\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2016 (2016), No. 142, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2016 Texas State University.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2016/142\hfil Lorentz estimates]
{Lorentz estimates for asymptotically regular elliptic
equations in quasiconvex domains}

\author[J. Zhang, S. Zheng \hfil EJDE-2016/142\hfilneg]
{Junjie Zhang, Shenzhou Zheng}

\address{Junjie Zhang \newline
Department of Mathematics,
Beijing Jiaotong University,
Beijing 100044, China}
\email{junjiezhang@bjtu.edu.cn}

\address{Shenzhou Zheng (corresponding author)\newline
Department of Mathematics,
Beijing Jiaotong University,
Beijing 100044, China}
\email{shzhzheng@bjtu.edu.cn}

\thanks{Submitted October 2, 2015. Published June 14, 2016.}
\subjclass[2010]{35J60, 35B65, 35D30}
\keywords{Lorentz estimate; Poisson kernel; Lorentz space; regularity}

\begin{abstract}
 We derive a global Lorentz estimate of the gradient of weak solutions
 to nonlinear elliptic problems with asymptotically regular nonlinearity
 in quasiconvex domains. Here, we prove its Lorentz estimate for such
 an asymptotically regular elliptic problem by constructing a regular
 problem via Poisson's formula, and quasiconvex domain locally approximated
 by convex domain.
\end{abstract}

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

\section{Introduction}

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with $n\geq 2$,
and $1<p<\infty$ be a fixed real number. The main purpose of this paper
is to attain a global estimate of the gradient of weak solutions in Lorentz
spaces for the following zero Dirichlet problem of nonlinear elliptic equations:
\begin{equation}\label{Dirichlet problem}
\begin{gathered}
\operatorname{div} \mathbf{a}(x,Du)
=\operatorname{div} (|\mathbf{f}|^{p-2}\mathbf{f}),
 \quad \text {in } \Omega,\\
u=0, \quad \text {on } \partial \Omega,
\end{gathered}
\end{equation}
where the vector-valued function $\mathbf{a}(x,Du):
\mathbb{R}^n\times \mathbb{R}^n \to \mathbb{R}^n$
is asymptotically regular (for details to
see Definition \ref{asymptotically regular}), and $\mathbf{f}$ is
any given vector-valued function in Lorentz spaces
$L^{\gamma,q}(\Omega,\mathbb{R}^n)$ with
$1<p\le \gamma<\infty$ and $0<q\le \infty$.
A weak solution of the Dirichlet problem \eqref{Dirichlet problem}
is understood in the distributional sense, if $u\in W^{1,p}_{0}(\Omega)$
satisfies
\begin{equation*}
 \int_{\Omega}\langle \mathbf{a}(x,Du),D\phi\rangle dx
=\int_{\Omega}\langle |\mathbf{f}|^{p-2}\mathbf{f}, D\phi\rangle dx, \quad
\text{for all } \phi\in W^{1,p}_{0}(\Omega).
\end{equation*}


Recently, there have been a lot of research activities about regular elliptic
problems, see the papers by Byun et al. \cite{ByW1,ByW2,ByW3,ByW4} and
references therein. We notice that these papers are concerned with the
Calder\'{o}n-Zygmund estimates or Orlicz estimates to elliptic and
parabolic equations defined in the domain of Reifenberg flat sense.
Lorentz spaces are a two-parameter scale of spaces which refine Lebesgue
spaces in some sense. Since the pioneering work of Talenti \cite{Tal}
based on symmetrization, there were a large of literature on the topic
 of Lorentz regularity to elliptic and parabolic PDEs. In particular,
 Mengesha-Phuc in \cite{MeP1} used a kind of geometrical approach to prove
the weighted Lorentz regularity of the gradient for quasilinear elliptic
$p$-Laplacian equations, and Zhang-Zhou \cite{ZhZ} extended their results
to the setting of quasilinear $p(x)$-Laplacian. Meanwhile, Baroni in
\cite{Bar1,Bar2} made use of another approach, which is called Large-M-inequality
principle introduced by Acerbi-Mingione in \cite{AcM1}, to prove the
Lorentz estimates of gradient for evolutionary $p$-Laplacian systems
and obstacle parabolic $p$-Laplacian, respectively.

