# Lipschitz regularity of almost minimizers in one-phase problems driven by
the $p$-Laplace operator

Serena Dipierro  Serena Dipierro: Department of Mathematics and Statistics,
University of Western Australia, 35 Stirling Hwy, Crawley WA 6009 Australia
<EMAIL_ADDRESS>,  Fausto Ferrari  Fausto Ferrari: Dipartimento di
Matematica
Università di Bologna
Piazza di Porta S.Donato 5
40126, Bologna-Italy<EMAIL_ADDRESS>,  Nicolò Forcillo  Nicolò
Forcillo: Dipartimento di Matematica
Università di Bologna
Piazza di Porta S.Donato 5
40126, Bologna-Italy<EMAIL_ADDRESS>and  Enrico Valdinoci  Enrico
Valdinoci: Department of Mathematics and Statistics, University of Western
Australia, 35 Stirling Hwy, Crawley WA 6009 Australia
<EMAIL_ADDRESS>
(Date: August 27, 2024)

###### Abstract.

We prove that, given $p>\max\left\\{\frac{2n}{n+2},1\right\\}$, the
nonnegative almost minimizers of the nonlinear free boundary functional

$J_{p}(u,\Omega):=\int_{\Omega}\Big{(}|\nabla
u(x)|^{p}+\chi_{\\{u>0\\}}(x)\Big{)}\,dx$

are Lipschitz continuous.

S.D. and E.V. are members of AustMS. F.F. and N.F. are members of INdAM. S.D.
has been supported by the Australian Research Council DECRA DE180100957 “PDEs,
free boundaries and applications”. F.F. and N.F. have been supported by 2022-
INDAM- GNAMPA project Regolarità locale e globale per problemi completamente
nonlineari. N.F. has been supported by GHAIA Horizon 2020 MCSA RISE programme
grant No 777822. E.V. has been supported by the Australian Laureate Fellowship
FL190100081 “Minimal surfaces, free boundaries and partial differential
equations”. Part of this work has been completed during a very pleasant visit
of N.F. to the University of Western Australia, that we thank for the warm
hospitality. The authors wish to thank Daniela De Silva and Ovidiu Savin for
useful discussions.

##  1\. Introduction

In this article we consider a nonlinear free boundary problem and we establish
the Lipschitz continuity of its almost minimizers. The classical motivations
for free boundary problems of these types stem from flows with jets and
cavities (see e.g. Section 1.1 in [MR2145284]). In this context, the
conditions arising from a free boundary problem can be seen as the variational
counterpart of Bernoulli’s law according to which pressure is prescribed on
the free streamline as a balance with the velocity (or the kinetic energy) of
the fluid.

The nonlinear feature of the corresponding differential operator aims at
modeling possibly non-Newtonian fluids, in which linear relations between
physical quantities are replaced by power-laws.

Similar models also appear in the study of electrical impedance tomography
(see [MR1607608]), optimal heat flows (see [MR473960]), electrochemical
machining (see [MR983738]) and high activation energy in combustion theory
(see [MR1044809, MR1976085, MR2237691]). Also, free boundary problems can be
used as a sharp-interface approximation of phase coexistence models (see e.g.
[MR2126143, MR2139200, MR2262703], roughly speaking as a replacing for phases
modeled by a smooth state parameter with a merely continuous one).

The setting of almost minimizers is also classical in the calculus of
variations (dating back, at least up to a certain extent, to a famous sentence
in Leibniz’s Specimen Geometriae Luciferae, probably written in the mid-1690s,
“pro minimis adhiberi possunt quasi minima”, that is “the almost minimizers
can be exploited in place of minimizers”).

More specifically, the mathematical setting that we consider here goes as
follows. Let $\Omega\subset\mathbb{R}^{n}$ be a given domain and
$p>\max\left\\{\frac{2n}{n+2},1\right\\}$. We consider the energy functional

(1.1) $J_{p}(u,\Omega):=\int_{\Omega}\Big{(}|\nabla
u(x)|^{p}+\chi_{\\{u>0\\}}(x)\Big{)}\,dx$

for all $u\in W^{1,p}(\Omega)$ with $u\geqslant 0$.

The condition that $u$ is nonnegative corresponds, in the framework of free
boundary problems, to considering “one-phase” solutions (solutions which may
change sign being related to “two-phase” problems).

The precise notion of almost minimizers that we use in this paper is the
following one:

######  Definition 1.1.

Let $\kappa\geqslant 0$ and $\beta>0$. We say that $u\in W^{1,p}(\Omega)$ is
an almost minimizer for $J_{p}$ in $\Omega$, with constant $\kappa$ and
exponent $\beta$, if $u\geqslant 0$ a.e. in $\Omega$ and

(1.2)
$J_{p}(u,B_{\varrho}(x))\leqslant(1+\kappa\varrho^{\beta})J_{p}(v,B_{\varrho}(x)),$

for every ball $B_{\varrho}(x)$ such that
$\overline{B_{\varrho}(x)}\subset\Omega$ and for every $v\in
W^{1,p}(B_{\varrho}(x))$ such that $v=u$ on $\partial B_{\varrho}(x)$ in the
sense of the trace.

In some sense, Definition 1.1 is one of the possible modern formalizations of
Leibniz’s initial intuition reported at the beginning of this paper: namely,
almost minimizers are natural objects to look at, for instance, to deal with
minimizers of “perturbed” functionals. As a concrete example, if we consider

$\widetilde{J}_{p}(u,\Omega):=J_{p}(u,\Omega)+\iint_{\Omega\times\Omega}\Phi(u(y))\,\Phi(u(z))\,\Phi(u(y)-u(z))\,dy\,dz$

for a function $\Phi:\mathbb{R}\to[0,1]$ with $\Phi=0$ in $(-\infty,0]$, we
readily see that $J_{p}(u,B_{r}(x))\leqslant\widetilde{J}_{p}(u,B_{r}(x))$ and

$\displaystyle\widetilde{J}_{p}(u,B_{r}(x))\leqslant
J_{p}(u,B_{r}(x))+\iint_{B_{r}(x)\times
B_{r}(x)}\chi_{\\{u>0\\}}(y)\,\chi_{\\{u>0\\}}(z)\,dy\,dz$
$\displaystyle\qquad\leqslant
J_{p}(u,B_{r}(x))+|B_{r}|\int_{B_{r}(x)}\chi_{\\{u>0\\}}(y)\,dy\leqslant(1+|B_{1}|\,r^{n})J_{p}(u,B_{r}(x)).$

Accordingly, a minimizer for the “complicated” functional $\widetilde{J}_{p}$
turns out to be an almost minimizer for the “simpler” functional $J_{p}$.

As usual, the constants depending only on $n$ and $p$ are called universal. If
$u$ is an almost minimizer, the structural constants may depend on $\kappa$
and $\beta$ as well.

Our main result establishes the Lipschitz regularity of the one-phase almost
minimizers as follows:

######  Theorem 1.2.

Let $p>\max\left\\{\frac{2n}{n+2},1\right\\}$ and $u$ be an almost minimizer
for $J_{p}$ in $B_{1}$ with constant $\kappa$ and exponent $\beta$.

Then,

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

where $C>0$ is a constant depending on $n$, $p$, $\kappa$ and $\beta$.

In addition, $u$ is uniformly Lipschitz continuous in a neighborhood of
$\left\\{u=0\right\\}$, namely if $u(0)=0$ then

$\left|\nabla u\right|\leqslant C\quad\mbox{in }B_{r_{0}},$

for some $C>0$, depending only on $n$, $p$, $\kappa$ and $\beta$, and
$r_{0}\in(0,1)$, depending on $n$, $p$, $\kappa$, $\beta$ and $\|\nabla
u\|_{L^{p}(B_{1})}$.

We stress that Theorem 1.2 is new for $p\neq 2$, the case $p=2$ being treated
in [DS].

We recall that, when $p=2$, minimizers of (1.1) were studied in [MR618549],
where the Lipschitz regularity of minimizers and the regularity of flat free
boundaries were established. In [MR990856, MR973745, MR1029856] Caffarelli
developed a viscosity approach to the free boundary problem for $p=2$ in the
two-phase setting (see also [D] and the references therein for related free
boundary regularity properties). For one-phase problems the viscosity approach
for operators governed by the $p$-Laplace or $p(x)-$Laplace operators has been
developed in [GleRic, FL] and in [FL2] as well, where Lipschitz regularity of
viscosity solutions of inhomogeneous one-phase problems, governed by the
$p(x)-$Laplace operator and some regularity properties of their free
boundaries were established.

The Lipschitz regularity of the minimizers of the functional in (1.1) has been
obtained in [MR3771123], which has also provided a proof of the Lipschitz
regularity when $p=2$ without using monotonicity formulae. We refer to
[MR3910196] as well, where a discrete version of the Weiss monotonicity
formula for the functional in (1.1) has been established for $p$ close to $2$.

Minimizers of the functional in (1.1) have been also considered in
[MR2133664], in which the regularity of the free boundary near flat points was
established. See also [MR2250499, MR2383537, MR2431665, MR2680176, MR2876248,
MR3168631, MR3249814, MR3390082, MR4266232] for regularity results on
$p$-Laplacian free boundary problems.

Regarding the setting of almost minimizers, the case $p=2$ has been
investigated in [MR3385167, MR3948692, DS]. The case of almost minimizers for
$p\neq 2$ was, to the best of our knowledge, not fully investigated, hence
Theorem 1.2 aims at starting a research line in this direction, and the
results presented in this article are part of the PhD thesis of the third
author [F]. In this paper we take inspiration from the approach used in [DS],
see also [MR4062979, MR4201786].

We point out that the case $p\neq 2$ provides significant technical
complications, especially due to the fact that the sum of two solutions is not
a solution any longer, thus making it difficult to develop an exhaustive
theory of harmonic replacements in the nonlinear scenario.

We highlight the fact that the regularity results put forth in this paper do
not follow from the classical ones in the calculus of variations, since the
integrand of the functional that we consider here is a discontinuous function,
due to the presence of the term $\chi_{\\{u>0\\}}$ (instead, the classical
cases dealt with require the integrand to be continuous, or even Lipschitz
continuous, see e.g. assumption (8.48) in [G]).

The paper is organized as follows. In Section 2 we recall the notion of
$p$-harmonic replacement and put forth some basic energy estimates needed in
our main arguments.

Then, in Section 3, we develop a dichotomy theory according to which, roughly
speaking, the average of the energy of an almost minimizer decreases in a
smaller ball, unless we are arbitrarily close to the case of linear functions.

In Section 4 we show that almost minimizers of $J_{p}$ are Lipschitz
continuous, namely we prove Theorem 1.2.

##  2\. The $p$-harmonic replacement

One of the classical ingredients in nonlinear partial differential equations
is the notion of $p$-harmonic replacement, which we now recall:

######  Definition 2.1.

Let $r>0$, $x_{0}\in\mathbb{R}^{n}$ and $u\in W^{1,p}(B_{r}(x_{0}))$. We say
that $v\in W^{1,p}(B_{r}(x_{0}))$ is the $p$-harmonic replacement of $u$ in
$B_{r}(x_{0})$ if

$\int_{B_{r}(x_{0})}\left|\nabla v\right|^{p}\,dx=\min_{u-w\in
W^{1,p}_{0}(B_{r}(x_{0}))}\int_{B_{r}(x_{0})}\left|\nabla w\right|^{p}\,dx.$

We remark that if $v$ is the $p$-harmonic replacement of $u$ in
$B_{r}(x_{0})$, then in particular it satisfies

(2.1) $\int_{B_{r}(x_{0})}\left|\nabla v(x)\right|^{p-2}\nabla
v(x)\cdot\nabla\varphi(x)\,dx=0$

for all $\varphi\in W^{1,p}_{0}(B_{r}(x_{0}))$, namely $v$ is a weak solution
of $\Delta_{p}v=0$ in $B_{r}(x_{0})$.

We now provide some energy estimates of classical flavor for the $p$-harmonic
replacement that we will use in the sequel.

######  Lemma 2.2.

Let $x_{0}\in\mathbb{R}^{n}$, $r>0$ and $u\in W^{1,p}(B_{r}(x_{0}))$. Let $v$
be the $p$-harmonic replacement of $u$ in $B_{r}(x_{0})$. Then,

  * (i)

if $1<p<2,$ then

(2.2) $\begin{split}&\int_{B_{r}(x_{0})}\left|\nabla u(x)-\nabla
v(x)\right|^{p}\,dx\\\ &\qquad\leqslant
C\left(\int_{B_{r}(x_{0})}\left(\left|\nabla u(x)\right|^{p}-\left|\nabla
v(x)\right|^{p}\right)\,dx\right)^{\frac{p}{2}}\left(\int_{B_{r}(x_{0})}\big{(}\left|\nabla
u(x)\right|+\left|\nabla
v(x)\right|\big{)}^{p}\,dx\right)^{1-\frac{p}{2}},\end{split}$

for some positive universal constant $C$;

  * (ii)

if $p\geqslant 2,$ then

(2.3) $\int_{B_{r}(x_{0})}\left|\nabla u(x)-\nabla
v(x)\right|^{p}\,dx\leqslant C\int_{B_{r}(x_{0})}\big{(}\left|\nabla
u(x)\right|^{p}-\left|\nabla v(x)\right|^{p}\big{)}\,dx,$

for some positive universal constant $C$.

