\psi\colon(0,\infty)\to(0,\infty)$ be a continuous
function. We define the generalized Campanato space

$\displaystyle\mathcal{L}_{p,\psi}(\Omega)\mathrel{\mathop{:}}=\left\\{f:\Omega\to\mathbb{R}:\,\,\|f\|_{\mathcal{L}_{p,\psi}(\Omega)}\mathrel{\mathop{:}}=\sup_{B_{r}(x_{0})\subset\Omega}\frac{1}{\psi(r)}\left(\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0ptB_{r}(x_{0})}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}(x_{0})}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}(x_{0})}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{B_{r}(x_{0})}}|f(x)-f_{B_{r}(x_{0})}|^{p}dx\right)^{\frac{1}{p}}<\infty\right\\}.$

######  Remark 4.6.

Observe that $\|f\|_{\mathcal{L}_{p,\psi}(\Omega)}$ is a norm (modulo constant
functions), thus $\mathcal{L}_{p,\psi}(\Omega)$ defines a Banach space.
Moreover, if $p=1$ and $\psi\equiv 1$, then
$\mathcal{L}_{1,1}(\Omega)=\mathrm{BMO}(\Omega)$. Finally, if $p=1$ and
$\psi(r)=r^{\gamma}$ (for some $0<\gamma\leq 1$), then
$\mathcal{L}_{1,r^{\gamma}}(\Omega)$ coincides with standard spaces
$C^{0,\gamma}(\Omega)$.

We also consider now the following functional space

$\text{BMO}^{\ast}(\Omega)\mathrel{\mathop{:}}=\left\\{u:\Omega\to\mathbb{R}:\,\,\,\displaystyle\|u\|_{\text{BMO}^{\ast}(\Omega)}\mathrel{\mathop{:}}=\sup_{x_{0}\in\Omega\atop{\rho>0}}\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0pt\Omega_{x_{0},\rho}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{\Omega_{x_{0},\rho}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{\Omega_{x_{0},\rho}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{\Omega_{x_{0},\rho}}}|u-u_{x_{0},\rho}|dx<\infty\right\\}.$

Now, we will suppose that $\Omega$ satisfies a $\hat{\mathrm{c}}_{0}-$uniform
positive density ($\hat{\mathrm{c}}_{0}-$UPD for short), i.e., there exists a
universal constant $\hat{\mathrm{c}}_{0}\in(0,1)$ such that

$\displaystyle\frac{|\Omega_{x_{0},\rho}|}{|B_{\rho}(x_{0})|}=\frac{|\Omega\cap
B_{\rho}(x_{0})|}{|B_{\rho}(x_{0})|}\geq\hat{\mathrm{c}}_{0}\quad\forall\,\,x_{0}\in\overline{\Omega}\,\,\text{and}\,\,\rho>0.$

Remember we are assuming that $\Omega$ is a bounded Lipschitz domain, thus it
fulfils $\hat{\mathrm{c}}_{0}-$UPD property for some
$\hat{\mathrm{c}}_{0}\in(0,1)$.

Finally, taking into account the above definitions and using Jensen’s
inequality for concave functions, we can conclude in the borderline case,
i.e., $\lambda=n$ (see (4.7) and (4.8)) the following:

$\begin{array}[]{rcl}\displaystyle\|u\|_{\text{BMO}^{\ast}(\Omega)}&\mathrel{\mathop{:}}=&\displaystyle\sup_{x_{0}\in\Omega\atop{\rho>0}}\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0pt\Omega_{x_{0},\rho}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{\Omega_{x_{0},\rho}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{\Omega_{x_{0},\rho}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{\Omega_{x_{0},\rho}}}|u-u_{x_{0},\rho}|dx\\\
&=&\displaystyle\sup_{x_{0}\in\Omega\atop{\rho>0}}\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0pt\Omega_{x_{0},\rho}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{\Omega_{x_{0},\rho}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{\Omega_{x_{0},\rho}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{\Omega_{x_{0},\rho}}}\mathrm{G}^{-1}\left(\mathrm{G}(|u-u_{x_{0},\rho}|)\right)dx\\\
&\leq&\displaystyle\sup_{x_{0}\in\Omega\atop{\rho>0}}\mathrm{G}^{-1}\left(\displaystyle\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0pt\Omega_{x_{0},\rho}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{\Omega_{x_{0},\rho}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{\Omega_{x_{0},\rho}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{\Omega_{x_{0},\rho}}}\mathrm{G}(|u-u_{x_{0},\rho}|)dx\right)\\\
&\leq&\mathrm{G}^{-1}\left(\displaystyle\sup_{x_{0}\in\Omega\atop{\rho>0}}\mathchoice{\mathop{\vrule
width=6.0pt,height=3.0pt,depth=-2.5pt\kern-8.0pt\intop}\nolimits_{\kern-6.0pt\Omega_{x_{0},\rho}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{\Omega_{x_{0},\rho}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{\Omega_{x_{0},\rho}}}{\mathop{\vrule
width=5.0pt,height=3.0pt,depth=-2.6pt\kern-6.0pt\intop}\nolimits_{\Omega_{x_{0},\rho}}}\mathrm{G}(|u-u_{x_{0},\rho}|)dx\right)\\\
&\leq&\mathrm{G}^{-1}\left(\hat{\mathrm{c}}_{0}^{-1}\omega_{n}^{-1}\displaystyle\sup_{x_{0}\in\Omega\atop{\rho>0}}\frac{1}{\rho^{n}}\int_{\Omega_{x_{0},\rho}}G(|u-u_{x_{0},\rho}|)dx\right)\\\
&\leq&\mathrm{G}^{-1}\left(\hat{\mathrm{c}}_{0}^{-1}\omega_{n}^{-1}\mathrm{C}_{\Phi}\right).\end{array}$

Therefore, $u\in\text{BMO}^{\ast}(\Omega)$ provided $\lambda=n$ and $\Omega$
fulfils a $\hat{\mathrm{c}}_{0}-$UPD property.

## Acknowledgments

The authors would like to thank Julián Fernández Bonder for lending his notes
on Morrey estimates for fractional Sobolev spaces, from which the ideas for
the Appendix were taken.

A. Silva was partially supported by ANPCyT under grant PICT 2019-3837, by
CONICET under grant PIP 11220210100238CO and by UNSL under grant PROICO
03-2023. J.V. da Silva has been partially supported by CNPq-Brazil under Grant
No. 307131/2022-0 and FAEPEX-UNICAMP 2441/23 Editais Especiais - PIND -
Projetos Individuais (03/2023). H. Vivas was partially supported by ANPCyT
under grant PICT 2019-03530.

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