The objective of this paper is mainly devoted to considering Lorentz
regularity of the gradient to the Dirchlet problems \eqref{Dirichlet problem}
by focusing on those two optimal conditions on the operator $\mathbf{a}(x,\xi)$
and $\partial\Omega$; that is to say, one is the smoothness on coefficients
and the other is the geometry of $\partial\Omega$. To the boundary geometry
of domain, the concept of Reifenberg flatness is already so general that
it includes very rough domains like Koch snowflake, see \cite{ByW2,ByW3,ByW4,DaT}
for the precise concept of Reifenberg flat domains. However, as indicated
in \cite{ByKSW,JiaLW1,JiaLW2} Reifenberg flatness excludes some geometrical
simple domains such as polygons. To this end, similar to the paper \cite{ByKSW}
 we introduce the concept of quasiconvex domain, roughly speaking, whose boundary
can be approximated from inside and outside by two convex surfaces in all scales,
rather than two hyperplanes for Reifenberg flat domains. Very recently,
there have been many interesting regularity problems to elliptic and
parabolic PDEs defined over a quasiconvex domain. For example, Jia-Li-Wang
 developed global regularity in Sobolev space $W^{1,p}$ and Orlicz space
$W^1_{0}L^{\psi}$ with $\psi\in\nabla_{2}\cap\triangle_{2}$ for linear
 divergence elliptic equations in \cite{JiaLW1} and \cite{JiaLW2},
respectively. Byun-Kwon-So-Wang \cite{ByKSW} extended the global
Calder\'{o}n-Zygmund estimates like
$\|Du\|_{L^{q}(\Omega)}\lesssim \|\mathbf{f}\|_{L^{q}(\Omega)}$
for all $q\in[p,\infty)$ in quasiconvex domains to the setting of
 $p$-Laplacian elliptic equations.

Another point in this paper is that $\mathbf{a}(x,\xi)$ is assumed an
asymptotically regular. Chipot and Evans \cite{ChE} first introduced the
notion of asymptotically regular in the elliptic framework, and
Raymond \cite{Ray} considered the Lipschitz regularity of solutions
to asymptotically regular problems with $p$-growth. Since then,
there is a large of literature on the topic of asymptotically regular.
 Scheven and Schmidt in \cite{ScS1,ScS2} obtained a local higher integrability
and a local partial Lipschitz continuity with a singular set of positive
measure for the gradient $Du$ to the system which exhibits a certain kind
of elliptic behavior near infinity, respectively. Furthermore, a global Lipschitz
regularity result was extended by Foss in \cite{Fos}.
Very recently, Byun-Oh-Wang \cite{ByOW} proved
 global Calder\'{o}n-Zygmund estimates for nonhomogeneous asymptotically
 regular elliptic and parabolic problems in divergence form in the Reifenberg
flat domain by covering the given asymptotically regular problems to suitable
regular problems. Later, Byun-Cho-Oh \cite{ByCO} extended the same conclusions
to the setting of nonlinear obstacle elliptic problems.
Zhang-Zheng \cite{ZhZhe} also further extended the work of Byun-Oh-Wang
\cite{ByOW} to the case of obstacle parabolic problems in the scale of Lorentz
spaces.