###### Proof.

For all $s\in[0,1]$, we consider the family of functions
$u_{s}(x):=su(x)+(1-s)v(x)$. Notice that $u_{0}=v$ and $u_{1}=u$. As a
consequence,

$\displaystyle\int_{B_{r}(x_{0})}\Big{(}\left|\nabla
u(x)\right|^{p}-\left|\nabla v(x)\right|^{p}\Big{)}\,dx$
$\displaystyle=\int_{B_{r}(x_{0})}\left(\int_{0}^{1}\frac{d}{ds}\left|\nabla
u_{s}(x)\right|^{p}\,ds\right)\,dx$
$\displaystyle=\int_{B_{r}(x_{0})}\left(\int_{0}^{1}p\left|\nabla
u_{s}(x)\right|^{p-2}\nabla u_{s}(x)\cdot\nabla(u-v)(x)\,ds\right)\,dx.$

This and (2.1) (used here with $\varphi:=u-v$) lead to

$\displaystyle\int_{B_{r}(x_{0})}\Big{(}\left|\nabla
u(x)\right|^{p}-\left|\nabla v(x)\right|^{p}\Big{)}\,dx$ $\displaystyle=\;$
$\displaystyle p\,\left[\int_{B_{r}(x_{0})}\left(\int_{0}^{1}\left|\nabla
u_{s}(x)\right|^{p-2}\nabla u_{s}(x)\cdot\nabla(u-v)(x)\,ds\right)\,dx\right.$
$\displaystyle\qquad\qquad\qquad\left.-\int_{0}^{1}\left(\int_{B_{r}(x_{0})}\left|\nabla
v(x)\right|^{p-2}\nabla v(x)\cdot\nabla(u-v)(x)\,dx\right)\,ds\right]$
$\displaystyle=\;$ $\displaystyle
p\int_{0}^{1}\bigg{(}\int_{B_{r}(x_{0})}\left(\left|\nabla
u_{s}(x)\right|^{p-2}\nabla u_{s}(x)-\left|\nabla v(x)\right|^{p-2}\nabla
v(x)\right)\cdot\nabla(u-v)(x)\,dx\bigg{)}\,ds.$

We also point out that

(2.4) $u_{s}(x)-v(x)=s(u(x)-v(x)),$

and therefore we arrive at

(2.5) $\begin{split}&\int_{B_{r}(x_{0})}\Big{(}\left|\nabla
u(x)\right|^{p}-\left|\nabla v(x)\right|^{p}\Big{)}\,dx\\\
=\;&p\int_{0}^{1}\frac{1}{s}\bigg{(}\int_{B_{r}(x_{0})}\left(\left|\nabla
u_{s}(x)\right|^{p-2}\nabla u_{s}(x)-\left|\nabla v(x)\right|^{p-2}\nabla
v(x)\right)\cdot\nabla(u_{s}-v)(x)\,dx\bigg{)}\,ds.\end{split}$

Now we recall the inequality

$\big{(}\left|\xi\right|^{p-2}\xi-\left|\zeta\right|^{p-2}\zeta\big{)}\cdot(\xi-\zeta)\geqslant\gamma\begin{cases}\left|\xi-\zeta\right|^{2}\left(\left|\xi\right|+\left|\zeta\right|\right)^{p-2}&\mbox{if
}1<p<2,\\\ \left|\xi-\zeta\right|^{p}&\mbox{if }p\geqslant 2,\end{cases}$

for any $\xi$, $\zeta\in\mathbb{R}^{n}\setminus\\{0\\}$, for some positive
universal constant $\gamma$, see e.g. page 100 in [MR2133664]. Thus, using
this inequality with $\xi:=\nabla u_{s}$ and $\zeta:=\nabla v$ into
(LABEL:lower-bound-integral-nabla-u-nabla-v-1), we obtain that

$\begin{split}&\int_{B_{r}(x_{0})}\Big{(}\left|\nabla
u(x)\right|^{p}-\left|\nabla v(x)\right|^{p}\Big{)}\,dx\\\
\geqslant\;&\begin{cases}\displaystyle
p\,\gamma\int_{0}^{1}\frac{1}{s}\left(\int_{B_{r}(x_{0})}\left|\nabla
u_{s}(x)-\nabla v(x)\right|^{2}(\left|\nabla u_{s}(x)\right|+\left|\nabla
v(x)\right|)^{p-2}\,dx\right)\,ds,&\;\mbox{if }1<p<2,\\\ \displaystyle
p\,\gamma\int_{0}^{1}\frac{1}{s}\left(\int_{B_{r}(x_{0})}\left|\nabla
u_{s}(x)-\nabla v(x)\right|^{p}\,dx\right)\,ds,&\;\mbox{if }p\geqslant
2.\end{cases}\end{split}$

Furthermore, recalling (2.4), we conclude that

(2.6) $\begin{split}&\int_{B_{r}(x_{0})}\Big{(}\left|\nabla
u(x)\right|^{p}-\left|\nabla v(x)\right|^{p}\Big{)}\,dx\\\
\geqslant\;&\begin{cases}\displaystyle
p\,\gamma\int_{0}^{1}{s}\left(\int_{B_{r}(x_{0})}\left|\nabla u(x)-\nabla
v(x)\right|^{2}(\left|\nabla u_{s}(x)\right|+\left|\nabla
v(x)\right|)^{p-2}\,dx\right)\,ds,&\;\mbox{if }1<p<2,\\\ \displaystyle
p\,\gamma\int_{0}^{1}{s}^{p-1}\left(\int_{B_{r}(x_{0})}\left|\nabla
u(x)-\nabla v(x)\right|^{p}\,dx\right)\,ds,&\;\mbox{if }p\geqslant
2.\end{cases}\end{split}$

Now, if $1<p<2$, since $s\in[0,1]$ we have that

$\left|\nabla u_{s}\right|+\left|\nabla v\right|\leqslant s\left|\nabla
u\right|+(1-s)\left|\nabla v\right|+\left|\nabla v\right|=s\left|\nabla
u\right|+(2-s)\left|\nabla v\right|{\leqslant}2\,\big{(}\left|\nabla
u\right|+\left|\nabla v\right|\big{)}.$

Hence, plugging this information into (LABEL:lower-bound-integral-nabla-u-
nabla-v-2), we find that

(2.7) $\begin{split}&\int_{B_{r}(x_{0})}\Big{(}\left|\nabla
u(x)\right|^{p}-\left|\nabla v(x)\right|^{p}\Big{)}\,dx\\\
\geqslant\;&p\,\gamma\,2^{p-2}\int_{0}^{1}{s}\left(\int_{B_{r}(x_{0})}\left|\nabla
u(x)-\nabla v(x)\right|^{2}\big{(}\left|\nabla u(x)\right|+\left|\nabla
v(x)\right|\big{)}^{p-2}\,dx\right)\,ds\\\
=\;&p\,\gamma\,2^{p-3}\int_{B_{r}(x_{0})}\left|\nabla u(x)-\nabla
v(x)\right|^{2}\big{(}\left|\nabla u(x)\right|+\left|\nabla
v(x)\right|\big{)}^{p-2}\,dx.\end{split}$

We now apply the Hölder’s inequality with Hölder exponent $2/p$ and conjugate
exponent

$\left(\frac{2}{p}\right)^{\prime}=\frac{2/p}{2/p-1}=\frac{2}{2-p},$

to see that

$\displaystyle\int_{B_{r}(x_{0})}\left|\nabla u(x)-\nabla v(x)\right|^{p}\,dx$
$\displaystyle=\;$ $\displaystyle\int_{B_{r}(x_{0})}\left|\nabla u(x)-\nabla
v(x)\right|^{p}\big{(}\left|\nabla u(x)\right|+\left|\nabla
v(x)\right|\big{)}^{\frac{p(p-2)}{2}}\big{(}\left|\nabla
u(x)\right|+\left|\nabla v(x)\right|\big{)}^{-\frac{p(p-2)}{2}}\,dx$
$\displaystyle\leqslant\;$ $\displaystyle\left(\int_{B_{r}(x_{0})}\left|\nabla
u(x)-\nabla v(x)\right|^{2}\big{(}\left|\nabla u(x)\right|+\left|\nabla
v(x)\right|\big{)}^{p-2}\,dx\right)^{\frac{p}{2}}$
$\displaystyle\qquad\qquad\times\left(\int_{B_{r}(x_{0})}\big{(}\left|\nabla
u(x)\right|+\left|\nabla v(x)\right|\big{)}^{p}\,dx\right)^{1-\frac{p}{2}}.$

This and (2.7) yield that

$\displaystyle\int_{B_{r}(x_{0})}\left|\nabla u(x)-\nabla v(x)\right|^{p}\,dx$
$\displaystyle\leqslant\;$ $\displaystyle
C\left(\int_{B_{r}(x_{0})}\big{(}\left|\nabla u(x)\right|^{p}-\left|\nabla
v(x)\right|^{p}\big{)}\,dx\right)^{\frac{p}{2}}\left(\int_{B_{r}(x_{0})}\big{(}\left|\nabla
u(x)\right|+\left|\nabla v(x)\right|\big{)}^{p}\,dx\right)^{1-\frac{p}{2}},$

for some universal constant $C>0$, which proves the desired result in (2.2)
when $1<p<2$.

If instead $p\geqslant 2$, we deduce from (LABEL:lower-bound-integral-nabla-u-
nabla-v-2) that

$\displaystyle\int_{B_{r}(x_{0})}\big{(}\left|\nabla
u(x)\right|^{p}-\left|\nabla v(x)\right|^{p}\big{)}\,dx$
$\displaystyle\geqslant
p\,\gamma\int_{0}^{1}{s}^{p-1}\left(\int_{B_{r}(x_{0})}\left|\nabla
u(x)-\nabla v(x)\right|^{p}\,dx\right)\,ds$
$\displaystyle=\gamma\int_{B_{r}(x_{0})}\left|\nabla u(x)-\nabla
v(x)\right|^{p}\,dx,$

which establishes (2.3) when $p\geqslant 2$ and completes the proof of Lemma
2.2. ∎

##  3\. Some dichotomy results

With the preliminary work carried out so far, we can now provide a dichotomy
statement. Roughly speaking, either the average of the energy of an almost
minimizer decreases in a smaller ball, or the distance of its gradient and a
suitable constant vector becomes as small as we wish (that is, linear
functions are the “only ones for which the average does not improve in small
balls”). The precise result goes as follows:

######  Proposition 3.1.

There exists $\varepsilon_{0}\in(0,1)$ such that for every
$\varepsilon\in(0,\varepsilon_{0})$ there exist $\eta\in(0,1)$, $M\geqslant 1$
and $\sigma_{0}\in(0,1)$, depending on $\varepsilon$, $n$ and $p$, such that
if $\sigma\in[0,\sigma_{0}]$ and $a\geqslant M$ then the following statement
holds true.

Let $u\in W^{1,p}(B_{1})$ be such that

(3.1) $J_{p}(u,B_{1})\leqslant(1+\sigma)J_{p}(v,B_{1})$

for all $v\in W^{1,p}(B_{1})$ such that $v=u$ on $\partial B_{1}$, with

(3.2) $a:=\left(\fint_{B_{1}}\left|\nabla u(x)\right|^{p}\,dx\right)^{1/p}.$

Then, either

(3.3) $\left(\fint_{B_{\eta}}\left|\nabla
u(x)\right|^{p}\,dx\right)^{1/p}\leqslant\frac{a}{2},$

or

(3.4) $\left(\fint_{B_{\eta}}\left|\nabla
u(x)-q\right|^{p}\,dx\right)^{1/p}\leqslant\varepsilon a,$

with $q\in\mathbb{R}^{n}$ such that

(3.5) $\frac{a}{4}<\left|q\right|\leqslant C_{0}\,a,$

and $C_{0}>0$ universal.

###### Proof.

Let $v$ be the $p$-harmonic replacement of $u$ in $B_{1}$. By Theorem 3.19 in
[MZ] we have that, for every $x\in B_{1/2}$,

$\displaystyle\left|\nabla v(x)\right|^{p}$
$\displaystyle\leqslant\sup_{B_{1/4}(x)}\left|\nabla v\right|^{p}\leqslant
C\fint_{B_{1/2}(x)}\left|\nabla v(y)\right|^{p}\,dy\leqslant
C\fint_{B_{1}}\left|\nabla u(y)\right|^{p}\,dy.$

As a consequence, for all $x\in B_{1/2}$,

(3.6) $\left|\nabla v(x)\right|\leqslant C_{0}\,a,$

for some positive universal constant $C_{0}$.

Accordingly, we denote by $q:=\nabla v(0)$ and we deduce from (3.6) that
$|q|\leqslant C_{0}\,a$. This, together with Theorem 2 in [Man], gives that,
for all $\eta\in(0,1/2]$,

(3.7) $\begin{split}\fint_{B_{\eta}}\left|\nabla
v(x)-q\right|^{p}\,dx\leqslant\;&\fint_{B_{\eta}}\bigg{(}C\bigg{(}\frac{\eta}{1/2}\bigg{)}^{\alpha}\left\|\nabla
v\right\|_{L^{\infty}(B_{1/2})}\bigg{)}^{p}\,dx\\\ \leqslant\;&C\,\eta^{\alpha
p}\left\|\nabla v\right\|_{L^{\infty}(B_{1/2})}^{p}\\\
\leqslant\;&C_{1}\,\eta^{\alpha p}\,a^{p},\end{split}$

for some $\alpha\in(0,1)$ and $C_{1}>0$ universal.

Furthermore, using (3.1) we have that

(3.8) $\begin{split}\int_{B_{1}}\big{(}\left|\nabla
u(x)\right|^{p}-\left|\nabla v(x)\right|^{p}\big{)}\,dx&\leqslant
J_{p}(u,B_{1})-\int_{B_{1}}\left|\nabla v(x)\right|^{p}\,dx\\\
&\leqslant(1+\sigma)J_{p}(v,B_{1})-\int_{B_{1}}\left|\nabla
v(x)\right|^{p}\,dx\\\ &\leqslant C\left(\sigma\int_{B_{1}}\left|\nabla
v(x)\right|^{p}\,dx+1\right)\\\ &\leqslant
C\left(\sigma\int_{B_{1}}\left|\nabla u(x)\right|^{p}\,dx+1\right)\end{split}$

We distinguish two cases, $p\geqslant 2$ and $1<p<2$.

If $p\geqslant 2$, then from (2.3) and (3.8) we deduce that

$\displaystyle\int_{B_{1}}\left|\nabla u(x)-\nabla v(x)\right|^{p}\,dx$
$\displaystyle\leqslant C\left(\sigma\int_{B_{1}}\left|\nabla
u(x)\right|^{p}\,dx+1\right).$

Taking the average over $B_{1}$, we thereby obtain that

$\fint_{B_{1}}\left|\nabla u(x)-\nabla v(x)\right|^{p}\,dx\leqslant C(\sigma
a^{p}+1).$

Using this and (3.7), we get that

(3.9) $\begin{split}\fint_{B_{\eta}}\left|\nabla
u(x)-q\right|^{p}\,dx&\leqslant 2^{p-1}\fint_{B_{\eta}}\big{(}\left|\nabla
u(x)-\nabla v(x)\right|^{p}+\left|\nabla v(x)-q\right|^{p}\big{)}\,dx\\\
&\leqslant
2^{p-1}\left(\frac{\left|B_{1}\right|}{\left|B_{\eta}\right|}\fint_{B_{1}}\left|\nabla
u(x)-\nabla v(x)\right|^{p}\,dx+C_{1}\,\eta^{\alpha p}\,a^{p}\right)\\\
&\leqslant 2^{p-1}\Big{(}C\eta^{-n}(\sigma a^{p}+1)+C_{1}a^{p}\eta^{\alpha
p}\Big{)},\\\ &=2^{p-1}C\eta^{-n}\sigma
a^{p}+2^{p-1}C\eta^{-n}+2^{p-1}C_{1}a^{p}\eta^{\alpha p}.\end{split}$

This yields that

(3.10) $\fint_{B_{\eta}}\left|\nabla u(x)\right|^{p}\,dx\leqslant
2^{2(p-1)}C\eta^{-n}\sigma
a^{p}+2^{2(p-1)}C\eta^{-n}+2^{2(p-1)}C_{1}a^{p}\eta^{\alpha
p}+2^{p-1}\left|q\right|^{p}.$

