\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 283, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2017/283\hfil Boundary regularity of solutions]
{Boundary regularity for quasi-linear elliptic equations with lower order term}

\author[G. Riey \hfil EJDE-2017/283\hfilneg]
{Giuseppe Riey}

\address{Giuseppe Riey \newline
Dipartimento di Matematica e Informatica,
Universit\`a della Calabria,
Ponte Pietro Bucci 31B, I-87036 Arcavacata di Rende,
Cosenza, Italy}
\email{riey@mat.unical.it}

\dedicatory{Communicated by Marco Squassina}

\thanks{Submitted July 15, 2017. Published November 14, 2017.}
\subjclass[2010]{35J92, 35B33, 35B06}
\keywords{Quasi-linear elliptic equations, $p$-Laplace operator, regularity}

\begin{abstract}
 We give results of summability up to the boundary for the
 solutions to quasi-linear elliptic equations involving the
 $p$-Laplace operator and a term depending on the gradient
 of the solution.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}\label{introduction}

In this article we extend the results in \cite{RiSc} to quasilinear elliptic
 equations involving lower order terms (see for instance
\cite{LePoRi, MeMoSc, MoScSq} and the references quoted there
for remarks on applications of these kind of equations).
Given $\Omega\subset\mathbb{R}^n$ bounded and with smooth boundary,
for fixed $p>1$ we consider the equation:
\begin{equation}\label{equazione forte}
\begin{gathered}
-\Delta_p u+H(x,\nabla u)=f(x,u) \quad\text{in }\Omega\\
u=0 \quad\text{on }\partial\Omega\,,
\end{gathered}
\end{equation}
where
$f=f(x,s):\bar\Omega\times\mathbb{R}\to\mathbb{R}$ is Lipshitz continuous 
on each compact subset of $\bar\Omega\times\mathbb{R}$ and 
$H=H(x,\xi):\Omega\times\mathbb{R}\to\mathbb{R}$
is a Carath\'eodory function
which is also locally Lipshitz continuous with respect to $x$ and
$C^1$ with respect to $\xi$ and satisfies the following property:
For each $M>0$, there exists $k=k(M)>0$ such that
\begin{equation}\label{prop H}
|H_\xi(x,\xi)|\leq k|\xi|^{\frac{p-2}{2}}\quad
\forall\xi:\,|\xi|\leq M\quad
\text{for a.e. } x\in\Omega.
\end{equation}

Let $u\in W^{1,p}_{0}(\Omega)$ be a weak solution to \eqref{equazione forte},
that means $u$ satisfies:
\begin{equation}\label{equazione debole}
\int_\Omega|\nabla u|^{p-2}\langle\nabla u,\nabla\varphi\rangle dx
+\int_\Omega H(x,\nabla u)\varphi dx
= \int_\Omega f(x,u)\varphi dx,
\quad\text{for all }\varphi\in C_c^{\infty}(\Omega).
\end{equation}
By \cite{Di,Li,T} we know that  $u\in C^{1,\alpha}(\overline{\Omega})$
for some $\alpha<1$.
We focus our attention on the summability of the second derivatives of $u$.
If the term $H(x,\nabla u)$ there is not in \eqref{equazione forte}, results on
the regularity of the second derivatives (in the interior of $\Omega$)
 can be found in \cite{DS1,Sci1,Sci2}.
Moreover, if the Hopf's boundary Lemma can be applied, the results hold up
to the boundary (in fact the solutions have no critical points there and
therefore the equation is no more degenerate).
Therefore we consider cases where Hopf's Lemma cannot be applied.
Our main result is the following.

\begin{theorem}\label{teorema principale}
Let $u\in C^{1,\alpha}(\overline{\Omega})$ be a weak solution to
\eqref{equazione forte}.
We have
\begin{itemize}
\item[(i)] if $p\leq 2$, then $u\in W^{2,2}(\Omega)$
\item[(ii)] if $p>2$, then $|\nabla u|^{p-1}\in W^{1,2}(\Omega)$.
\end{itemize}
\end{theorem}

The main tool in our proofs is a weighted estimate for the second
derivatives of the solution, achieved by means of the linearized
operator of a transformed equation (using the techniques developed in
\cite{DS1,DS3,MRS,Sci1,Sci2}).