Our consideration is inspired by \cite{ByKSW,ByOW,MeP1} regarding the Lorentz
scales by refining Lebesgue spaces and the minimal smooth assumptions
imposed on the nonlinearity "coefficients" and the geometry of domain.
More precisely, our aim is to prove a global Lorentz estimate of the gradient
for nonlinear elliptic problem with asymptotically regular nonlinearity
in a quasiconvex domain as mentioned above. That is a natural refined
outgrowth of Byun-Oh-Wang's paper \cite{ByOW} and Byun-Kwon-So-Wang's
paper \cite{ByKSW} in the following two aspects,
Indeed, the Lebesgue space $L^{\gamma}$ is a special case of Lorentz
space $L^{\gamma,q}$ when $q=\gamma$ and the $(\delta,R)$-Reifenberg
flat domain in \cite{ByOW} is also a special case of
$(\delta,\sigma,R)$-quasiconvex domain. To attain our aim, some ideas
from the papers \cite{ByKSW,ByOW} are employed in our main proof.
For example, to get the global Lorentz estimate
we will make use of an equivalent representation of Lorentz norm, the
Hardy-Littlewood maximal functions, and the Poisson formula by constructing
a regular problem from the given irregular problem. Before stating the
main result, let us give some basic concepts and facts.

We first recall that the Lorentz space
$L^{\gamma,q}(\Omega)$ with $1\leq \gamma<\infty, 0<q<\infty$ is the
set of measurable function $g:\Omega\to \mathbb{R}$ such that
\begin{equation*}
 \|g\|^{q}_{L^{\gamma,q}(\Omega)}:=q\int^{\infty}_{0}
\Big(\mu^{\gamma}|\{\xi \in \Omega: |g(\xi)|>\mu\}|\Big)^{q/\gamma}
\frac{d\mu}{\mu} <+\infty.
\end{equation*}
While the Lorentz space $L^{\gamma,\infty}$ for
$1\leq \gamma<\infty, q=\infty$ is set to be the usual Marcinkiewicz space
$\mathcal{M}^{\gamma}(\Omega)$ with
quasinorm
\begin{equation*}
 \|g\|_{L^{\gamma,\infty}}=\|g\|_{\mathcal{M}^{\gamma}(\Omega)}
:=\underset{\mu>0}{\sup}\Big(\mu^{\gamma}|\{\xi \in \Omega:
|g(\xi)|>\mu\}|\Big)^{1/\gamma}<+\infty.
\end{equation*}
The local variant of such spaces is defined in the usual way.
Moreover, we note that by Fubini's theorem there holds
\begin{equation*}
 \|g\|^{\gamma}_{L^{\gamma}(\Omega)}
=\gamma\int^{\infty}_{0}\Big(\mu^{\gamma}|\{\xi \in \Omega:
 |g(\xi)|>\mu\}|\Big)\frac{d\mu}{\mu}=\|g\|^{\gamma}_{L^{\gamma,\gamma}(\Omega)},
\end{equation*}
so that $L^{\gamma}(\Omega)=L^{\gamma,\gamma}(\Omega)$;
cf. \cite{Bar1,Bar2,Bar3}.

Asymptotically regular $\mathbf{a}(x,\xi)$ says the case that it is getting
closer to some vector-valued function $\mathbf{b}(x,\xi)$ as
$|\xi|$ goes to infinity, where $\mathbf{b}(x,\xi)$ satisfies the following
assumptions:
\begin{itemize}
 \item[(H1)] $\mathbf{b}(x,\xi): \mathbb{R}^n\times
\mathbb{R}^n\to \mathbb{R}^n$ is measurable in $x$ and differential in
$\xi$, and satisfies the ellipticity and growth conditions:
 \begin{equation}\label{ellipticity and growth conditions}
 \begin{gathered}
 \langle\partial_{\xi}\mathbf{b}(x,\xi)\eta,\eta\rangle
\geq \lambda|\xi|^{p-2}|\eta|^{2}, \\
 |\mathbf{b}(x,\xi)|+|\xi||\partial_{\xi}\mathbf{b}(x,\xi)|
\leq \Lambda|\xi|^{p-1},
 \end{gathered}
 \end{equation}
 for almost every $x\in\Omega$ and all $\xi,\eta\in \mathbb{R}^n$, where the structural constants satisfy $0<\lambda\leq 1\leq \Lambda\leq \infty$.
 