If instead $1<p<2$, by virtue of (2.2) and (3.8), we obtain that

$\displaystyle\int_{B_{1}}\left|\nabla u(x)-\nabla v(x)\right|^{p}\,dx$
$\displaystyle\leqslant\;$ $\displaystyle
C\left(\sigma\int_{B_{1}}\left|\nabla
u(x)\right|^{p}\,dx+1\right)^{\frac{p}{2}}\left(\int_{B_{1}}\big{(}\left|\nabla
u(x)\right|+\left|\nabla v(x)\right|\big{)}^{p}\,dx\right)^{1-\frac{p}{2}}$
$\displaystyle\leqslant\;$ $\displaystyle
C\left(\sigma^{\frac{p}{2}}\left(\int_{B_{1}}\left|\nabla
u(x)\right|^{p}\,dx\right)^{\frac{p}{2}}+1\right)\left(\int_{B_{1}}\big{(}\left|\nabla
u(x)\right|^{p}+\left|\nabla
v(x)\right|^{p}\big{)}\,dx\right)^{1-\frac{p}{2}}.$

Thus, since $v$ is the $p$-harmonic replacement of $u$ in $B_{1}$,

$\displaystyle\int_{B_{1}}\left|\nabla u(x)-\nabla v(x)\right|^{p}\,dx$
$\displaystyle\leqslant\;$ $\displaystyle
C\left(\sigma^{\frac{p}{2}}\left(\int_{B_{1}}\left|\nabla
u(x)\right|^{p}\,dx\right)^{\frac{p}{2}}+1\right)\left(\int_{B_{1}}\left|\nabla
u(x)\right|^{p}\,dx\right)^{1-\frac{p}{2}}$ $\displaystyle=\;$ $\displaystyle
C\left(\sigma^{\frac{p}{2}}\int_{B_{1}}\left|\nabla
u(x)\right|^{p}\,dx+\left(\int_{B_{1}}\left|\nabla
u(x)\right|^{p}\,dx\right)^{1-\frac{p}{2}}\right).$

Consequently, taking the average integral, we have that

$\fint_{B_{1}}\left|\nabla u(x)-\nabla v(x)\right|^{p}\,dx\leqslant
C\left(\sigma^{\frac{p}{2}}a^{p}+|B_{1}|^{-\frac{p}{2}}a^{p\left(1-\frac{p}{2}\right)}\right)\leqslant
C\left(\sigma^{\frac{p}{2}}a^{p}+a^{p\left(1-\frac{p}{2}\right)}\right).$

From this and (3.7), we obtain that

(3.11) $\begin{split}\fint_{B_{\eta}}\left|\nabla
u(x)-q\right|^{p}\,dx\leqslant\;&2^{p-1}\fint_{B_{\eta}}\big{(}\left|\nabla
u(x)-\nabla v(x)\right|^{p}+\left|\nabla v(x)-q\right|^{p}\big{)}\,dx\\\
\leqslant\;&2^{p-1}\left(\frac{\left|B_{1}\right|}{\left|B_{\eta}\right|}\fint_{B_{1}}\left|\nabla
u(x)-\nabla v(x)\right|^{p}\,dx+C_{1}\,\eta^{\alpha p}\,a^{p}\right)\\\
\leqslant\;&2^{p-1}C\eta^{-n}\sigma^{\frac{p}{2}}a^{p}+2^{p-1}C\eta^{-n}a^{p\left(1-\frac{p}{2}\right)}+2^{p-1}C_{1}a^{p}\eta^{\alpha
p}.\end{split}$

This gives that

(3.12) $\fint_{B_{\eta}}\left|\nabla u(x)\right|^{p}\,dx\leqslant
2^{2(p-1)}C\eta^{-n}\sigma^{\frac{p}{2}}a^{p}+2^{2(p-1)}C\eta^{-n}a^{p\left(1-\frac{p}{2}\right)}+2^{2(p-1)}C_{1}a^{p}\eta^{\alpha
p}+2^{p-1}\left|q\right|^{p}.$

Now, given $\varepsilon_{0}\in(0,1/4]$, we claim that for every
$\varepsilon\in(0,\varepsilon_{0})$ there exists $\eta$ small enough
(depending on $\varepsilon$) such that if $\sigma$ is chosen sufficiently
small and $a$ sufficiently large (depending on $\eta$, and thus on
$\varepsilon$) then

(3.13) $\begin{cases}2^{2(p-1)}C\eta^{-n}\sigma
a^{p}+2^{2(p-1)}C\eta^{-n}+2^{2(p-1)}C_{1}a^{p}\eta^{\alpha p}\leqslant
2^{p-1}\varepsilon^{p}a^{p}\leqslant\frac{a^{p}}{2^{p+1}}\\\ \mbox{if
}p\geqslant 2,\\\ \\\
2^{2(p-1)}C\eta^{-n}\sigma^{\frac{p}{2}}a^{p}+2^{2(p-1)}C\eta^{-n}a^{p\left(1-\frac{p}{2}\right)}+2^{2(p-1)}C_{1}a^{p}\eta^{\alpha
p}\leqslant 2^{p-1}\varepsilon^{p}a^{p}\leqslant\frac{a^{p}}{2^{p+1}}\\\
\mbox{if }1<p<2.\end{cases}$

To prove this we distinguish two cases. If $p\geqslant 2,$ we pick $\eta>0$
sufficiently small such that
$\varepsilon^{p}-2^{p-1}C\eta>2^{p-1}C_{1}\eta^{\alpha p}$. This allows us to
define

$M:=\left(\frac{2^{p-1}C\eta^{-n}}{\varepsilon^{p}-2^{p-1}C\eta-2^{p-1}C_{1}\eta^{\alpha
p}}\right)^{1/p}.$

Note also that we can suppose $M\geqslant 1$ by taking $\eta$ small enough.
Let also

$\sigma_{0}:=\eta^{n+1}.$

With this setting, we obtain that, for every $a\geqslant M$ and for every
$0<\sigma\leqslant\sigma_{0},$

$\displaystyle 2^{2(p-1)}C\eta^{-n}\sigma
a^{p}+2^{2(p-1)}C\eta^{-n}+2^{2(p-1)}C_{1}a^{p}\eta^{\alpha p}$
$\displaystyle\leqslant$ $\displaystyle
a^{p}\Big{(}2^{2(p-1)}C\eta+2^{2(p-1)}C_{1}\eta^{\alpha
p}\Big{)}+2^{2(p-1)}C\eta^{-n}$ $\displaystyle=$ $\displaystyle
a^{p}\Big{(}2^{2(p-1)}C\eta+2^{2(p-1)}C_{1}\eta^{\alpha
p}\Big{)}+2^{p-1}\,M^{p}\Big{(}\varepsilon^{p}-2^{p-1}C\eta-2^{p-1}C_{1}\eta^{\alpha
p}\Big{)}$ $\displaystyle\leqslant$ $\displaystyle
a^{p}\Big{(}2^{2(p-1)}C\eta+2^{2(p-1)}C_{1}\eta^{\alpha
p}\Big{)}+2^{p-1}\,a^{p}\Big{(}\varepsilon^{p}-2^{p-1}C\eta-2^{p-1}C_{1}\eta^{\alpha
p}\Big{)}$ $\displaystyle=$ $\displaystyle
2^{p-1}\varepsilon^{p}a^{p}+a^{p}\Big{(}2^{2(p-1)}C\eta+2^{2(p-1)}C_{1}\eta^{\alpha
p}-2^{2(p-1)}C\eta-2^{2(p-1)}C_{1}\eta^{\alpha p}\Big{)}$ $\displaystyle=$
$\displaystyle 2^{p-1}\varepsilon^{p}a^{p},$

which proves (3.13) when $p\geqslant 2$.

If instead $1<p<2,$ we pick $\eta>0$ small enough such that
$\varepsilon^{p}>2^{p-1}C\eta+2^{p-1}C_{1}\eta^{\alpha p}.$ In this way, we
can define

$M:=\left(\frac{C2^{p-1}\eta^{-n}}{\varepsilon^{p}-2^{p-1}C\eta-2^{p-1}C_{1}\eta^{\alpha
p}}\right)^{2/p^{2}}.$

Let also $\sigma_{0}:=\eta^{(n+1)\frac{2}{p}}.$ Then, whenever $a\geqslant M$
and $0<\sigma\leqslant\sigma_{0}$ it follows that

$\displaystyle
2^{2(p-1)}C\eta^{-n}\sigma^{\frac{p}{2}}a^{p}+2^{2(p-1)}C\eta^{-n}a^{p\left(1-\frac{p}{2}\right)}+2^{2(p-1)}C_{1}a^{p}\eta^{\alpha
p}$ $\displaystyle\leqslant$ $\displaystyle 2^{2(p-1)}C\eta
a^{p}+2^{2(p-1)}C\eta^{-n}a^{p\left(1-\frac{p}{2}\right)}+2^{2(p-1)}C_{1}a^{p}\eta^{\alpha
p}$ $\displaystyle=$ $\displaystyle
2^{p-1}a^{p}\Big{(}2^{p-1}C\eta+2^{p-1}C\eta^{-n}a^{-\frac{p^{2}}{2}}+2^{p-1}C_{1}\eta^{\alpha
p}\Big{)}$ $\displaystyle\leqslant$ $\displaystyle
2^{p-1}a^{p}\Big{(}2^{p-1}C\eta+2^{p-1}C\eta^{-n}M^{-\frac{p^{2}}{2}}+2^{p-1}C_{1}\eta^{\alpha
p}\Big{)}$ $\displaystyle=$ $\displaystyle
2^{p-1}a^{p}\Big{(}2^{p-1}C\eta+\varepsilon^{p}-2^{p-1}C\eta-2^{p-1}C_{1}\eta^{\alpha
p}+2^{p-1}C_{1}\eta^{\alpha p}\Big{)}$ $\displaystyle=$ $\displaystyle
2^{p-1}\varepsilon^{p}a^{p},$

which establishes (3.13) in the case $1<p<2$ as well.

In order to complete the proof of Proposition 3.1, we now distinguish two
cases according to the size of $\left|q\right|.$ More precisely, we first
suppose that

$\left|q\right|\leqslant\frac{a}{4}.$

Then, we use either (3.10) (if $p\geqslant 2$) or (3.12) (if $1<p<2$), and
(3.13) to conclude that

$\displaystyle\fint_{B_{\eta}}\left|\nabla
u(x)\right|^{p}\,dx\leqslant\frac{a^{p}}{2^{p+1}}+2^{p-1}\frac{a^{p}}{2^{2p}}=\frac{a^{p}}{2^{p+1}}+\frac{a^{p}}{2^{p+1}}=\frac{a^{p}}{2^{p}},$

and thus

$\left(\fint_{B_{\eta}}\left|\nabla
u(x)\right|^{p}\,dx\right)^{1/p}\leqslant\frac{a}{2},$

which is the first alternative in (3.3).

Otherwise, it holds that

$\frac{a}{4}<\left|q\right|\leqslant C_{0}\,a,$

and therefore, by either (3.9) (if $p\geqslant 2$) or (3.11) (if $1<p<2$), and
(3.13), we have that

$\left(\fint_{B_{\eta}}\left|\nabla
u(x)-q\right|^{p}\,dx\right)^{1/p}\leqslant\varepsilon a,$

which is the second alternative in (3.4). The proof of Proposition 3.1 is
thereby complete. ∎

We will now show that the alternative in (3.4) can be “improved” when
$\varepsilon$ and $\sigma$ are sufficiently small. This result is the
counterpart of Lemma 2.3 in [DS] in the more general setting dealt with in
this paper. The main difficulty here with respect to Lemma 2.3 in [DS] relies
on the fact that the problem is not linear when $p\neq 2$, and therefore, if
$v_{1}$ and $v_{2}$ are the $p$-harmonic replacements of $u_{1}$ and $u_{2}$
in $B_{1}$, then it is not true that $v_{1}+v_{2}$ is the $p$-harmonic
replacement of $u_{1}+u_{2}$, unless $p=2$.

We will overcome this difficulty by showing that a “uniformly elliptic”
equation is still satisfied from the sum of $p$-harmonic replacements. More
precisely, we will show the following result:

######  Lemma 3.2.

Let $q\in\mathbb{R}^{n}$ and let $\eta:\mathbb{R}^{n}\to\mathbb{R}^{n}$ be
such that $|\eta(x)|<\frac{|q|}{2}$. Let $F(z):=|z|^{p-2}z$ and

$A(x):=\int_{0}^{1}DF\big{(}q+t\eta(x)\big{)}\,dt.$

Then,

$\lambda\,|q|^{p-2}\leqslant A\xi\cdot\xi\leqslant\Lambda\,|q|^{p-2},$

for all $\xi\in\partial B_{1}$, for some $\Lambda\geqslant\lambda>0$,
depending on $p$.

###### Proof.

We notice that, for all $t\in(0,1)$,

(3.14)
$\begin{split}&|q+t\eta|\leqslant|q|+|\eta|<|q|+\frac{|q|}{2}=\frac{3|q|}{2}\\\
{\mbox{and
}}\quad&|q+t\eta|\geqslant|q|-|\eta|>|q|-\frac{|q|}{2}=\frac{|q|}{2}.\end{split}$

Furthermore, we observe that

$DF(z)=(p-2)|z|^{p-4}z\otimes z+|z|^{p-2}{\text{Id}},$

where Id denotes the identity matrix in dimension $n$, and therefore, for all
$\xi\in\partial B_{1}$,

$\displaystyle DF(z)\xi\cdot\xi$ $\displaystyle=$
$\displaystyle(p-2)|z|^{p-4}(z\cdot\xi)^{2}+|z|^{p-2}|\xi|^{2}$
$\displaystyle\geqslant$ $\displaystyle-(2-p)^{+}|z|^{p-2}+|z|^{p-2}|\xi|^{2}$
$\displaystyle=$ $\displaystyle\big{[}1-(2-p)^{+}\big{]}\,|z|^{p-2}|\xi|^{2}$
$\displaystyle=$ $\displaystyle\big{[}1-(2-p)^{+}\big{]}\,|z|^{p-2}.$

Consequently,

$\displaystyle
A\xi\cdot\xi\geqslant\big{[}1-(2-p)^{+}\big{]}\int_{0}^{1}|q+t\eta|^{p-2}\,dt.$

Accordingly, we can use (LABEL:sod30v9b54y8b67vb985_78by75y89) to find that

$A\xi\cdot\xi\geqslant\frac{1-(2-p)^{+}}{2^{p-2}}\,|q|^{p-2}$

if $p\geqslant 2$, and

$A\xi\cdot\xi\geqslant\big{[}1-(2-p)^{+}\big{]}\left(\frac{3}{2}\right)^{p-2}|q|^{p-2}$

if $p\in(1,2)$.

Similarly, since

$\displaystyle DF(z)\xi\cdot\xi$ $\displaystyle\leqslant$
$\displaystyle(p-2)^{+}|z|^{p-2}+|z|^{p-2}|\xi|^{2}$ $\displaystyle=$
$\displaystyle\big{[}1+(p-2)^{+}\big{]}\,|z|^{p-2}|\xi|^{2}$ $\displaystyle=$
$\displaystyle\big{[}1+(p-2)^{+}\big{]}\,|z|^{p-2},$

we have that

$\displaystyle
A\xi\cdot\xi\leqslant\big{[}1+(p-2)^{+}\big{]}\int_{0}^{1}|q+t\eta|^{p-2}\,dt.$

Hence, making again use of (LABEL:sod30v9b54y8b67vb985_78by75y89),

$A\xi\cdot\xi\leqslant\big{[}1+(p-2)^{+}\big{]}\left(\frac{3}{2}\right)^{p-2}|q|^{p-2}$

if $p\geqslant 2$, and

$A\xi\cdot\xi\leqslant\frac{1+(p-2)^{+}}{2^{p-2}}\,|q|^{p-2}$

if $p\in(1,2)$.