Regularity results in cases, where Hopf's Lemma does not apply,
can be found also in \cite{MRS},
where the summability of the second derivatives is studied for $p$ close to two.
An example of application of the above techniques to
cases where the Hopf's Lemma fails, can be found also in
\cite{LePoRi,MoScSq}.
Note that the study of the regularity of solutions
to $p$-Laplace equations, which is interesting in itself,
is also very much related to the study of the qualitative properties 
of the solutions
(see for instance \cite{DS1, DS2, FaMoSc, FaMoRiSc, MoRiSc, Sci3}).

\section{Notation and preliminary results}

Given a matrix $A$, $A^T$ is the transposed of $A$
and we set
$|A|=\big(\sum_{i,j=1}^n|a_{ij}|^2\big)^{1/2}$.
For $r>0$ and $\overline{x}\in\mathbb{R}^n$ we set 
$B_r(\overline{x})=\{x\in\mathbb{R}^n:|x-\overline{x}|<r\}$.
For $\overline{x}\in\partial\Omega$ and $r>0$ small enough, we set
 $B^+:=B_r(\overline{x})\cap\Omega$.
We consider a diffeomorphism $\Phi:\mathbb{R}^n\to\mathbb{R}^n$
(also called \emph{flattening operator})
such that $\Phi^{-1}(B_r(\overline{x})\cap\partial\Omega)\subset \{y_n=0\}$.
We set:
$\mathcal B^+=\Phi^{-1}(B^+)$ and we denote by $\mathcal B^-$ the
reflection of $\mathcal B^+$ with respect to the hyperplane
$\{y_n=0\}$. We set $\mathcal B=\mathcal B^+\cup\mathcal
B^-\cup\Phi^{-1}\left(\partial\Omega\cap B_r(\overline{x})\right)$.


We set $\Psi=\Phi^{-1}$ and we use the change of variable:
$ x=\Phi(y)$, $y=\Psi(x)$, so that we have
  \[
  w(y)=u(\Phi(y))\quad \text{in}\quad \mathcal{B}^{+}\,.
\]
Denoting by $J\Phi(y)$ the Jacobian matrix of $\Phi$,
equation \eqref{equazione debole} becomes
\begin{equation}\label{turu}
\begin{aligned}
&\int_{\mathcal{B}^{+}}
|\nabla_x u(\Phi(y))|^{p-2}\langle\nabla_x u(\Phi(y)),\nabla_x\varphi(\Phi(y))
\rangle |\det  J\Phi(y)|dy \\
 &+\int_{\mathcal{B}^{+}}H(\Phi(y),\nabla_x u(\Phi(y)))\psi(y)|\det  J\Phi(y)|dy \\
 &= \int_{\mathcal{B}^{+}}g(y,w(y))\psi(y)|\det  J\Phi(y)|dy
\end{aligned}
\end{equation}
where $\det J\Phi(y)$ denotes the determinant of $J\Phi(y)$
and we have set
$$
g(y,w)=f(\Phi(y),w),\quad \psi(y)=\varphi(\Phi(y))\,.
$$
Hence we have
$$
\nabla_y w(y)= J\Phi(y)^T\nabla_x u(\Phi(y))\,,\quad
\nabla_y \psi(y)= J\Phi(y)^T\nabla_x \varphi(\Phi(y))
$$
and \eqref{turu} gives
\begin{equation}\label{equazione debole dominio sopra}
\begin{split}
&\int_{\mathcal{B}^{+}}
\big|[J\Phi(y)^T]^{-1}\nabla w(y)\big|^{p-2}
\big\langle\big([J\Phi(y)^T]^{-1}\big)^T
[J\Phi(y)^T]^{-1}\nabla w(y),
\nabla\psi(y)\big\rangle\\
&\times |\det J\Phi(y)|dy
 +\int_{\mathcal{B}^{+}}L(y,\nabla w(y))\psi(y)|\det  J\Phi(y)|dy\\
&=\int_{\mathcal{B}^{+}} g(y,w(y)) \psi(y)|\det J(\Phi(y))|dy\,,
\end{split}
\end{equation}
where we have set
$$
L(y,\nabla w)=H\left(\Phi(y),[J(\Phi(y))^T]^{-1}\nabla w\right).
$$

\begin{remark}\label{prop L} \rm
From the properties of $\Phi$ it follows that  $L$ enjoys
the same regularity properties of $H$ and in particular
it satisfies \eqref{prop H}.
\end{remark}

Setting
\begin{equation}\label{def A H}
A(y)=[J(\Phi(y))^T]^{-1},\quad
K(y)=A(y)^T A(y),\quad
\rho(y)=|\det J(\Phi(y))|
\end{equation}
it follows that $w(y)$ weakly satisfies
\begin{equation}\label{equazione forte dominio sopra}
-\operatorname{div}\left(\rho(y)|A(y) \nabla w(y)|^{p-2}
K(y)\nabla w(y)\right)+L(y,\nabla w(y))\rho(y)
=g(y,w(y))\rho(y)
\end{equation}
in $\mathcal{B}^{+}$.