\item[(H2)] ($(\delta,R)$-vanishing)\ $\mathbf{b}(x,\xi)$ is 
$(\delta,R)$-vanishing if we have
 \begin{equation*}
 \omega_{\mathbf{b}}(R):=\sup_{0<r\leq R}  \sup_{x_{0}\in \Omega}
-\hspace{-3.5mm}\int_{B_{r}(x_{0})\cap \Omega}\beta(\mathbf{b},B_{r}(x_{0}))(x)dx\leq\delta,
 \end{equation*}
 where
 \begin{equation*}
 \beta(\mathbf{b},B_{r}(x_{0}))(x):=\sup_{\xi\in \mathbb{R}^n}
\frac{|\mathbf{b}(x,\xi)-\overline{\mathbf{b}}_{B_{r}(x_{0})}(\xi)|}{(1+|\xi|)^{p-1}},
 \quad \overline{\mathbf{b}}_{B_{r}(x_{0})}(\xi)=-\hspace{-3.5mm}\int_{B_{r}(x_{0})
\cap \Omega}\mathbf{b}(x,\xi)dx.
 \end{equation*}
\end{itemize}


\begin{definition}[Asymptotically $\delta$-Regular] \label{asymptotically regular} \rm
Let $\mathbf{b}(x,\xi)$ satisfies the assumption (H1). 
Then we say that $\mathbf{a}(x,\xi)$ is asymptotically $\delta$-regular 
with $\mathbf{b}(x,\xi)$ if there exists a uniformly bounded nonnegative 
function $\theta:[0,\infty)\to [0,\infty]$ such that
\begin{equation*}
 \limsup_{\rho\to 0}  \theta(\rho)\leq \delta
\end{equation*}
and
\begin{equation*}
 |\mathbf{a}(x,\xi)-\mathbf{b}(x,\xi)|\leq \theta(|\xi|)(1+|\xi|^{p-1})
\end{equation*}
for almost every $x\in \Omega$ and all $\xi\in\mathbb{R}^n$.
\end{definition}



\begin{remark} \label{rmk1.2} \rm
(i) From Definition \ref{asymptotically regular}, we can easily conclude that
\begin{equation}\label{a-small-regu}
 \lim_{|\xi|\to \infty} \sup_{x\in \Omega}
\frac{|\mathbf{a}(x,\xi)-\mathbf{b}(x,\xi)|}{|\xi|^{p-1}}\leq 2\delta,
\end{equation}
namely, for any sufficiently small $\delta>0$, $\mathbf{a}(x,\xi)$ is in a 
regular range as $|\xi|$ is near infinity. Throughout the paper we always
 assume that $\mathbf{a}(x,\xi)$ is asymptotically $\delta$-regular with 
$\mathbf{b}(x,\xi)$ satisfying the assumption H1, where $\delta$ is to be 
determined later.

(ii) The above assumption \eqref{ellipticity and growth conditions} implies 
that the following monotonicity condition: 
for all $\xi,\eta\in\mathbb{R}^n$ and
for almost every $x\in\mathbb{R}^n$,
\begin{equation*}
\langle \mathbf{b}(x,\xi)-\mathbf{b}(x,\eta),\xi-\eta\rangle
\geq
\begin{cases}
\nu(n,p,\lambda)(|\xi|+|\eta|)^{p-2}|\xi-\eta|^{2}, & \text {if $1<p<2$},\\
\nu(n,p,\lambda)|\xi-\eta|^{p}, & \text {if $p\geq 2$}.
\end{cases}
\end{equation*}

(iii) By Browder-Minty Theorem, it is well known that under the basic 
assumption H1, the problem \eqref{Dirichlet problem} has a unique weak 
solution provided $\mathbf{f}\in L^{p}(\Omega,\mathbb{R}^n)$ and
$|\Omega|<\infty$, with the estimate
\begin{equation}\label{classical Lp estimate}
 \|Du\|_{L^{p}(\Omega)}\leq C(\lambda, p)\|\mathbf{f}\|_{L^{p}(\Omega)}.
\end{equation}

(iv) The assumption that $\mathbf{b}(x,\xi)$ is $(\delta,R)$-vanishing refines 
the assumption that $\mathbf{b}(x,\xi)$ is $VMO_{x}$, that is to say the 
nonlinearity $\mathbf{b}(x,\xi)$ has small BMO semi-norm uniformly with 
respect to the independent variables.
\end{remark}