These considerations complete the proof of Lemma 3.2. ∎

With this, we can now state the following result:

######  Lemma 3.3.

Let $a_{1}>a_{0}>0$ and

(3.15) $p>\max\left\\{\frac{2n}{n+2},1\right\\}.$

There exist $\alpha_{0}\in(0,1]$ and $\widetilde{C}>0$, depending on $n$ and
$p$, such that for every $\alpha\in(0,\alpha_{0})$ there exist

  * •

$\rho\in(0,1)$, depending on $n$, $p$ and $\alpha$,

  * •

$\varepsilon_{0}\in(0,1)$, depending on $n$, $p$, $a_{0}$, $a_{1}$ and
$\alpha$,

  * •

$c_{0}>0$, depending on $n$, $p$, $a_{0}$, $a_{1}$ and $\alpha$,

such that, if $\varepsilon\in(0,\varepsilon_{0}]$ and
$\sigma\in(0,c_{0}\varepsilon^{P}]$, with $P:=\max\\{p,2\\}$, then the
following statement holds true.

Let $u\in W^{1,p}(B_{1})$ be such that $u\geqslant 0$ in $B_{1}$ and

(3.16) $J_{p}(u,B_{1})\leqslant(1+\sigma)J_{p}(v,B_{1})$

for all $v\in W^{1,p}(B_{1})$ such that $v=u$ on $\partial B_{1}$.

Let

(3.17) $a:=\left(\fint_{B_{1}}\left|\nabla u(x)\right|^{p}\,dx\right)^{1/p}$

and suppose that

(3.18) $a\in[a_{0},a_{1}].$

Assume also that

(3.19) $\left(\fint_{B_{1}}\left|\nabla
u(x)-q\right|^{p}\,dx\right)^{1/p}\leqslant\varepsilon a,$

for some $q\in\mathbb{R}^{n}$ such that

(3.20) $\frac{a}{8}<|q|\leqslant 2C_{0}a,$

where $C_{0}>0$ is the universal constant given by Proposition 3.1.

Then

(3.21) $\left(\fint_{B_{\rho}}\left|\nabla
u(x)-\widetilde{q}\right|^{p}\,dx\right)^{1/p}\leqslant\rho^{\alpha}\varepsilon
a,$

with $\widetilde{q}\in\mathbb{R}^{n}$ such that

(3.22) $\left|q-\widetilde{q}\right|\leqslant\widetilde{C}\varepsilon a.$

The quantity $\alpha_{0}$ in the statement of Lemma 3.3 measures the
oscillation of $\nabla u,$ see [Ur], [MR709038], [T] or, specifically for our
case, Theorem 2 in [Man]. When $p=2$, we have that $\alpha_{0}=1$, according
to the full regularity of harmonic functions, but when $p\not=2$ in general we
only know that $\alpha_{0}\in(0,1)$ (but when $n=2$ a sharp regularity
exponent has been determined in [IwMa]).

###### Proof of Lemma 3.3.

Let $\bar{v}$ denote the $p$-harmonic replacement of $u$ in $B_{9/10}$ and let
$v$ be defined as

(3.23) $v:=\begin{cases}\bar{v}&\mbox{in }B_{9/10},\\\ u&\mbox{in
}B_{1}\setminus B_{9/10}.\end{cases}$

Then, since $v=u$ on $\partial B_{1}$ and $v\in W^{1,p}(B_{1})$, we can use
(3.16) to see that

$J_{p}(u,B_{1})\leqslant(1+\sigma)J_{p}(v,B_{1}).$

Consequently,

$\begin{split}J_{p}(u,B_{9/10})=\;&J_{p}(u,B_{1})-J_{p}(u,B_{1}\setminus
B_{9/10})\\\ \leqslant\;&J_{p}(v,B_{9/10})+J_{p}(v,B_{1}\setminus
B_{9/10})+\sigma J_{p}(v,B_{1})-J_{p}(u,B_{1}\setminus B_{9/10})\\\
=\;&J_{p}(v,B_{9/10})+J_{p}(u,B_{1}\setminus B_{9/10})+\sigma
J_{p}(v,B_{1})-J_{p}(u,B_{1}\setminus B_{9/10})\\\
=\;&J_{p}(v,B_{9/10})+\sigma J_{p}(v,B_{1}).\end{split}$

Hence, recalling the definition of $J_{p}$ in (1.1),

$\displaystyle\int_{B_{9/10}}\left|\nabla
u(x)\right|^{p}\,dx+\left|\left\\{u>0\right\\}\cap
B_{9/10}\right|\leqslant\int_{B_{9/10}}\left|\nabla
v(x)\right|^{p}\,dx+\left|B_{9/10}\right|+\sigma J_{p}(v,B_{1}),$

which yields that

(3.24) $\int_{B_{9/10}}\big{(}\left|\nabla u(x)\right|^{p}-\left|\nabla
v(x)\right|^{p}\big{)}\,dx\leqslant\left|\left\\{u=0\right\\}\cap
B_{9/10}\right|+\sigma J_{p}(v,B_{1}).$

Moreover, recalling (3.23), we point out that $v$ is the $p$-harmonic
replacement of $u$ in $B_{9/10}$, and therefore

(3.25) $\begin{split}J_{p}(v,B_{1})=\;&\int_{B_{1}}\Big{(}|\nabla
v(x)|^{p}+\chi_{\\{v>0\\}}(x)\Big{)}\,dx\\\ \leqslant\;&\int_{B_{9/10}}|\nabla
v(x)|^{p}\,dx+\int_{B_{1}\setminus B_{9/10}}|\nabla v(x)|^{p}\,dx+|B_{1}|\\\
\leqslant\;&\int_{B_{1}}\left|\nabla
u\right|^{p}\,dx+\left|B_{1}\right|\leqslant
a^{p}+\left|B_{1}\right|.\end{split}$

Furthermore, if $p\geqslant 2$, by (2.3) and (3.24) we deduce that

$\int_{B_{9/10}}\left|\nabla u(x)-\nabla v(x)\right|^{p}\,dx\leqslant
C\left|\left\\{u=0\right\\}\cap B_{9/10}\right|+C\sigma J_{p}(v,B_{1}),$

for some positive universal constant $C$. Consequently, exploiting (3.25), we
obtain that

(3.26) $\int_{B_{9/10}}\left|\nabla u(x)-\nabla v(x)\right|^{p}\,dx\leqslant
C\left|\left\\{u=0\right\\}\cap B_{9/10}\right|+C\sigma(a^{p}+1),$

up to renaming $C$.

If instead $p<2$, we use (2.2), (3.24) and (3.25) to write that

(3.27) $\begin{split}&\int_{B_{9/10}}\left|\nabla u(x)-\nabla
v(x)\right|^{p}\,dx\\\ \leqslant\;&C\left(\int_{B_{9/10}}\left(\left|\nabla
u(x)\right|^{p}-\left|\nabla
v(x)\right|^{p}\right)\,dx\right)^{\frac{p}{2}}\left(\int_{B_{9/10}}\big{(}\left|\nabla
u(x)\right|+\left|\nabla v(x)\right|\big{)}^{p}\,dx\right)^{1-\frac{p}{2}}\\\
\leqslant\;&C\Big{(}\left|\left\\{u=0\right\\}\cap B_{9/10}\right|+\sigma
J_{p}(v,B_{1})\Big{)}^{\frac{p}{2}}\left(\int_{B_{9/10}}2^{p}\left|\nabla
u(x)\right|^{p}\,dx\right)^{1-\frac{p}{2}}\\\
\leqslant\;&C\Big{(}\left|\left\\{u=0\right\\}\cap
B_{9/10}\right|+\sigma(a^{p}+1)\Big{)}^{\frac{p}{2}}a^{p\left(1-\frac{p}{2}\right)}.\end{split}$

Now we claim that

(3.28) $\left|B_{9/10}\cap\left\\{u=0\right\\}\right|\leqslant
C_{1}\varepsilon^{p+\delta},$

for some $C_{1}>0$ and $\delta>0$. To prove this, we consider the linear
function

(3.29) $\ell(x):=b+q\cdot x,\quad{\mbox{ with }}\quad
b:=\fint_{B_{1}}u(x)\,dx.$

We remark that

$\displaystyle\fint_{B_{1}}\big{(}u(x)-\ell(x)\big{)}\,dx=b-\left(b+\fint_{B_{1}}q\cdot
x\,dx\right)=-\fint_{B_{1}}q\cdot x\,dx=0.$

As a consequence, denoting by

$(u-\ell)_{B_{1}}:=\fint_{B_{1}}\big{(}u(x)-\ell(x)\big{)}\,dx$

and using the Poincaré inequality, we see that

$\left\|u-\ell-(u-\ell)_{B_{1}}\right\|_{L^{p}(B_{1})}=\left\|u-\ell\right\|_{L^{p}(B_{1})}\leqslant
C\left\|\nabla(u-\ell)\right\|_{L^{p}(B_{1})},$

for some $C>0$ universal.

This and (3.19) entail that

(3.30) $\fint_{B_{1}}\left|u(x)-\ell(x)\right|^{p}\,dx\leqslant
C\int_{B_{1}}|\nabla(u-\ell)(x)|^{p}\,dx\leqslant C\varepsilon^{p}a^{p}.$

We also observe that, since $u\geqslant 0,$ it holds that
$\ell^{-}\leqslant\left|u-\ell\right|$. Using this information into (3.30), we
obtain that

(3.31) $\fint_{B_{1}}(\ell^{-}(x))^{p}\,dx\leqslant C\varepsilon^{p}a^{p}.$

Now we claim that, if $\varepsilon$ is sufficiently small,

(3.32) $\ell\geqslant c_{1}a\quad\mbox{in }B_{9/10},$

for some $c_{1}>0$. To check this, we argue by contradiction assuming that

$\min_{x\in\overline{B_{9/10}}}\ell(x)<ca$

for all $c>0$. We notice that, for every $x\in B_{9/10}$,

$-\frac{9|q|}{10}\leqslant\ell(x)-b\leqslant\frac{9|q|}{10}.$

and therefore

$ca>\min_{x\in\overline{B_{1/2}}}\ell(x)\geqslant b-\frac{9|q|}{10}.$

This leads to

(3.33) $b\leqslant ca+\frac{9|q|}{10}.$

Now we define

${\mathcal{B}}:=\left\\{x=-\frac{tq}{|q|}+\eta,\;{\mbox{ for some
}}t\in\left[\frac{38}{40},\frac{39}{40}\right]{\mbox{ and }}\eta\in
B_{1/40}\right\\}.$

Notice that if $x\in{\mathcal{B}}$ then

$|x|\leqslant t+|\eta|<\frac{39}{40}+\frac{1}{40}=1,$

and this shows that

(3.34) ${\mathcal{B}}\subseteq B_{1}.$

Furthermore, from (3.20) and (3.33) we see that, if $x\in{\mathcal{B}}$,

$\displaystyle\ell(x)=b-t|q|+\eta\cdot q\leqslant
ca+\frac{9|q|}{10}-\frac{38|q|}{40}+\frac{|q|}{40}=ca-\left(\frac{38}{40}-\frac{9}{10}-\frac{1}{40}\right)|q|$
$\displaystyle\qquad\qquad=ca-\frac{|q|}{40}<ca-\frac{a}{320}.$

Hence, taking $c\in\left(0,\frac{1}{640}\right)$, we infer that

$\ell(x)\leqslant-\frac{a}{640}.$

Accordingly, using this and (3.34) into (3.31), we obtain that

$\displaystyle
C\,|B_{1}|\,\varepsilon^{p}a^{p}\geqslant\int_{B_{1}}(\ell^{-}(x))^{p}\,dx\geqslant\int_{{\mathcal{B}}}(\ell^{-}(x))^{p}\,dx\geqslant\int_{{\mathcal{B}}}\left(\frac{a}{80\cdot
2^{(3p+1)/p}}\right)^{p}\,dx\geqslant\overline{c}a^{p},$

for some positive universal constant $\overline{c}$. This establishes the
desired contradiction if $\varepsilon$ is sufficiently small, and thus the
proof of (3.32) is complete.

Now, we distinguish the three following cases: $p\in(1,n),$ $p=n$ and $p>n$.

Firstly, if $p<n,$ recalling the Poincaré-Sobolev inequality (see e.g. Theorem
2 on page 265 of [MR1625845], or Theorem 7.30 in [GilT]) and applying the same
argument of formula (7.40) in [GilT], we obtain that

(3.35)
$\left(\int_{B_{1}}\left|u(x)-\ell(x)\right|^{p^{*}}\,dx\right)^{1/p^{*}}\leqslant
C\left(\int_{B_{1}}\left|\nabla\big{(}u(x)-\ell(x)\big{)}\right|^{p}\,dx\right)^{1/p}$

for some $C>0$ universal, where

$p^{*}:=\frac{np}{n-p}.$

Then, by virtue of (3.19), (3.29) and (3.35), we get that

$\displaystyle\left(\displaystyle\int_{B_{1}}\left|u(x)-\ell(x)\right|^{p^{*}}\,dx\right)^{1/p^{*}}\leqslant
C\left(\int_{B_{1}}\left|\nabla\big{(}u(x)-\ell(x)\big{)}\right|^{p}\,dx\right)^{1/p}$
$\displaystyle\qquad\qquad=C\left(\int_{B_{1}}\left|\nabla
u(x)-q\right|^{p}\,dx\right)^{1/p}\leqslant C\varepsilon a.$

This and (3.32) entail that

$\displaystyle C\varepsilon
a\geqslant\left(\int_{B_{9/10}\cap\left\\{u=0\right\\}}\left|\ell(x)\right|^{p^{*}}\,dx\right)^{1/p^{*}}\geqslant
c_{1}a\left|B_{9/10}\cap\left\\{u=0\right\\}\right|^{1/p^{*}},$

and thus, up to renaming constants,

(3.36) $\left|B_{9/10}\cap\left\\{u=0\right\\}\right|\leqslant
C\varepsilon^{p^{*}}.$

Now we notice that

$p^{*}=\frac{np+p^{2}-p^{2}}{n-p}=p+\frac{p^{2}}{n-p}.$

Therefore, setting

(3.37) $\delta:=\frac{p^{2}}{n-p}>0,$

we obtain (3.28) from (3.36) in the case $p<n$.