We define the odd extension of $w(y)$
and the even extension of $\rho(y)$ and $A(y)$ (and hence of $K$) as follows:
\begin{gather}\label{def estensione 1}
\bar{w}(y)=\begin{cases}
w(y), & \text{if } y_n\geq 0,\\
-w(y_1,\dots,y_{n-1},-y_n), & \text{if } y_n<0;
\end{cases} \\
\label{def estensione 2}
\bar{\rho}(y)=\begin{cases}
\rho(y), & \text{if } y_n\geq 0,\\
\rho(y_1,\dots,y_{n-1},-y_n), & \text{if } y_n<0;
\end{cases} \\
\label{def estensione 3}
\bar{A}(y)=\begin{cases}
A(y), & \text{if } y_n\geq 0,\\
A(y_1,\dots,y_{n-1},-y_n), & \text{if } y_n<0.
\end{cases}
\end{gather}

For the function $g(y,t)$ we consider
the mixed extension (odd with respect to $t$ and even with respect to $y_n$):
\begin{equation}\label{def estensione 4}
\bar{g}(y,t)=\begin{cases}
g(y,t), & \text{if } y_n\geq 0,\,t\geq 0,\\
-g(y,-t), & \text{if } y_n\geq 0,\,t<0,\\
g(y_1,\dots,y_{n-1},-y_n,t), & \text{if } y_n<0,\,t\geq 0,\\
-g(y_1,\dots,y_{n-1},-y_n,-t), & \text{if } y_n<0,\,t<0\,.
\end{cases}
\end{equation}

For $L(y,\xi)$ we consider the extension (even with respect to $y_n$):
\begin{equation}\label{def estensione 5}
\bar{L}(y)=\begin{cases}
L(y,\xi), & \text{if } y_n\geq 0,\\
L(y_1,\dots,y_{n-1},-y_n,\xi), & \text{if } y_n<0.
\end{cases}
\end{equation}

In this way, $\bar{w}(y)$ satisfies the equation
\begin{equation}\label{equazione forte dominio esteso}
-\operatorname{div}(\bar{\rho}(y)|\bar{A}(y)\nabla \bar{w}(y)|^
{p-2}\bar{K}(y)\nabla \bar{w}(y))+L(y,\nabla\bar w(y))\bar\rho(y)
=\bar{g}(y,\bar{w})\bar\rho(y)
\end{equation}
in $\mathcal B$.


We remark that Standard $C^{1,\alpha}$ regularity results (in $\mathcal{B}$)
(see for instance \cite{Di,Li,T})
can be applied to \eqref{equazione forte dominio esteso}.

We assume that $\partial\Omega$ is smooth enough and,
without loss of generality, we can assume that $\Psi(\bar x)=0$.
Therefore
we can construct $\Phi$ in such a way that:
\begin{equation}\label{diffeo 1}
\Phi(y)=y+F(y)
\end{equation}
with $F$ such that $F(0)=\bar x$ and
$\sup_{|y|<\tau_1}|JF(y)|<\tau_2$
for suitable $\tau_1$ and $\tau_2$ small enough.
An explicit representation of $\Phi$ can be found for
instance in \cite{AzMaMoPe}.
By \eqref{diffeo 1} we have:
\begin{equation}\label{diffeo 3}
J\Phi(y)=I+JF(y),
\end{equation}
where $I$ is the identity matrix and $JF$ is the Jacobian matrix
of $F$.