Next we introduce the definition of quasiconvex domain, see
\cite[Definition 1.3]{ByKSW}.

\begin{definition}\label{quasiconvex domain} \rm
A bounded domain $\Omega$ is said to be $(\delta,\sigma,R)$-quasiconvex if 
for all $x\in \partial\Omega, 0<r\leq R$, the following properties hold:
\begin{itemize}
\item[(i)] there exists a ball $B_{\sigma r}(x_{0}) \subset \Omega_{r}(x)$, 
where $x_{0}$ is relative to $x$ and $\sigma\in (0,\frac{1}{4})$ is a 
uniform constant;

\item[(ii)] there exist a hyperplane $A(x,r)$ containing $x$, a unit normal 
vector $\mathbf{n}(x,r)$ to $A(x,r)$, and a half space 
$H(x,r)=\{y+t\mathbf{n}(x,r):y\in L(x,r), t\geq -\delta r\}$ such that
\begin{equation*}
 \Omega_{r}(x)\subset H(x,r)\cap B_{r}(x).
\end{equation*}
\end{itemize}
\end{definition}
We would like to remark two points. The constant $\delta$ here is to be 
chosen in the range $(0,\frac{1}{2^{n+1}})$. By scaling
the problem \eqref{Dirichlet problem}, we can take $R=1$ or any number 
bigger than 1, while $\delta$ is invariant under such scaling, see 
\cite[Lemma 2.6]{ByKSW}.

Let us summarize our main result as follows.

\begin{theorem}\label{main result}
 Assume $1<p\le \gamma<\infty, 0<q\leq \infty$ and $0<\sigma <\frac{1}{4}$. 
Let $u\in W^{1,p}_{0}(\Omega)$ be the solution to Dirichlet problem
 \eqref{Dirichlet problem} with the vector-valued function $\mathbf{a}(x,\xi)$
and $\mathbf{f}\in L^{\gamma,q}(\Omega)$. Then there exists a small 
$\delta=\delta(\sigma,n,p,\gamma,q,\lambda,\Lambda)>0$ such that if
 $\mathbf{a}(x,\xi)$ is asymptotically $\delta$-regular with 
$\mathbf{b}(x,\xi)$ satisfying the assumptions H1 and H2, and $\Omega$ 
is $(\delta,\sigma,R)$-quasiconvex, then $Du\in L^{\gamma,q}(\Omega)$ 
with the estimate
\begin{equation}\label{main Lorentz estimate}
 \|Du\|_{L^{\gamma,q}(\Omega)}\leq C\|F\|_{L^{\gamma,q}(\Omega)},
\end{equation}
for some positive constant $C=C(n,\lambda,\Lambda,p,\gamma,q,\theta)$ 
(except in the case $q=\infty$, where it depends only on 
$n,\lambda,\Lambda,p,\gamma,\theta$).
\end{theorem}

The rest of this article is organized as follows. 
In section 2, we state some properties of quasiconvex domains, Lorentz 
spaces and Hardy-Littlewood maximal function.
Section 3 is devoted to proving Theorem \ref{main result}. 
On the basis of global Lorentz regularity for a regular problem, we prove 
our main result by taking a transformation from given asymptotically 
regular problem to a suitable regular problem.

\section{Preliminaries}

We begin this section by introducing some properties of quasiconvex domains. Set
\begin{equation*}
 \Omega_{r}(x)=\Omega\cap B_{r}(x), \quad 
\partial_{\omega}\Omega(x)=\partial\Omega\cap B_{r}(x),
\end{equation*}
and by
$$
D(E,F)=\max\{\sup_{x\in E}dist(x,F),\sup_{y\in F}dist(y,E)\}
$$
we denote the Hausdorff distance between two sets $E$ and $F$ in $\mathbb{R}^n$.
It is clear that the quasiconvex domains are $W^{1,p}$ extension domains 
(see \cite{HajM}) in which the extension theorem and Sobolev embedding theorem 
are available. The property $(ii)$ in Definition \ref{quasiconvex domain} 
implies that quasiconvex domains are locally approximated by convex domains 
in the following sense, see \cite[Lemma 3.3]{ByKSW}.