If instead $p>n,$ we recall Theorem 7.17 as well as (7.12) and (7.16) in
[GilT], to see that

$\sup\limits_{B_{1}}\left|u-\ell\right|\leqslant
C\left(\int_{B_{1}}\left|\nabla\big{(}u(x)-\ell(x)\big{)}\right|^{p}\,dx\right)^{1/p}$

for some $C>0$ universal. From this, and making again use of (3.19) and
(3.29), we deduce that

$\displaystyle\sup_{B_{1}}\left|u-\ell\right|\leqslant
C\left(\int_{B_{1}}\left|\nabla(u(x)-\ell(x))\right|^{p}\,dx\right)^{1/p}\leqslant
C\varepsilon a.$

As a consequence of this and (3.32), for all $x\in B_{9/10}$,

$\displaystyle C\varepsilon
a\geqslant\sup_{B_{1}}\left|u-\ell\right|\geqslant\sup_{B_{9/10}}(\ell-u)\geqslant\ell(x)-u(x)\geqslant
c_{1}a-u(x),$

and thus $u(x)\geqslant c_{1}a-C\varepsilon a>0$ for all $x\in B_{9/10}$, as
long as $\varepsilon$ is sufficiently small. Accordingly, it follows that

(3.38) $|B_{9/10}\cap\left\\{u=0\right\\}|=0,$

and this gives (3.28) in the case $p>n$ as well.

It remains to analyze the case $p=n$. For this purpose, we point out that
$u-\ell\in W^{1,n}(B_{1})$, and so we can apply Theorem 7.35 and Theorem 7.16
in [GilT] to obtain that

$\int_{B_{1}}\exp\left(\frac{\left|u(x)-\ell(x)\right|}{C_{2}\left\|\nabla(u-\ell)\right\|_{L^{n}(B_{1})}}\right)^{\frac{n}{n-1}}\,dx\leqslant
C_{3}\left|B_{1}\right|,$

where $C_{2}$ and $C_{3}$ are positive universal constants.

Also, for all $x\in B_{1}$, we have that $e^{x}\geqslant C_{4}x^{n}$ for some
$C_{4}>0$, and thus, recalling (3.19) and (3.29),

(3.39) $\int_{B_{1}}\left|u(x)-\ell(x)\right|^{\frac{n^{2}}{n-1}}\,dx\leqslant
C\left\|\nabla u-q\right\|_{L^{n}(B_{1})}^{\frac{n^{2}}{n-1}}\leqslant
C\varepsilon^{\frac{n^{2}}{n-1}}a^{\frac{n^{2}}{n-1}},$

for some $C>0,$ from which we deduce that, using (3.32) and (3.39),

$\displaystyle
C\varepsilon^{\frac{n^{2}}{n-1}}a^{\frac{n^{2}}{n-1}}\geqslant\int_{B_{1}}\left|u(x)-\ell(x)\right|^{\frac{n^{2}}{n-1}}\,dx\geqslant\int_{B_{9/10}\cap\\{u=0\\}}\left|\ell(x)\right|^{\frac{n^{2}}{n-1}}\,dx$
$\displaystyle\qquad\qquad\geqslant
c_{1}^{\frac{n^{2}}{n-1}}a^{\frac{n^{2}}{n-1}}|B_{9/10}\cap\\{u=0\\}|.$

We point out that

$\frac{n^{2}}{n-1}=\frac{n^{2}-n+n}{n-1}=n+\frac{n}{n-1},$

and therefore, choosing

$\delta:=\frac{n}{n-1}>0,$

we establish (3.28) when $p=n$ as well.

Gathering together (3.26) and (3.28) we thus conclude that, if $p\geqslant 2$,

(3.40) $\int_{B_{9/10}}\left|\nabla u(x)-\nabla v(x)\right|^{p}\,dx\leqslant
C_{1}\varepsilon^{p+\delta}+C\sigma(a^{p}+1).$

Instead, when $1<p<2$, we make use of (3.27) and (3.28) and we obtain that

(3.41) $\int_{B_{9/10}}\left|\nabla u(x)-\nabla v(x)\right|^{p}\,dx\leqslant
C\Big{(}C_{1}\varepsilon^{p+\delta}+\sigma(a^{p}+1)\Big{)}^{\frac{p}{2}}a^{p\left(1-\frac{p}{2}\right)}.$

Our aim is now to show that, even if $v-q\cdot x$ is not the $p$-harmonic
replacement of $u-q\cdot x,$ as in the classical case (see Lemma 2.3 in [DS]),
it satisfies a “nice” equation in $B_{9/10}$.

To this end, we observe that if $p\geqslant 2$, using (3.19) and (3.40),

(3.42) $\begin{split}\int_{B_{9/10}}\left|\nabla
v(x)-q\right|^{p}\,dx\leqslant\;&2^{p-1}\left(\int_{B_{9/10}}\left|\nabla
u(x)-q\right|^{p}\,dx+\int_{B_{9/10}}\left|\nabla v(x)-\nabla
u(x)\right|^{p}\,dx\right)\\\
\leqslant\;&C_{2}\varepsilon^{p}a^{p}+C_{1}\varepsilon^{p+\delta}+C\sigma(a^{p}+1),\end{split}$

up to renaming constants.

If instead $1<p<2$, from (3.19) and (3.41) we have that

(3.43) $\begin{split}\int_{B_{9/10}}\left|\nabla
v(x)-q\right|^{p}\,dx\leqslant\;&2^{p-1}\left(\int_{B_{9/10}}\left|\nabla
u(x)-q\right|^{p}\,dx+\int_{B_{9/10}}\left|\nabla v(x)-\nabla
u(x)\right|^{p}\,dx\right)\\\
\leqslant\;&C_{2}\varepsilon^{p}a^{p}+C\Big{(}C_{1}\varepsilon^{p+\delta}+\sigma(a^{p}+1)\Big{)}^{\frac{p}{2}}a^{p\left(1-\frac{p}{2}\right)}.\end{split}$

Suppose now that $p\geqslant 2$ and $\sigma\leqslant c_{0}\varepsilon^{p}$,
with $c_{0}$ to be made precise later. Recalling (3.18) and (3.42), we infer
that

(3.44) $\int_{B_{9/10}}\left|\nabla v(x)-q\right|^{p}\,dx\leqslant
C_{2}\varepsilon^{p}a^{p}+C_{1}\varepsilon^{p+\delta}+Cc_{0}\varepsilon^{p}(a^{p}+1)\leqslant
C\varepsilon^{p}a^{p},$

up to relabeling $C>0$.

Similarly, if $1<p<2$, we take $\sigma\leqslant c_{0}\varepsilon^{2}$, with
$c_{0}$ to be made precise later. In this case, we deduce from (3.43) that

(3.45) $\begin{split}&\int_{B_{9/10}}\left|\nabla
v(x)-q\right|^{p}\,dx\leqslant
C_{2}\varepsilon^{p}a^{p}+C\Big{(}C_{1}\varepsilon^{p+\delta}+c_{0}\varepsilon^{2}(a^{p}+1)\Big{)}^{\frac{p}{2}}a^{p\left(1-\frac{p}{2}\right)}\\\
&\qquad\qquad\leqslant
C_{1}\varepsilon^{(p+\delta)\frac{p}{2}}a^{p}+C_{2}\varepsilon^{p}a^{p},\end{split}$

up to renaming the constants.

We stress that we have not used the assumption on $p$ given in (3.15) so far.
We will use it next to reabsorb the term
$\varepsilon^{(p+\delta)\frac{p}{2}}a^{p}$ appearing in (3.45) into the term
$\varepsilon^{p}a^{p}$. More precisely, we point out that

(3.46) ${\mbox{if\leavevmode\nobreak\
$p\in\left(\max\left\\{\frac{2n}{n+2},1\right\\},\,2\right)$,\quad then
}}(p+\delta)\frac{p}{2}>p.$

Indeed, when $n=1$ we are in the situation in which $n=1<p$, and therefore we
can take $\delta=1$ in (3.28) (recall also (3.38)), thus obtaining that
$\frac{p+1}{2}>1$, which gives (3.46) in this case.

Also, if $n\geqslant 2$, then we are in the case $p<n$, and thus, recalling
the value of $\delta$ given in (3.37), we obtain that

$p+\delta=p+\frac{p^{2}}{n-p}=\frac{np}{n-p}>2,$

which gives (3.46).

As a consequence of (3.45) and (3.46), we obtain that, if
$p\in\left(\max\left\\{\frac{2n}{n+2},1\right\\},\,2\right)$,

$\int_{B_{9/10}}\left|\nabla v(x)-q\right|^{p}\,dx\leqslant
C\varepsilon^{p}a^{p},$

for some $C>0$.

Putting together this and (3.44), we conclude that, if
$p>\max\left\\{\frac{2n}{n+2},1\right\\}$,

(3.47) $\int_{B_{9/10}}\left|\nabla v(x)-q\right|^{p}\,dx\leqslant
C\varepsilon^{p}a^{p}.$

Furthermore, we claim that, if $\varepsilon$ is sufficiently small, for all
$x\in B_{1/2}$,

(3.48) $\left|\nabla v(x)-q\right|\leqslant C(\varepsilon a)^{\nu},$

for some $C>0$ and $\nu\in(0,1)$. Indeed, suppose by contradiction that

(3.49) $\max_{x\in\overline{B_{1/2}}}|\nabla v(x)-q|>C(\varepsilon a)^{\nu}$

for all $C>0$ and $\nu\in(0,1)$. We recall that $v$ is $p$-harmonic in
$B_{9/10}$ and thus the maximum in (3.49) is achieved, see e.g. [MR709038] or
[T]. Moreover, let $\overline{x}\in\overline{B_{1/2}}$ be such that

(3.50) $|\nabla v(\overline{x})-q|=\max_{x\in\overline{B_{1/2}}}|\nabla
v(x)-q|>C(\varepsilon a)^{\nu}.$

Then, for all $x\in B_{1/8}(\overline{x})$,

$\big{|}\nabla v(x)-\nabla
v(\overline{x})\big{|}\leqslant\overline{C}|x-\overline{x}|^{\alpha},$

for some $\overline{C}>0$ and $\alpha\in(0,1]$, with $\alpha$ depending only
on $n$ and $p$, and $\overline{C}$ bounded by a constant depending only on $n$
and $p$ times $\|\nabla v\|_{L^{\infty}(B_{1/4}(\overline{x}))}$, see Theorem
2 in [Man]. Actually, since, by formula (3.44) in [MZ],

$\|\nabla v\|_{L^{\infty}(B_{1/4}(\overline{x}))}\leqslant\|\nabla
v\|_{L^{\infty}(B_{3/4})}\leqslant\hat{C}\|\nabla
v\|_{L^{p}(B_{9/10})}\leqslant\hat{C}\|\nabla
u\|_{L^{p}(B_{9/10})}\leqslant\widetilde{C}a_{1},$

for some $\hat{C}$ and $\widetilde{C}>0$ depending only on $n$ and $p$, we
infer that $\overline{C}$ only depends on $n$, $p$ and $a_{1}$.

As a consequence of this and (3.50), for all $x\in
B_{\left(\frac{C(\varepsilon
a)^{\nu}}{2\overline{C}}\right)^{1/\alpha}}(\overline{x})$,

$\displaystyle|\nabla v(x)-q|\geqslant|\nabla
v(\overline{x})-q|-\big{|}(\nabla v(x)-q)-(\nabla v(\overline{x})-q)\big{|}$
$\displaystyle\qquad\qquad\geqslant C(\varepsilon
a)^{\nu}-\overline{C}|x-\overline{x}|^{\alpha}\geqslant C(\varepsilon
a)^{\nu}-\frac{C(\varepsilon a)^{\nu}}{2}=\frac{C(\varepsilon a)^{\nu}}{2}.$

Notice also that, if $\varepsilon$ is sufficiently small, then
$B_{\left(\frac{C(\varepsilon
a)^{\nu}}{2\overline{C}}\right)^{1/\alpha}}(\overline{x})\subseteq B_{9/10}$.
Accordingly,

$\displaystyle\int_{B_{9/10}}\left|\nabla
v(x)-q\right|^{p}\,dx\geqslant\int_{B_{\left(\frac{C(\varepsilon
a)^{\nu}}{2\overline{C}}\right)^{1/\alpha}}(\overline{x})}\left|\nabla
v(x)-q\right|^{p}\,dx\geqslant\left(\frac{C(\varepsilon
a)^{\nu}}{2}\right)^{p}\left|B_{\left(\frac{C(\varepsilon
a)^{\nu}}{2\overline{C}}\right)^{1/\alpha}}(\overline{x})\right|$
$\displaystyle\qquad\qquad\left(\frac{C(\varepsilon
a)^{\nu}}{2}\right)^{p}\left(\frac{C(\varepsilon
a)^{\nu}}{2\overline{C}}\right)^{n/\alpha}|B_{1}|=\frac{C^{p+\frac{n}{\alpha}}|B_{1}|}{2^{p+\frac{n}{\alpha}}\,\overline{C}^{\frac{n}{\alpha}}}\,(\varepsilon
a)^{\nu\left(p+\frac{n}{\alpha}\right)}.$

Hence, exploiting (3.47), we find that

$C\varepsilon^{p}a^{p}\geqslant\frac{C^{p+\frac{n}{\alpha}}|B_{1}|}{2^{p+\frac{n}{\alpha}}\,\overline{C}^{\frac{n}{\alpha}}}\,(\varepsilon
a)^{\nu\left(p+\frac{n}{\alpha}\right)},$

which yields the desired contradiction as soon as $\varepsilon$ is chosen
sufficiently small and $\nu\in\left(0,\frac{p}{p+\frac{n}{\alpha}}\right).$
The proof of (3.48) is thus complete.

Let us define now the function $F:\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n}$
as $F(z):=\left|z\right|^{p-2}z$. We have that

$\displaystyle\left|\nabla v\right|^{p-2}\nabla
v-\left|q\right|^{p-2}q=F(\nabla v)-F(\nabla(q\cdot x))=F(\nabla v)-F(q)$
$\displaystyle\qquad\qquad=\int_{0}^{1}\frac{d}{dt}F(t\nabla
v+(1-t)q)\,dt=\int_{0}^{1}DF(t\nabla v+(1-t)q)(\nabla v-q)\,dt.$

Hence, taking the divergence of both sides, we obtain that

(3.51) $\displaystyle\operatorname*{div}\big{(}A(\nabla
v-q)\big{)}=0\quad\mbox{ in }B_{9/10},$

with

$A(x):=\int_{0}^{1}DF\big{(}t\nabla v(x)+(1-t)q\big{)}\,dt.$

Notice that we are in the setting of Lemma 3.2 with $\eta:=\nabla v-q$.
Indeed, we point out that, in light of (3.20) and (3.48), for all $x\in
B_{1/4}$,

$|\nabla v(x)-q|\leqslant C(\varepsilon
a)^{\nu}\leqslant\frac{a}{16}<\frac{|q|}{2},$

as long as $\varepsilon$ is sufficiently small.