By classical results of linear algebra and the regularity
properties of $\Phi$ it follows
that, there exist $\delta=\delta(\tau_1)$ and $c_1>0,c_2>0,c_3>0,c_4>0$
such that:
\begin{gather}\label{stima A}
c_1|v|\leq |A(y)v|\leq c_2|v|\quad\forall v\in\mathbb{R}^n\,,
\forall y\in \mathcal B_\delta(0), \\
\label{stima K}
c_3|v|\leq |K(y)v|\leq c_4|v|\quad\forall v\in\mathbb{R}^n\,,\forall
 y\in \mathcal B_\delta(0)\,.
\end{gather}

\section{Main results}

We first compute a linearized version of \eqref{equazione forte dominio esteso}.
To simplify notation, we will omit the bar over the functions defined in
\eqref{def estensione 1},
\eqref{def estensione 2},\eqref{def estensione 3}, \eqref{def estensione 4},
 \eqref{def estensione 5}.
For a function $v=v(y)$
in the sequel $v_j$ is the derivative with respect to $y_j$
and for the functions $g=g(y,t)$ and $L(y,\xi)$
we denote by $g_j$ and $L_j$ the derivative with respect to $y_j$
and by $g'(y,t)$ and $L_{\xi}$ the derivative of $g$ with respect to $t$
and the gradient of $L$ with respect to $\xi$.

We briefly recall definition and basic properties about some weighted
Sobolev spaces useful to
write a linearized equation associated
to equation \eqref{equazione forte dominio esteso}
(for more details see for instance \cite{MeSe,MuSt,Tr}).

Let $U\subset\mathbb{R}^n$ be a bounded smooth domain.
For $\mu\in L^1(U)$ the weighted Sobolev space $H^1_\mu(U)$
(with respect to the weight $\mu$) is
defined as
the completion of $C^{\infty}(\overline U)$
with respect to the norm
\begin{equation}\label{norma sobolev peso}
 \| v \| =\Big(\int_U |v|^2\Big)^{1/2}+
\Big(\int_U |\nabla v|^2\mu\Big)^{1/2},
\end{equation}
where $\nabla v$ is the distributional derivative.
The space $H^1_{0,\mu}(U)$
is defined as the closure of $C^\infty_c(U)$ in $H^1_\mu(U)$.

We set $Z=\{x\in\mathcal B:\nabla w(x)=0\}$ and we consider
$\psi\in C^\infty_c(\mathcal B\setminus Z)$.
For any $j=1,\dots,n$, we use $\psi_j$ as test function
in the weak formulation of \eqref{equazione forte dominio esteso}
and, since $w\in C^2(\mathcal B\backslash Z)$, we can integrate by parts obtaining:
\begin{equation}\label{linearizzato}
\begin{aligned}
 & \int_{\mathcal{B}} \rho_j(y)|A(y)\nabla w(y)|^{p-2}\langle K(y)\nabla w(y),
\nabla\psi(y)\rangle dy\\
 &+  (p-2)\int_{\mathcal{B}}\rho(y)|A(y)\nabla w(y)|^{p-4}
  \langle A_j(y)^T A(y)\nabla w(y),\nabla w(y)\rangle \\
&\times   \langle K(y)\nabla w(y),\nabla\psi(y)\rangle dy\\
&+  (p-2)\int_{\mathcal{B}}\rho(y)|A(y)\nabla w(y)|^{p-4}
  \langle K(y)\nabla w(y),\nabla w_j(y)\rangle \\
&\times   \langle K(y)\nabla w(y),\nabla\psi(y)\rangle dy\\
&+\int_{\mathcal{B}} \rho(y)|A(y)\nabla w(y)|^{p-2}
  \langle K_j(y)\nabla w(y),\nabla\psi(y)\rangle dy\\
&+\int_{\mathcal{B}} \rho(y)|A(y)\nabla w(y)|^{p-2}
  \langle K(y)\nabla w_j(y),\nabla\psi(y)\rangle dy\\
&+ \int_{\mathcal{B}} \rho(y)L_j(y,\nabla w(y))\psi(y) dy
 +\int_{\mathcal{B}} \rho(y)
 \langle L_{\xi}(y,\nabla w(y)),\nabla w_j (y)\rangle\psi(y) dy\\
&= \int_{\mathcal{B}} \left[g_j(y,w(y))\rho(y)+g'(y,w(y))w_j(y)\rho(y)
 +g(y,w(y))\rho_j(y)\right]\psi dy\,.
\end{aligned}
\end{equation}
By a density argument \eqref{linearizzato} holds for any
$\psi \in H^1_\mu(\mathcal B) \cap L^\infty(\mathcal B)$
with compact support in $\mathcal B\setminus Z$.
The main tool to achieve our estimates is the following result.

\begin{proposition}[Hessian estimate]\label{stima hessiano locale}
For $p\in (1,\infty)$ fixed we consider a weak solution
$w\in W^{1,\infty}_{\rm loc}(\mathcal B)$ of
\eqref{equazione forte dominio esteso}.
For $y_0\in \mathcal B$, let $r>0$ be such that
 $B_{2r}(y_0)\subset\mathcal B$.
It holds
\begin{equation}\label{eq stima hessiano locale}
\int_{B_r(y_0)}|\nabla w|^{p-2}|D^2 w|^2 dy \leq C\,,
\end{equation}
where $C= C(y_0,r,p,n,\|w\|_{W^{1,\infty}},g,L)$.
\end{proposition}

\begin{proof}
Let $G_\alpha:\mathbb{R}\to\mathbb{R}$ be defined as
$$
G_\alpha(s)=\begin{cases}
s & \text{if } |s| \geq 2 \alpha, \\
2[s- \alpha \frac{s}{|s|}]  & \text{if }  \alpha< |s|< 2\alpha, \\
0 & \text{if } |s|\leq \alpha,
\end{cases}
$$
and let $\varphi$ be a cut-off
function such that
\begin{equation}\label{psi}
\varphi\in C^\infty_c(B_{2r}(y_0)),\; \varphi\equiv 1\text{ in }
 B_r(y_0)\quad \text{and}\quad |D \varphi|\leq\frac{2}{r},
\end{equation}
with $2r <\operatorname{dist}(y_0,\partial \Omega)$.
We set
\begin{equation}\label{test}
\psi (y)=G_\varepsilon(w_j(y)) \varphi^2(y).
\end{equation}
In the sequel we omit the dependence on $y$.
We put $\psi$ as test function in \eqref{linearizzato},
and we obtain
\begin{equation}\label{linearizzato con psi}
\begin{aligned}
 &\int_{\mathcal{B}}\rho_j|A\nabla w|^{p-2}2\varphi G_\varepsilon(w_j)\langle K\nabla w,\nabla\varphi\rangle\,dy\\
 &+\int_{\mathcal{B}}\rho_j|A\nabla w|^{p-2}\varphi^2G'_\varepsilon(w_j)\langle K\nabla w,\nabla w_j\rangle\,dy\\
 &+ (p-2)\int_{\mathcal{B}}\rho|A\nabla w|^{p-4}2\varphi G_\varepsilon(w_j)
 \langle A_j^T A\nabla w,\nabla w\rangle\cdot\langle K\nabla w,\nabla\varphi\rangle\,dy\\
 &+ (p-2)\int_{\mathcal{B}}\rho|A\nabla w|^{p-4}\varphi^2 G'_\varepsilon(w_j)\langle A_j^T A\nabla w,
 \nabla w\rangle\cdot\langle K\nabla w,\nabla w_j\rangle\,dy\\
 &+ (p-2)\int_{\mathcal{B}}\rho|A\nabla w|^{p-4}2\varphi G_\varepsilon(w_j)\langle K\nabla w,
 \nabla w_j\rangle\cdot\langle K\nabla w,\nabla\varphi\rangle\,dy\\
 &+ (p-2)\int_{\mathcal{B}}\rho|A\nabla w|^{p-4}\varphi^2 G'_\varepsilon(w_j)\langle K\nabla w,\nabla w_j\rangle^2\,dy\\
 &+ \int_{\mathcal{B}}\rho|A\nabla w|^{p-2}2\varphi G_\varepsilon(w_j)\langle K_j\nabla w,\nabla\varphi\rangle\,dy\\
 &+ \int_{\mathcal{B}}\rho|A\nabla w|^{p-2}\varphi^2 G'_\varepsilon(w_j)\langle K_j\nabla w,\nabla w_j\rangle\,dy\\
 &+ \int_{\mathcal{B}}\rho|A\nabla w|^{p-2}2\varphi G_\varepsilon(w_j)\langle K\nabla w_j,\nabla\varphi\rangle\,dy\\
 &+ \int_{\mathcal{B}}\rho|A\nabla w|^{p-2}\varphi^2 G'_\varepsilon(w_j)\langle K\nabla w_j,\nabla w_j\rangle\,dy\\
 &+ \int_{\mathcal{B}} \rho\left[ L_j(y,D\nabla w)+\langle L_{\xi}(y,\nabla  w),
 \nabla w_j\rangle \right]\  G_\varepsilon(w_j)\varphi^2 dy\\
 &= \int_{\mathcal{B}}\left[g_j\rho+g'w_j\rho+g\rho_j\right]\varphi^2 G_\varepsilon(w_j)\,dy\,.
\end{aligned}
\end{equation}
In the sequel $c$ and $C$ will be positive constants
(possibly depending on $r$, $y_0$, $\|w\|_{W^{1,\infty}(B_{2r}(\bar x))}$)
whose value can varies from line to line.