Accordingly, exploiting Lemma 3.2 we find that

$\lambda\,|q|^{p-2}\leqslant A\xi\cdot\xi\leqslant\Lambda\,|q|^{p-2},$

for all $\xi\in\partial B_{1}$, for some $\Lambda\geqslant\lambda>0$,
depending only on $p$. Recalling (3.20), this gives that

$\lambda\,a^{p-2}\leqslant A\xi\cdot\xi\leqslant\Lambda\,a^{p-2},$

up to renaming $\lambda$ and $\Lambda$, that now may depend on $n$ and $p$.

We point out that we are in the setting of Section 3.2 in [MZ] (see in
particular formula (3.43) there, with $\lambda$ and $\Lambda$ replaced by
$\lambda\,a^{p-2}$ and $\Lambda\,a^{p-2}$ here, respectively). Hence, we are
in the position of applying Theorem 3.19 in [MZ], and so we obtain that, for
every $x\in B_{1/2}$,

(3.52) $\left|\nabla v(x)-q\right|^{p}\leqslant\sup_{B_{1/8}(x)}\left|\nabla
v-q\right|^{p}\leqslant C\fint_{B_{1/4}(x)}\left|\nabla
v(y)-q\right|^{p}\,dy\leqslant C\varepsilon^{p}a^{p},$

for some $C>0$, depending on $n$ and $p,$ where we have also used (3.47).

Consequently, for all $x\in B_{1/2}$,

(3.53) $\left|\nabla v(x)-q\right|\leqslant C\varepsilon a,$

up to renaming $C$, depending on $n$ and $p$.

Also, denoting $\bar{q}:=\nabla v(0)-q$, we have that

(3.54) $|\bar{q}|=|\nabla v(0)-q|\leqslant C\varepsilon a.$

Therefore, from this and (3.53), we deduce that, for all $x\in B_{1/2}$,

$\left|\nabla v(x)-q-\bar{q}\right|\leqslant C\varepsilon a.$

As a consequence, exploiting Theorem 2 in [Man], we obtain that, for every
$\rho\in(0,1/2)$,

(3.55) $\fint_{B_{\rho}}\left|\nabla
v(x)-q-\bar{q}\right|^{p}\,dx\leqslant\fint_{B_{\rho}}\left(C\left(\frac{\rho}{1/2}\right)^{\mu}\left\|\nabla
v-q-\bar{q}\right\|_{L^{\infty}(B_{1/2})}\right)^{p}\,dx\leqslant
C_{2}\,\rho^{\mu p}\,\varepsilon^{p}a^{p},$

for some $\mu\in(0,1]$ and $C_{2}>0$ depending on $n$ and $p$.

Then, if $p\geqslant 2$, putting together (3.40) and (3.55), we arrive at

(3.56) $\begin{split}&\fint_{B_{\rho}}\left|\nabla
u(x)-q-\bar{q}\right|^{p}\,dx\\\
\leqslant\;&2^{p-1}\left(\fint_{B_{\rho}}\left|\nabla u(x)-\nabla
v(x)\right|^{p}\,dx+\fint_{B_{\rho}}\left|\nabla
v(x)-q-\bar{q}\right|^{p}\,dx\right)\\\
\leqslant\;&2^{p-1}C_{1}\varepsilon^{p+\delta}\rho^{-n}+2^{p-1}{C}\sigma(a^{p}+1)\rho^{-n}+2^{p-1}C_{2}\rho^{\mu
p}\varepsilon^{p}a^{p}.\end{split}$

Thus, setting $\alpha_{0}:=\mu$ and, for every $\alpha\in(0,\alpha_{0})$,

(3.57)
$\rho:=(2^{p+1}C_{2})^{\frac{1}{(\alpha-\alpha_{0})p}},\qquad\varepsilon_{0}:=\left(\frac{\rho^{\alpha
p+n}a_{0}^{p}}{2^{p+1}C_{1}}\right)^{\frac{1}{\delta}}\quad{\mbox{and}}\quad
c_{0}:=\frac{\rho^{\alpha p+n}a_{0}^{p}}{2^{p+1}\,{C}(a_{1}^{p}+1)},$

we have that, for every $\varepsilon\in(0,\varepsilon_{0}]$ and
$\sigma\in(0,c_{0}\varepsilon^{p}]$,

$\displaystyle 2^{p-1}C_{2}\rho^{{\mu}p}\leqslant\frac{1}{4}\rho^{\alpha p},$
$\displaystyle
2^{p-1}C_{1}\varepsilon^{p+\delta}\rho^{-n}\leqslant\frac{1}{4}\rho^{\alpha
p}\varepsilon^{p}a^{p}$ and $\displaystyle
2^{p-1}{C}\sigma(a^{p}+1)\rho^{-n}\leqslant\frac{1}{4}\rho^{\alpha
p}\varepsilon^{p}a^{p}.$

As a consequence of this and (3.56),

$\displaystyle\fint_{B_{\rho}}\left|\nabla
u(x)-q-\bar{q}\right|^{p}\,dx\leqslant\frac{1}{4}\rho^{\alpha
p}\varepsilon^{p}a^{p}+\frac{1}{4}\rho^{\alpha
p}\varepsilon^{p}a^{p}+\frac{1}{4}\rho^{\alpha
p}\varepsilon^{p}a^{p}\leqslant\rho^{\alpha p}\varepsilon^{p}a^{p},$

which gives the desired result in (3.21) by setting
$\widetilde{q}:=q+\bar{q}$.

Moreover, from (3.54) we have that $|q-\widetilde{q}|=|\bar{q}|\leqslant
C\varepsilon a$, which establishes (3.22). This completes the proof of Lemma
3.3 when $p\geqslant 2$.

In the case $\max\left\\{\frac{2n}{n+2},1\right\\}<p<2$, we use (3.41) and
(3.55) and we see that

$\displaystyle\fint_{B_{\rho}}|\nabla u(x)-q-\bar{q}|^{p}\,dx$
$\displaystyle\leqslant$ $\displaystyle
2^{p-1}\left(\fint_{B_{\rho}}\left|\nabla u(x)-\nabla
v(x)\right|^{p}\,dx+\fint_{B_{\rho}}|\nabla v(x)-q-\bar{q}|^{p}\,dx\right)$
$\displaystyle\leqslant$ $\displaystyle
2^{p-1}C\Big{(}C_{1}\varepsilon^{p+\delta}+\sigma(a^{p}+1)\Big{)}^{\frac{p}{2}}a^{p\left(1-\frac{p}{2}\right)}\rho^{-n}+2^{p-1}C_{2}\rho^{\mu
p}\varepsilon^{p}a^{p}$ $\displaystyle\leqslant$ $\displaystyle
2^{p-1}C_{1}\varepsilon^{\frac{(p+\delta)p}{2}}a^{p\left(1-\frac{p}{2}\right)}\rho^{-n}+2^{p-1}{C}\sigma^{\frac{p}{2}}(a^{p}+1)^{\frac{p}{2}}a^{p\left(1-\frac{p}{2}\right)}\rho^{-n}+2^{p-1}C_{2}\rho^{\mu
p}\varepsilon^{p}a^{p}$ $\displaystyle\leqslant$ $\displaystyle
2^{p-1}C_{1}\varepsilon^{p+\widetilde{\delta}}a^{p\left(1-\frac{p}{2}\right)}\rho^{-n}+2^{p-1}{C}\sigma^{\frac{p}{2}}(a^{p}+1)^{\frac{p}{2}}a^{p\left(1-\frac{p}{2}\right)}\rho^{-n}+2^{p-1}C_{2}\rho^{\mu
p}\varepsilon^{p}a^{p},$

where $\widetilde{\delta}:=(p+\delta)p/2-p>0$, thanks to (3.46) (and up to
renaming $C$ and $C_{1}$, depending on $n$ and $p$).

We set $\alpha_{0}:=\mu$ and, for all $\alpha\in(0,\alpha_{0})$, we take
$\rho$ as in (3.57) and

$\varepsilon_{0}:=\left(\frac{\rho^{\alpha
p+n}a_{0}^{\frac{p^{2}}{2}}}{2^{p+1}C_{1}}\right)^{\frac{1}{\widetilde{\delta}}}\quad{\mbox{and}}\quad
c_{0}:=\frac{\rho^{2\alpha+\frac{n}{p}}a_{0}^{p}}{4^{\frac{p+1}{p}}\,{C}^{\frac{2}{p}}(a_{1}^{p}+1)},$

obtaining that, for all $\varepsilon\in(0,\varepsilon_{0}]$ and
$\sigma\in(0,c_{0}\varepsilon^{2}]$,

$\fint_{B_{\rho}}|\nabla u(x)-q-\bar{q}|^{p}\,dx\leqslant\rho^{\alpha
p}\varepsilon^{p}a^{p}.$

Hence, setting $\widetilde{q}:=q+\bar{q}$, the desired results in (3.21) and
(3.22) follow from this and (3.54).

The proof of Lemma 3.3 is thereby complete. ∎

Iterating Lemma 3.3 we obtain the following estimates:

######  Corollary 3.4.

Let $a_{1}>a_{0}>0$ and $p>\max\left\\{\frac{2n}{n+2},1\right\\}$. Let $u$ be
an almost minimizer for $J_{p}$ in $B_{1}$ (with constant $\kappa$ and
exponent $\beta$) and

$a:=\left(\fint_{B_{1}}\left|\nabla u(x)\right|^{p}\,dx\right)^{1/p}.$

Suppose that

(3.58) $a\in[a_{0},a_{1}]$

and that $u$ satisfies (3.5) and (3.19).

Then there exist $\varepsilon_{0}$, $\kappa_{0}$ and $\gamma\in(0,1)$,
depending on $n$, $p$, $\beta$, $a_{0}$ and $a_{1}$, such that, for every
$\varepsilon\in(0,\varepsilon_{0}]$ and
$\kappa\in(0,\kappa_{0}\varepsilon^{P}]$, with $P:=\max\\{p,2\\}$, then

(3.59) $\left\|u-\ell\right\|_{C^{1,\gamma}(B_{1/2})}\leqslant C\varepsilon
a.$

The positive constant $C$ depends only on $n$ and $p$, and $\ell$ is a linear
function of slope $q$.

Moreover,

(3.60) $\left\|\nabla u\right\|_{L^{\infty}(B_{1/2})}\leqslant\widetilde{C}a,$

with $\widetilde{C}>0$ depending only on $n$ and $p$.

######  Remark 3.5.

We point out that a consequence of (3.59) in Corollary 3.4 is that, if
$\varepsilon$ is sufficiently small,

(3.61) $\nabla u\neq 0\quad{\mbox{in }}B_{1/2}.$

Indeed, by (3.59), we have that, for all $x\in B_{1/2}$,

$\left|\nabla u(x)-q\right|\leqslant C\varepsilon a,$

which gives that

$\left|\nabla u(x)\right|\geqslant|q|-\left|\nabla
u(x)-q\right|\geqslant\left|q\right|-C\varepsilon a.$

As a consequence, from (3.5), we get

$\left|\nabla u(x)\right|\geqslant\frac{a}{4}-C\varepsilon a>0,$

as soon as $\varepsilon$ is sufficiently small, which yields (3.61).

Furthermore,

(3.62) $u>0\quad{\mbox{in }}B_{1/2}.$

To check this, we recall that $u\geqslant 0$ and we suppose by contradiction
that there exists a point $x_{0}\in B_{1/2}$ such that $u(x_{0})=0$. As a
consequence, since $u\in C^{1,\gamma}(B_{1/2})$, we see that $\nabla
u(x_{0})=0$, and this contradicts (3.61), thus proving (3.62).

In order to prove Corollary 3.4 we recall the definition of Campanato spaces
and a result which we will use in the proof of the corollary, see [G].

######  Definition 3.6 (Campanato spaces).

Let $\Omega$ be a bounded open set in $\mathbb{R}^{n}$, $1\leqslant p<+\infty$
and $\lambda\geqslant 0$. We denote by $\mathcal{L}^{p,\lambda}(\Omega)$ the
space of functions $u\in L^{p}(\Omega)$ such that

(3.63)
$[u]^{p}_{\mathcal{L}^{p,\lambda}(\Omega)}:=\sup_{\stackrel{{\scriptstyle
x_{0}\in\Omega}}{{\rho>0}}}\rho^{-\lambda}\int_{\Omega_{x_{0},\rho}}\left|u(x)-u_{x_{0},\rho}\right|^{p}\,dx<+\infty,$

where

$\Omega_{x_{0},\rho}:=\Omega\cap B_{\rho}(x_{0})\qquad{\mbox{and}}\qquad
u_{x_{0},\rho}:=\fint_{\Omega_{x_{0},\rho}}u(x)\,dx.$

We point out that the quantity $[u]_{\mathcal{L}^{p,\lambda}(\Omega)}$ is a
seminorm in $\mathcal{L}^{p,\lambda}(\Omega)$ and

(3.64) ${\mbox{it is equivalent to the quantity
}}\quad\left(\sup_{\stackrel{{\scriptstyle
x_{0}\in\Omega}}{{\rho>0}}}\rho^{-\lambda}\inf_{\xi\in\mathbb{R}^{n}}\int_{\Omega_{x_{0},\rho}}\left|u(x)-\xi\right|^{p}\,dx\right)^{1/p}.$

We also define the norm in $\mathcal{L}^{p,\lambda}(\Omega)$ as

(3.65)
$\left\|u\right\|_{\mathcal{L}^{p,\lambda}(\Omega)}:=\left\|u\right\|_{L^{p}(\Omega)}+[u]_{\mathcal{L}^{p,\lambda}(\Omega)}.$

######  Theorem 3.7.

Let $x\in\mathbb{R}^{n}$ and $r>0$. Let $1\leqslant p<+\infty$ and
$\lambda\geqslant 0$. Suppose that $n<\lambda\leqslant n+p$.

Then, the space $\mathcal{L}^{p,\lambda}(B_{r}(x))$ is isomorphic to
$C^{0,\alpha}(B_{r}(x)),$ with $\alpha:=\frac{\lambda-n}{p}.$

We also point out the following scaling property of almost minimizers:

######  Lemma 3.8.

Let $u$ be an almost minimizer for $J_{p}$ in $B_{1}$ with constant $\kappa$
and exponent $\beta$. For any $r\in(0,1)$, let

(3.66) $u_{r}(x):=\frac{u(rx)}{r}.$

Then, $u_{r}$ is an almost minimizer for $J_{p}$ in $B_{1/r}$ with constant
$\kappa r^{\beta}$ and exponent $\beta$, namely

(3.67) $J_{p}(u_{r},B_{\varrho}(x_{0}))\leqslant(1+\kappa
r^{\beta}\varrho^{\beta})J_{p}(v,B_{\varrho}(x_{0})),$

for every ball $B_{\varrho}(x_{0})$ such that
$\overline{B_{\varrho}(x_{0})}\subset B_{1/r}$ and for every $v\in
W^{1,p}(B_{\varrho}(x_{0}))$ such that $v=u_{r}$ on $\partial
B_{\varrho}(x_{0})$ in the sense of the trace.