We set:
\begin{gather}\label{I1}
\begin{aligned}
I_1&=\int_{\mathcal{B}}\rho_j|A\nabla w|^{p-2}2\varphi G_\varepsilon(w_j)\langle K\nabla w,\nabla\varphi\rangle\,dy\\
 &\quad+ (p-2)\int_{\mathcal{B}}\rho|A\nabla w|^{p-4}2\varphi G_\varepsilon(w_j)\langle A_j^T A\nabla w,
 \nabla w\rangle\cdot\langle K\nabla w,\nabla\varphi\rangle\,dy\\
 &\quad + \int_{\mathcal{B}}\rho|A\nabla w|^{p-2}2\varphi G_\varepsilon(w_j)\langle K_j\nabla w,\nabla\varphi\rangle\,dy\,;
\end{aligned}\\
\label{I2}
\begin{aligned}
I_2 &=(p-2)\int_{\mathcal{B}}\rho|A\nabla w|^{p-4}2\varphi G_\varepsilon(w_j)\langle K\nabla w,\nabla w_j\rangle
 \langle K\nabla w,\nabla\varphi\rangle\,dy\\
 &\quad + \int_{\mathcal{B}}\rho|A\nabla w|^{p-2}2\varphi G_\varepsilon(w_j)\langle K\nabla w_j,\nabla\varphi\rangle\,dy\,;
\end{aligned} \\
\label{I3}
\begin{aligned}
I_3 &=\int_{\mathcal{B}}\rho_j|A\nabla w|^{p-2}\varphi^2G'_\varepsilon(w_j)\langle K\nabla w,\nabla w_j\rangle\,dy\\
 &\quad+ (p-2)\int_{\mathcal{B}}\rho|A\nabla w|^{p-4}\varphi^2 G'_\varepsilon(w_j)
 \langle A_j^T A\nabla w,\nabla w\rangle \langle K\nabla w,\nabla w_j\rangle\,dy\\
 &\quad + \int_{\mathcal{B}}\rho|A\nabla w|^{p-2}\varphi^2 G'_\varepsilon(w_j)\langle K_j\nabla w,\nabla w_j\rangle\,dy\,;
\end{aligned}\\
\label{I4}
I_4=(p-2)\int_{\mathcal{B}}\rho|A\nabla w|^{p-4}\varphi^2 G'_\varepsilon(w_j)\langle K\nabla w,\nabla w_j\rangle^2\,dy\,; \\
\label{I5}
I_5=\int_{\mathcal{B}}\rho|A\nabla w|^{p-2}\varphi^2 G'_\varepsilon(w_j)\langle K\nabla w_j,\nabla w_j\rangle\,dy\,; \\
\label{I6}
I_6=\int_{\mathcal{B}}\rho L_j(y,D\nabla w) G_\varepsilon(w_j)\varphi^2 dy\,; \\
\label{I7}
I_7=\int_{\mathcal{B}}\rho \langle L_{\xi} (y,\nabla  w),\nabla w_j\rangle G_\varepsilon(w_j)\varphi^2 dy\,;\\
\label{I8}
I_8=\int_{\mathcal{B}}\left[g_j\rho+g'w_j\rho+g\rho_j\right] G _\varepsilon(w_j)\varphi^2\,dy\,.
\end{gather}
If $p\geq 2$, then $I_4$ is positive and hence
\begin{equation}\label{I6I7 a}
I_4+I_5\geq I_5\,.
\end{equation}

If $p<2$, then by the definition of $K$ we have
\begin{equation}\label{I6I7 b}
|A\nabla w|^{p-4}\langle K\nabla w,\nabla w_j\rangle^2=|A\nabla w|^{p-4}
\langle A\nabla w,A\nabla w_j\rangle^2\leq |A\nabla w|^{p-2}|A\nabla w_j|^2\,,
\end{equation}
which implies
\begin{equation}\label{I6I7 c}
(p-2)|A\nabla w|^{p-4}\langle K\nabla w,\nabla w_j\rangle^2\geq (p-2)|A\nabla w|^{p-2}|A\nabla w_j|^2
\end{equation}
and hence
\begin{equation}\label{I6I7 d}
I_4+I_5\geq (p-1)I_5\,.
\end{equation}
By \eqref{I6I7 a} and \eqref{I6I7 d} we have that, for every $p>1$, it holds
\begin{equation}\label{I6I7 e}
I_4+I_5\geq \min\{1,p-1\}I_5\,.
\end{equation}

By \eqref{linearizzato con psi} and \eqref{I6I7 e} we infer
\begin{equation}\label{diseq linearizzato 1}
\min\{1,p-1\}I_5\leq I_4+I_5\leq \sum_{i=1}^7|I_i|+I_8\,.
\end{equation}
By the properties of $\Phi$ there exist $c,M>0$
such that
\begin{equation}\label{stima rho}
c\leq\rho(y)\leq M, \quad
|\rho_j(y)|\leq M
\quad\forall y\in\mathcal B,\; \forall j=1,\ldots,n.
\end{equation}

By \eqref{stima A}, \eqref{stima K}, \eqref{diseq linearizzato 1} and
\eqref{stima rho},
estimating the terms in the righthand side of \eqref{diseq linearizzato 1},
we obtain
\begin{equation}\label{diseq linearizzato 2}
\begin{aligned}
& \int_{\mathcal{B}} |\nabla w|^{p-2}|\nabla w_j|^2 G'_\varepsilon(w_j)H_{\delta,z}\varphi^2 dy\\
 &\leq c\int_{\mathcal{B}} |G_\varepsilon(w_j)||\nabla w|^{p-1}|\nabla\varphi|\varphi dy\\
 &\quad + c\int_{\mathcal{B}} |G_\varepsilon(w_j)||\nabla w|^{p-2}|\nabla w_j||\nabla\varphi|\varphi dy\\
 &\quad + c\int_{\mathcal{B}} |G'_\varepsilon(w_j)||\nabla w|^{p-1}|\nabla w_j|\varphi^2 dy\\
 &\quad + c\int_{\mathcal{B}} |L_\xi(y,\nabla w)||\nabla w_j||G_\varepsilon(w_j)|\varphi^2 dy\\
 &\quad + c\int_{\mathcal{B}} [(|L_j|+|g_j|+|g'||w_j|)\rho+|g\rho_j|]|G_\varepsilon(w_j)|\varphi^2 dy\,.
\end{aligned}
\end{equation}
We set
\begin{gather*}
 J_1=\int_{\mathcal{B}} |G_\varepsilon(w_j)||\nabla w|^{p-1}|\nabla\varphi|\varphi dy\\
 J_2=\int_{\mathcal{B}} |G_\varepsilon(w_j)||\nabla w|^{p-2}|\nabla w_j||\nabla\varphi|\varphi dy\\
 J_3=\int_{\mathcal{B}} |G'_\varepsilon(w_j)||\nabla w|^{p-1}|\nabla w_j|\varphi^2 dy
  = \int_{\mathcal B\cap \{w_j>\varepsilon\}} |G'_\varepsilon(w_j)||\nabla w|^{p-1}|\nabla w_j|\varphi^2 dy\\
 J_4 = \int_{\mathcal{B}} |L_\xi(y,\nabla w)||\nabla w_j||G_\varepsilon(w_j)|\varphi^2 dy\\
 J_5 = \int_{\mathcal{B}} [(|L_j|+|g_j|+|g'||w_j|)\rho+|g\rho_j|]|G_\varepsilon(w_j)|\varphi^2 dy\,.
\end{gather*}
From definition of $G_\varepsilon$ it follows that
\begin{gather}\label{stima Te1}
|G_\varepsilon(w_j)|\leq 2|w_j|, \\
\label{stima Te2}
|G'_\varepsilon(w_j)|\leq C.
\end{gather}
Recalling that $g$ and $L$ are locally Lipshitz continuous,
by properties of $\rho$ and the regularity of $w$ we have:
\begin{equation}\label{stima J6}
J_5\leq C\,.
\end{equation}
We recall that for $a,b\in\mathbb{R}$ and $\theta>0$ there holds the Young inequality
\begin{equation}\label{young}
ab\leq\theta a^2+\frac{1}{4\theta}b^2\,.
\end{equation}
Using \eqref{young}, we obtain
\begin{equation}\label{stima J4}
J_3\leq c\int_{\mathcal{B}} |\nabla w|^{p-2}|\nabla w_j|\varphi^2\chi_{\{w_j>\varepsilon\}}
\leq\theta\int_{\mathcal{B}} |\nabla w_|^{p-2}|\nabla w_j|^2\varphi^2\chi_{\{w_j>\varepsilon\}}dy+C\,.
\end{equation}
Recalling that $|\nabla\varphi|\leq\frac{2}{\rho}$, we also have
\begin{gather}\label{stima J1}
J_1\leq C, \\
\label{stima J3}
J_2\leq\theta\int_{\mathcal{B}} |\nabla w|^{p-2}|\nabla w_j|^2\varphi^2dy+C\,.
\end{gather}
Recalling Remark \ref{prop L} and using \eqref{young}, we obtain
\begin{equation}\label{stima J4b}
J_4\leq c\int_{\mathcal{B}}|\nabla w|^{\frac{p-2}{2}}|\nabla w_j||G_\varepsilon(w_j)|\varphi^2 dy\leq
\theta\int_{\mathcal{B}} |\nabla w|^{p-2}|\nabla w_j|^2 G_\varepsilon(w_j)\varphi^2 dy+C\,.
\end{equation}
After setting $\vartheta=c\theta$,
we choose $\theta$ such that $\vartheta<1$.
By the above estimates we obtain
\begin{equation}\label{stima hess 1}
\int_{\mathcal B\cap \{w_j>\varepsilon\}} |\nabla w|^{p-2}|\nabla w_j|^2
\left(
G'_\varepsilon(w_j)-\vartheta\right)\varphi^2 dy\leq c\,.
\end{equation}