###### Proof.

By definition of almost minimizers, we know that

(3.68)
$J_{p}(u,B_{\vartheta}(y_{0}))\leqslant(1+\kappa\vartheta^{\beta})J_{p}(w,B_{\vartheta}(y_{0}))$

for every ball $B_{\vartheta}(y_{0})$ such that
$\overline{B_{\vartheta}(y_{0})}\subset B_{1/r}$ and for every $w\in
W^{1,p}(B_{\vartheta}(y_{0}))$ such that $w=u$ on $\partial
B_{\vartheta}(y_{0})$ in the sense of the trace.

Now, given $x_{0}\in B_{1/r}$, we take $\varrho$ and $v$ as in the statement
of Lemma 3.8 and we define

(3.69) $w(x):=rv\left(\frac{x}{r}\right).$

Then, using the notation $y_{0}:=rx_{0}$ and $\vartheta:=r\varrho$, for all
$x\in\partial B_{\vartheta}(y_{0})$, we have that $\frac{x}{r}\in\partial
B_{\varrho}(x_{0})$ and therefore, in the sense of the trace,

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

Accordingly, we can use $w$ as a competitor for $u$ in (3.68), thus obtaining
that

(3.70) $\int_{B_{r\varrho}(y_{0})}\Big{(}\left|\nabla
u(y)\right|^{p}+\chi_{\left\\{u>0\right\\}}(y)\Big{)}\,dy\leqslant(1+\kappa
r^{\beta}\varrho^{\beta})\int_{B_{r\varrho}(y_{0})}\Big{(}\left|\nabla
w(y)\right|^{p}+\chi_{\left\\{w>0\right\\}}(y)\Big{)}\,dy.$

Furthermore, using, consistently with (3.66), the notation
$w_{r}(x):=\frac{w(rx)}{r}$, with the change of variable $x:=\frac{y}{r}$ we
see that

(3.71) $\begin{split}&\int_{B_{r\varrho}(y_{0})}\Big{(}\left|\nabla
w(y)\right|^{p}+\chi_{\left\\{w>0\right\\}}(y)\Big{)}\,dy={r^{n}}\int_{B_{\varrho}(x_{0})}\Big{(}\left|\nabla
w(rx)\right|^{p}+\chi_{\left\\{w>0\right\\}}(rx)\Big{)}\,dx\\\
&\qquad\qquad=r^{n}\int_{B_{\varrho}(x_{0})}\Big{(}\left|\nabla
w_{r}(x)\right|^{p}+\chi_{\left\\{w_{r}>0\right\\}}(x)\Big{)}\,dx\end{split}$

and a similar identity holds true with $u$ and $u_{r}$ replacing $w$ and
$w_{r}$.

Also, recalling (3.69), we observe that $v=w_{r}$. Plugging this information
and (3.71) into (3.70), we obtain the desired result in (3.67).∎

###### Proof of Corollary 3.4.

Up to scaling, we suppose that

(3.72) $u$ is an almost minimizer for $J_{p}$ in $B_{2}$ (with constant
$\kappa$ and exponent $\beta$).

We prove that we can iterate Lemma 3.3 indefinitely with
$\alpha:=\min\left\\{\frac{\alpha_{0}}{2},\frac{\beta}{P}\right\\}$, where
$P:=\max\\{p,2\\}$ and $\alpha_{0}$ is given in Lemma 3.3.

More precisely, we claim that, for all $k\geqslant 0$, there exists
$q_{k}\in\mathbb{R}^{n}$ such that

(3.73)
$\begin{split}&|q_{k}|\in\left[\frac{a}{4}-\frac{\widetilde{C}\varepsilon
a\left(1-\rho^{k\alpha}\right)}{1-\rho^{\alpha}},\,C_{0}a+\frac{\widetilde{C}\varepsilon
a\left(1-\rho^{k\alpha}\right)}{1-\rho^{\alpha}}\right],\\\
&\left(\fint_{B_{\rho^{k}}}\left|\nabla
u(x)-q_{k}\right|^{p}\,dx\right)^{1/p}\leqslant\rho^{{k\alpha}}\varepsilon
a\\\ {\mbox{and }}\quad&\left(\fint_{B_{\rho^{k}}}\left|\nabla
u(x)\right|^{p}\,dx\right)^{1/p}\in\left[\frac{|q|}{2},2|q|\right],\end{split}$

where $\rho\in(0,1)$, $C_{0}>0$ and $\widetilde{C}>0$ are universal
quantities.

To prove this, we argue by induction. When $k=0$, we pick $q_{0}:=q$. Then, in
this case, the desired claims in (3.73) follow from (3.5), (3.17) and (3.19).

Now we perform the inductive step by assuming that (3.73) is satisfied for $k$
and we establish the claim for $k+1$. To this end, we let $r:=\rho^{k}$ and
utilize the rescaling $u_{r}$ defined in (3.66). In this setting, the
inductive assumption gives that

$\left(\fint_{B_{1}}\left|\nabla
u_{r}(x)-q_{k}\right|^{p}\,dx\right)^{1/p}\leqslant r^{\alpha}\varepsilon
a=\varepsilon_{k}a,$

with $\varepsilon_{k}:=r^{\alpha}\varepsilon=\rho^{k\alpha}\varepsilon$.

Notice that the inductive assumption also yields (3.20), as soon as
$\varepsilon$ is chosen conveniently small. Therefore, thanks to Lemma 3.8 as
well, we are in position of using Lemma 3.3 on the function $u_{r}$ with
$\sigma:=\kappa r^{\beta}$. We stress that the structural condition
$\sigma\leqslant c_{o}\varepsilon^{P}$ in Lemma 3.3 translates here into
$\kappa\leqslant c_{0}\varepsilon^{P}$, which is precisely the requirement in
the statement of Corollary 3.4 (by taking $\kappa_{0}$ there less than or
equal to $c_{0}$). In this way, we deduce from (3.21) and (3.22) that there
exists $q_{k+1}\in\mathbb{R}^{n}$ such that

$\begin{split}\left(\fint_{B_{\rho}}\left|\nabla
u_{r}(x)-q_{k+1}\right|^{p}\,dx\right)^{1/p}\leqslant\rho^{\alpha}\varepsilon_{k}a\qquad{\mbox{and}}\qquad\left|q_{k}-{q}_{k+1}\right|\leqslant\widetilde{C}\varepsilon_{k}a.\end{split}$

Scaling back, we find that

$\left(\fint_{B_{\rho^{k+1}}}\left|\nabla
u(x)-q_{k+1}\right|^{p}\,dx\right)^{1/p}\leqslant\rho^{\alpha}\rho^{{k\alpha}}\varepsilon
a=\rho^{{(k+1)\alpha}}\varepsilon a.$

Furthermore,

$\displaystyle|q_{k+1}|\leqslant|q_{k}-q_{k+1}|+|q_{k}|\leqslant\widetilde{C}\varepsilon_{k}a+C_{0}a+\frac{\widetilde{C}\varepsilon
a\left(1-\rho^{k\alpha}\right)}{1-\rho^{\alpha}}$
$\displaystyle\qquad=C_{0}a+\frac{\widetilde{C}\varepsilon
a\left(1-\rho^{k\alpha}\right)}{1-\rho^{\alpha}}+\widetilde{C}\rho^{k\alpha}\varepsilon
a=C_{0}a+\frac{\widetilde{C}\varepsilon
a\left(1-\rho^{(k+1)\alpha}\right)}{1-\rho^{\alpha}}$

and

$\displaystyle|q_{k+1}|\geqslant|q_{k}|-|q_{k}-q_{k+1}|\geqslant\frac{a}{4}-\frac{\widetilde{C}\varepsilon
a\left(1-\rho^{k\alpha}\right)}{1-\rho^{\alpha}}-\widetilde{C}\varepsilon_{k}a$
$\displaystyle\qquad=\frac{a}{4}-\frac{\widetilde{C}\varepsilon
a\left(1-\rho^{k\alpha}\right)}{1-\rho^{\alpha}}-\widetilde{C}\rho^{k\alpha}\varepsilon
a=\frac{a}{4}-\frac{\widetilde{C}\varepsilon
a\left(1-\rho^{(k+1)\alpha}\right)}{1-\rho^{\alpha}}.$

In addition,

$\displaystyle\left|\left(\fint_{B_{\rho^{k+1}}}\left|\nabla
u(x)\right|^{p}\,dx\right)^{1/p}-q_{k+1}\right|=\frac{1}{|B_{\rho^{k+1}}|^{1/p}}\left|\left(\int_{B_{\rho^{k+1}}}\left|\nabla
u(x)\right|^{p}\,dx\right)^{1/p}-q_{k+1}|B_{\rho^{k+1}}|^{1/p}\right|$
$\displaystyle\qquad=\frac{1}{|B_{\rho^{k+1}}|^{1/p}}\left|\left\|\nabla
u\right\|_{L^{p}(B_{\rho^{k+1}})}-\left\|q_{k+1}\right\|_{L^{p}(B_{\rho^{k+1}})}\right|\leqslant\frac{1}{|B_{\rho^{k+1}}|^{1/p}}\left\|\nabla
u-q_{k+1}\right\|_{L^{p}(B_{\rho^{k+1}})}$
$\displaystyle\qquad=\left(\fint_{B_{\rho^{k+1}}}\left|\nabla
u(x)-q_{k+1}\right|^{p}\,dx\right)^{1/p}\leqslant\varepsilon a,$

which yields that

$\displaystyle\left|\left(\fint_{B_{\rho^{k+1}}}\left|\nabla
u(x)\right|^{p}\,dx\right)^{1/p}-q\right|=\left|\left(\fint_{B_{\rho^{k+1}}}\left|\nabla
u(x)\right|^{p}\,dx\right)^{1/p}-q_{0}\right|$
$\displaystyle\qquad\leqslant\left|\left(\fint_{B_{\rho^{k+1}}}\left|\nabla
u(x)\right|^{p}\,dx\right)^{1/p}-q_{k+1}\right|+\sum_{j=0}^{k}|q_{j+1}-q_{j}|$
$\displaystyle\qquad\leqslant\varepsilon
a+\widetilde{C}a\sum_{j=0}^{k}\varepsilon_{j}\leqslant\varepsilon
a+\widetilde{C}\varepsilon
a\sum_{j=0}^{+\infty}\rho^{j\alpha}=\left(1+\frac{\widetilde{C}}{1-\rho^{\alpha}}\right)\varepsilon
a$
$\displaystyle\qquad\leqslant\left(1+\frac{\widetilde{C}}{1-\rho^{\alpha}}\right)\varepsilon|q|\leqslant\frac{|q|}{2}.$

These observations conclude the proof of the inductive step and establish
(3.73).

By scaling the relations in (3.73) (and recalling (3.72)), we have that, for
all $x_{0}\in B_{3/4}$,

(3.74) $\left(\fint_{B_{\rho^{k}}(x_{0})}\left|\nabla
u(x)-q_{k,x_{0}}\right|^{p}\,dx\right)^{1/p}\leqslant
C\rho^{k\alpha}\varepsilon a,$

for some positive constant $C$, depending only on $n$ and $p$, and suitable
points $q_{k,x_{0}}$ such that

$|q_{k,x_{0}}|\in\left[\frac{a}{4}-\frac{\widetilde{C}\varepsilon
a\left(1-\rho^{k\alpha}\right)}{1-\rho^{\alpha}},\,C_{0}a+\frac{\widetilde{C}\varepsilon
a\left(1-\rho^{k\alpha}\right)}{1-\rho^{\alpha}}\right].$

We now want to exploit the Campanato estimate in Theorem 3.7, here applied to
the gradient of $u$ minus $q$, and used with $\Omega:=B_{1/2}$ and
$\lambda:=n+\alpha p$ (note that, with this choice, $\lambda\in(n,n+p]$ and we
are therefore under the structural assumptions of Theorem 3.7). For this
purpose, we claim that, for all $x_{0}\in B_{3/4}$ and $\tau>0$,

(3.75)
$\left(\tau^{-\lambda}\inf_{\xi\in\mathbb{R}^{n}}\int_{B_{\tau}(x_{0})\cap
B_{1/2}}\left|\nabla u(x)-q-\xi\right|^{p}\,dx\right)^{1/p}\leqslant
C\varepsilon a,$

up to renaming $C>0$.

To prove this, we distinguish two cases, either $\tau\geqslant 1$ or
$\tau\in(0,1)$.

If $\tau\geqslant 1$, we use (3.19) and we get that

$\displaystyle\left(\tau^{-\lambda}\inf_{\xi\in\mathbb{R}^{n}}\int_{B_{\tau}(x_{0})\cap
B_{1/2}}\left|\nabla
u(x)-q-\xi\right|^{p}\,dx\right)^{1/p}\leqslant\left(\int_{B_{1/2}}\left|\nabla
u(x)-q\right|^{p}\,dx\right)^{1/p}$
$\displaystyle\qquad\leqslant\left(|B_{1}|\fint_{B_{1}}\left|\nabla
u(x)-q\right|^{p}\,dx\right)^{1/p}\leqslant|B_{1}|^{1/p}\,\varepsilon a,$

which gives (3.75) in this case.

If instead $\tau\in(0,1)$, we pick $k\in\mathbb{N}$ such that
$\rho^{k+1}<\tau\leqslant\rho^{k}$ and we make use of (3.74) to infer that

$\displaystyle\left(\tau^{-\lambda}\inf_{\xi\in\mathbb{R}^{n}}\int_{B_{\tau}(x_{0})\cap
B_{1/2}}\left|\nabla
u(x)-q-\xi\right|^{p}\,dx\right)^{1/p}\leqslant\left(\rho^{-\lambda(k+1)}\int_{B_{\rho^{k}}(x_{0})}\left|\nabla
u(x)-q_{k,x_{0}}\right|^{p}\,dx\right)^{1/p}$
$\displaystyle\qquad=\left(\rho^{-\lambda(k+1)+kn}\,|B_{1}|\fint_{B_{\rho^{k}}(x_{0})}\left|\nabla
u(x)-q_{k,x_{0}}\right|^{p}\,dx\right)^{1/p}$ $\displaystyle\qquad\leqslant
C|B_{1}|^{1/p}\,\rho^{[-(n+\alpha p)(k+1)+kn+k\alpha p]/p}\,\varepsilon a$
$\displaystyle\qquad\leqslant C|B_{1}|^{1/p}\,\rho^{[-n-\alpha
p]/p}\,\varepsilon a,$

which gives (3.75) up to renaming $C$. The proof of (3.75) is thereby
complete.

By (3.64) and (3.75), we obtain that

$[\nabla u-q]_{\mathcal{L}^{p,\lambda}(B_{1/2})}\leqslant C\varepsilon a,$

up to renaming $C$. Since also

$\left\|\nabla u-q\right\|_{L^{p}(B_{1/2})}\leqslant C\varepsilon a,$

thanks to (3.19), we deduce from (3.65) that $\left\|\nabla
u-q\right\|_{\mathcal{L}^{p,\lambda}(B_{1/2})}\leqslant C\varepsilon a$, up to
renaming $C$. Then, in view of Theorem 3.7, we find that

(3.76) $\left\|\nabla u-q\right\|_{C^{0,\gamma}(B_{1/2})}\leqslant
C\varepsilon a,$

with $\gamma=\frac{\lambda-n}{p}=\alpha$.