From the definition of $G_\varepsilon$ it follows that for all $s>0$
$G'_\varepsilon(s)$ converges to $1$
as $\varepsilon$ goes to $0$. Therefore by Fatou's Lemma we have
\begin{equation}\label{stima hess 2}
\int_{\mathcal B\setminus\{w_j=0\}} |\nabla w|^{p-2}|\nabla w_j|^2 \varphi^2 dy\leq c
\end{equation}
and hence
\begin{equation}\label{stima hess 3}
\int_{\mathcal B\setminus Z} |\nabla w|^{p-2}|\nabla w_j|^2\varphi^2 dy\leq c\,,
\end{equation}
where $c$ depends on $y_0,r, n,p, g,\Phi,\|w\|_{W^{1,\infty}(B_{2r}(\bar x))}$.
From the properties of $\varphi$ we infer
$$
\int_{B_r(y_0)\setminus Z}|\nabla w|^{p-2}|D^2 w |^2 dy\leq c\,.
$$
\end{proof}

We now prove Theorem \ref{teorema principale}.
Since
$$
\int_{B_r(y_0)}|\nabla w|^{p-2} | D^2 w |^2dy \leq C,
$$
by the properties of $\Phi$
the same estimate holds for $u$ in $\Phi(\mathcal B)$.
Hence, recalling that $\overline\Omega$ is compact,
we get that there exists $C>0$ such that
\begin{equation}\label{stima principale derivate seconde u}
\int_\Omega |\nabla u|^{p-2}|D^2 u|^2 dx\leq C\,.
\end{equation}
If $p\leq 2$, it immediately follows that
$$
\int_\Omega |D^2 u|^2 dx\leq\int_\Omega |\nabla u|^{p-2}|D^2 u|^2 dx
$$
and hence by \eqref{stima principale derivate seconde u}
the thesis (i) of Theorem \ref{teorema principale} follows.

If $p>2$, since for suitable $c>0$ it holds
$$
|\nabla \left(|\nabla u|^{p-1}\right)|\leq c|\nabla u|^{p-2}|D^2 u|,
$$
recalling that $\nabla u$ is bounded, there exists $C>0$:
$$
|\nabla \left(|\nabla u|^{p-1}\right)|^2\leq
c^2|\nabla u|^{2(p-2)}|D^2 u|^2\leq C|\nabla u|^{p-2}|D^2 u|^2
$$
and hence statment (ii) of Theorem \ref{teorema principale} follows by
\eqref{stima principale derivate seconde u},
provided that, arguing as in \cite{RiSc},
it can be shown that, for every $i\in\{1,\dots,n\}$,
the $i$-th generalized derivative of $|\nabla u|^{p-1}$
coincides with the classical one.

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\end{document}