Now we define $\ell(x):=u(0)+q\cdot x$. For all $x\in B_{1/2}$ we have that

$|u(x)-\ell(x)|=|u(x)-u(0)-q\cdot x|=\left|\int_{0}^{1}\big{(}\nabla
u(tx)-q\big{)}\cdot x\,dt\right|\leqslant C\varepsilon a,$

thanks to (3.76), up to renaming $C$, and therefore
$\|u-\ell\|_{L^{\infty}(B_{1/2})}\leqslant C\varepsilon a$. This and (3.76)
establish (3.59), as desired. ∎

##  4\. Lipschitz continuity of almost minimizers and proof of Theorem 1.2

We are now in position of establishing the Lipschitz regularity result in
Theorem 1.2.

###### Proof of Theorem 1.2.

In the light of Lemma 3.8, up to a rescaling, we can assume that $u$ is an
almost minimizer with constant

(4.1) $\widetilde{\kappa}:=\kappa s^{\beta},$

which can be made arbitrarily small by an appropriate choice of $s>0$.

We let $P:=\max\\{p,2\\}$. Let also $\alpha_{0}\in(0,1]$ be the structural
constant given by Lemma 3.3 and define

$\alpha:=\frac{1}{2}\min\left\\{\alpha_{0},\frac{\beta}{P}\right\\}.$

We also consider $\varepsilon_{0}$ as given by Proposition 3.1 and take
$\eta\in(0,1)$ and $M\geqslant 1$ as in Proposition 3.1 (corresponding here to
choice $\varepsilon:=\varepsilon_{0}/2$).

Let us define

(4.2) $a(\tau):=\bigg{(}\fint_{B_{\tau}}\left|\nabla
u(x)\right|^{p}\,dx\bigg{)}^{1/p}.$

We claim that, for every $r\in(0,\eta]$,

(4.3) $a(r)\leqslant C(M,\eta)(1+a(1)),$

for some $C(M,\eta)>0$, possibly depending on $n$ and $p$ as well.

To prove this, we consider the set
${\mathcal{K}}\subseteq\mathbb{N}=\\{0,1,2,\dots\\}$ containing all the
integers $k\in\mathbb{N}$ such that

(4.4) $a(\eta^{k})\leqslant C(\eta)M+2^{-k}a(1),$

where

(4.5) $C(\eta):=2\eta^{-n/p}.$

We stress that for $k=0$ formula (4.4) is clearly true, hence

(4.6) $0\in{\mathcal{K}}\neq\varnothing.$

We then distinguish two cases, namely whether (4.4) holds for every $k$ (i.e.
${\mathcal{K}}=\mathbb{N}$) or not (i.e.
${\mathcal{K}}\subsetneqq\mathbb{N}$).

To start with, let us assume that we are in the first situation. Thus, for
every $r\in(0,\eta],$ we pick $k_{0}\in\mathbb{N}={\mathcal{K}}$ such that
$\eta^{k_{0}+1}<r\leqslant\eta^{k_{0}}.$ Hence, according to (4.2) and (4.4),
we get that

$\begin{split}&a(r)\leqslant\bigg{(}\frac{1}{|B_{1}|\,\eta^{(k_{0}+1)n}}\int_{B_{\eta^{k_{0}}}}\left|\nabla
u(x)\right|^{p}\,dx\bigg{)}^{1/p}=\eta^{-n/p}a(\eta^{k_{0}})\leqslant\eta^{-n/p}(C(\eta)M+a(1))\\\
&\qquad\qquad\leqslant\eta^{-n/p}\max\left\\{C(\eta)M,1\right\\}(1+a(1))\leqslant
C(M,\eta)(1+a(1)),\end{split}$

provided that $C(M,\eta)\geqslant\eta^{-n/p}\max\left\\{C(\eta)M,1\right\\}$.
The proof of (4.3) is thereby complete in this case.

We now consider the second case, that is the one in which
${\mathcal{K}}\subsetneqq\mathbb{N}$. Then, by (4.6), there exists
$k_{0}\in\mathbb{N}$ such that $\\{0,\dots,k_{0}\\}\in{\mathcal{K}}$ and

(4.7) $k_{0}+1\not\in{\mathcal{K}}.$

We notice that, by (4.2) and (4.5),

$\eta^{-n/p}M<C(\eta)M\leqslant
C(\eta)M+2^{-(k_{0}+1)}a(1)<a(\eta^{k_{0}+1})\leqslant\eta^{-n/p}a(\eta^{k_{0}})$

and therefore

(4.8) $a(\eta^{k_{0}})>M.$

Furthermore, from (4.7),

(4.9)
$a(\eta^{k_{0}+1})>C(\eta)M+2^{-(k_{0}+1)}a(1)\geqslant\frac{C(\eta)M+2^{-k_{0}}a(1)}{2}\geqslant\frac{a(\eta^{k_{0}})}{2}.$

Now, we claim that

(4.10) $\begin{split}&\left(\fint_{B_{\eta^{k_{0}+1}}}\left|\nabla
u(x)-q\right|^{p}\,dx\right)^{1/p}\leqslant\varepsilon a(\eta^{k_{0}}),\\\
&{\mbox{for some\leavevmode\nobreak\ $q\in\mathbb{R}^{n}$ such that
}}\quad\frac{a(\eta^{k_{0}})}{4}<\left|q\right|\leqslant
C_{0}a(\eta^{k_{0}}),\end{split}$

being $C_{0}$ the constant given by Proposition 3.1.

To prove this, we apply the dichotomy result in Proposition 3.1 rescaled in
the ball $B_{\eta^{k_{0}}}$. In this way, we deduce from (3.3), (3.4) and
(3.5) that (4.10) holds true, unless

$a(\eta^{k_{0}+1})=\left(\fint_{B_{\eta^{k_{0}+1}}}\left|\nabla
u(x)\right|^{p}\,dx\right)^{1/p}\leqslant\frac{a(\eta^{k_{0}})}{2},$

in contradiction with (4.9). This ends the proof of (4.10).

Now we apply Corollary 3.4 rescaled in the ball $B_{\eta^{k_{0}+1}}$: namely,
we use here Corollary 3.4 with $B_{1}$ replaced by $B_{\eta^{k_{0}+1}}$ and
$a$ replaced by $a(\eta^{k_{0}+1})$. To this end, we need to verify that the
assumptions of Corollary 3.4 are fulfilled in this rescaled situation.
Specifically, we note that, by (4.8) and (4.9),

$a(\eta^{k_{0}+1})\geqslant\frac{M}{2}.$

Also, since $k_{0}\in{\mathcal{K}}$,

$a(\eta^{k_{0}+1})\leqslant\eta^{-n/p}a(\eta^{k_{0}})\leqslant\eta^{-n/p}\big{(}C(\eta)M+2^{-k_{0}}a(1)\big{)}\leqslant\eta^{-n/p}\big{(}C(\eta)M+a(1)\big{)}.$

These observations give that (3.58) is satisfied in this rescaled setting with

(4.11) $a_{0}:=\frac{M}{2}\qquad{\mbox{ and }}\qquad
a_{1}:=\eta^{-n/p}\big{(}C(\eta)M+a(1)\big{)}.$

Moreover, by (4.9) and (4.10),

$\left(\fint_{B_{\eta^{k_{0}+1}}}\left|\nabla
u(x)-q\right|^{p}\,dx\right)^{1/p}\leqslant 2\varepsilon a(\eta^{k_{0}+1}),$

showing that (3.19) is satisfied (here with $2\varepsilon$ instead of
$\varepsilon$).

We also deduce from (4.9) and (4.10) that

$\frac{\eta^{n/p}a(\eta^{k_{0}+1})}{4}\leqslant\frac{a(\eta^{k_{0}})}{4}<\left|q\right|\leqslant
C_{0}a(\eta^{k_{0}})\leqslant 2C_{0}a(\eta^{k_{0}+1}),$

which gives that (3.5) is satisfied here (though with different structural
constants).

We can thereby exploit Corollary 3.4 in a rescaled version to obtain the
existence of $\varepsilon_{0}$ (possibly different from the one given by
Proposition 3.1) and $\kappa_{0}$, depending on $n$, $p$, $\beta$, $a_{0}$ and
$a_{1}$, such that, if

(4.12) $\widetilde{\kappa}\in(0,\kappa_{0}\varepsilon_{0}^{P}],$

we have that

(4.13) $\left\|\nabla
u\right\|_{L^{\infty}(B_{\eta^{k_{0}+1}/2})}\leqslant\bar{C}a(\eta^{k_{0}}),$

for some structural constant $\bar{C}>0$.

We remark that condition (4.12) is satisfied by taking $s$ in (4.1)
sufficiently small, namely taking
$s:=\left(\frac{\kappa_{0}\varepsilon_{0}^{P}}{2\kappa}\right)^{\frac{1}{\beta}}$.
Notice in particular that

(4.14) $s$ depends on $n$, $p$, $\kappa$, $\beta$ and $\|\nabla
u\|_{L^{p}(B_{1})}$,

due to (4.11).

As a result of (4.13), for all $r\in\left(0,\frac{\eta^{k_{0}+1}}{2}\right)$,

(4.15) $\begin{split}&a(r)=\left(\frac{1}{|B_{r}|}\int_{B_{r}}|\nabla
u(x)|^{p}\,dx\right)^{\frac{1}{p}}\leqslant\bar{C}a(\eta^{k_{0}})\leqslant\bar{C}\big{(}C(\eta)M+2^{-k_{0}}a(1)\big{)}\\\
&\qquad\qquad\leqslant\bar{C}\big{(}C(\eta)M+a(1)\big{)}\leqslant
C(M,\eta)(1+a(1)),\end{split}$

as long as $C(M,\eta)\geqslant\bar{C}(C(\eta)M+1)$.

Moreover, if $r\in\left[\frac{\eta^{k_{0}+1}}{2},\eta\right)$ then we take
$k_{r}\in\mathbb{N}$ such that $\eta^{k_{r}+1}<r\leqslant\eta^{k_{r}}$. Note
that

$\frac{1}{\eta^{k_{r}}}\leqslant\frac{1}{r}\leqslant\frac{2}{\eta^{k_{0}+1}},$

whence

(4.16) $k_{r}\leqslant k_{0}+C_{\star},$

where $C_{\star}:=1+\frac{\log 2}{\log(1/\eta)}$.

We now distinguish two cases: if $k_{r}\in\\{0,\dots,k_{0}\\}$ then
$k_{r}\in{\mathcal{K}}$ and therefore

$a(\eta^{k_{r}})\leqslant C(\eta)M+2^{-k_{r}}a(1).$

From this we obtain that

(4.17)
$\begin{split}&a(r)\leqslant\left(\frac{1}{|B_{\eta^{k_{r}+1}}|}\int_{B_{\eta^{k_{r}}}}|\nabla
u(x)|^{p}\,dx\right)^{\frac{1}{p}}=\eta^{-n/p}a(\eta^{k_{r}})\\\
&\qquad\qquad\leqslant\eta^{-n/p}\big{(}C(\eta)M+2^{-k_{r}}a(1)\big{)}\leqslant
C(M,\eta)(1+a(1)).\end{split}$

If instead $k_{r}>k_{0}$, we employ (4.16) to see that

$\displaystyle
a(r)\leqslant\left(\frac{1}{|B_{\eta^{k_{0}+C_{\star}+1}}|}\int_{B_{\eta^{k_{0}}}}|\nabla
u(x)|^{p}\,dx\right)^{\frac{1}{p}}=\eta^{-n(C_{\star}+1)/p}a(\eta^{k_{0}})$
$\displaystyle\qquad\qquad\quad\leqslant\eta^{-n(C_{\star}+1)/p}\big{(}C(\eta)M+2^{-k_{0}}a(1)\big{)}\leqslant\eta^{-n(C_{\star}+1)/p}\big{(}C(\eta)M+a(1)\big{)}$
$\displaystyle\quad\qquad\qquad\leqslant C(M,\eta)(1+a(1)),$

as long as $C(M,\eta)$ is chosen large enough.

This and (4.17) give that for all
$r\in\left[\frac{\eta^{k_{0}+1}}{2},\eta\right)$, we have that $a(r)\leqslant
C(M,\eta)(1+a(1))$. Combining this with (4.15), we deduce that (4.3) holds
true, as desired.

Now, up to scaling and translations, we can extend (4.3) to all balls with
center $x_{0}$ in $B_{1/2}$ and sufficiently small radius. Namely, we have
that, for all $r\in(0,\eta]$,

$a(r,x_{0})\leqslant C(M,\eta)(1+a(1)),$

where

$a(r,x_{0}):=\left(\fint_{B_{r}(x_{0})}\left|\nabla
u(x)\right|^{p}\,dx\right)^{1/p}.$

Therefore, according to the Lebesgue Differentiation Theorem, recalling that
$u\in W^{1,p}(B_{1}),$ we find that, for all $x_{0}\in B_{1/2}$,

$\displaystyle\left|\nabla u(x_{0})\right|=\lim_{r\rightarrow
0}a(r,x_{0})\leqslant C(M,\eta)(1+a(1))=C\Big{(}1+\left\|\nabla
u\right\|_{L^{p}(B_{1})}\Big{)},$

for some $C>0$, which yields

(4.18) $\left\|\nabla u\right\|_{L^{\infty}(B_{1/2})}\leqslant
C\Big{(}1+\left\|\nabla u\right\|_{L^{p}(B_{1})}\Big{)}.$

So, the first claim in Theorem 1.2 is proved.

We now show that the second claim in Theorem 1.2 holds true. For this, we can
assume that

(4.19) $\\{u=0\\}\cap B_{s/100}\neq\varnothing$

and we take a Lebesgue point $\bar{x}\in B_{s/100}$ for $\nabla u$. Up to a
translation, we can suppose that $\bar{x}=0$ and change our assumption (4.19)
into

(4.20) $\\{u=0\\}\cap B_{s/50}\neq\varnothing.$

We claim that, in this situation,

(4.21) ${\mathcal{K}}=\mathbb{N}.$

Indeed, suppose not and let $k_{0}$ as above (recall (4.7)). Then, in light of
(4.10), we can apply Corollary 3.4 (rescaled as before) and conclude, by
(3.62), that $u>0$ in $B_{s/2}$, in contradiction with our hypothesis in
(4.20). This establishes (4.21).

Therefore, in view of (4.4) and (4.21), for all $k\in\mathbb{N}$,

$a(\eta^{k})\leqslant C(\eta)M+2^{-k}a(1),$

and as a result

$|\nabla u(\bar{x})|=|\nabla
u(0)|=\lim_{k\to+\infty}a(\eta^{k})\leqslant\lim_{k\to+\infty}\big{(}C(\eta)M+2^{-k}a(1)\big{)}=C(\eta)M.$

Recalling also (4.14), the proof of the second claim in Theorem 1.2 is thereby
complete. ∎

## References