\begin{lemma}\label{approximately quasiconvex domain}
If $\Omega$ is a $(\delta,\sigma,R)$-quasiconvex domain, then for each 
$x\in \partial\Omega$ and for every $r\in (0,\frac{R}{2})$, 
there exist two convex domains $F_{r}(x)$ and $F^{\ast}_{r}(x)$ such that
\begin{equation}\label{two approximately quasiconvex domain}
 F^{\ast}_{r}(x)\subset \Omega_{r}(x) \subset F_{r}(x) \quad 
D(F^{\ast}_{r}(x),F_{r}(x))\leq \frac{34\delta r}{\sigma^{3}}.
\end{equation}
\end{lemma}

It is worthwhile noting that
\begin{gather*}
F_{r}(x)=\cap_{y\in \partial_{\omega}\Omega_{r}(x)} H(y,2r)\cap B_{r}(x), \\
F^{\ast}_{r}(x)=\big\{x_{0}+\big(1-\frac{16r\delta}{\sigma^{3}}\big)(y-y_{0}):
y\in F_{r}(x)\big\},
\end{gather*}
where $H(y,2r)$ and $x_{0}\in \Omega_{r}(x)$ are given in Definition 
\ref{quasiconvex domain}.

The next lemma states some useful embedding relations in Lorentz spaces, 
see \cite{MeP1}.

\begin{lemma}\label{Lorentz spaces embedding}
Let $\Omega$ be a bounded measurable subset of $\mathbb{R}^n$.
Then the following holds:
\begin{enumerate}
\item  If $0<q_{1},q_{2}\leq \infty$ and $p <\eta <\gamma<\infty$, 
then $L^{\gamma,q_{1}}(\Omega)\subset L^{\eta,q_{2}}(\Omega)$;

\item  If $0<q_{1}<q_{2}\leq \infty$ and $p<\gamma<\infty$, then 
$L^{\gamma,q_{1}}(\Omega)\subset L^{\gamma,q_{2}}(\Omega)
 \subset L^{\gamma,\infty}(\Omega)\subset L^{\gamma-\varepsilon}(\Omega)$ 
for any $\varepsilon>0$ such that $\gamma-\varepsilon>p$.
\end{enumerate}
\end{lemma}


One of the main tools in our approach is the Hardy-Littlewood maximal function, 
which allows us to control the local behavior of a function. For a function 
$g\in L^1_{\rm loc}(\mathbb{R}^n)$, the Hardy-Littlewood maximal function
of $g$ is defined by
\begin{equation*}
 \mathcal{M}g(x)=\sup_{r>0} -\hspace{-3.5mm}\int_{B_{r}(x)}|g(y)|dy.
\end{equation*}
Further, for a function defined on a bounded domain $U\subset \mathbb{R}^n$,
we can define the Hardy-Littlewood maximal function locally by
\begin{equation*}
 \mathcal{M}_{U}g:=\mathcal{M}(g\chi_{U}),
\end{equation*}
where $\chi$ is the standard characteristic function on $U$. 
We recall two basic properties of the Hardy-LIttlewood maximal function as follows:
\begin{gather*}
 |\{x\in \mathbb{R}^n: \mathcal{M}g(x)\geq \mu\}|
\leq \frac{C(n)}{\mu}\|g\|_{L^1(\mathbb{R}^n)}, \quad \text{for } \forall t>0,\\
 \|\mathcal{M}g\|_{L^{p}(\mathbb{R}^n)}
\leq C(n,p)\|g\|_{L^{p}(\mathbb{R}^n)}, \quad \text{for }  1<p\leq \infty.
\end{gather*}
Recently, this boundedness in $L^{p}$ has been extended to Lorentz space 
by Mengesha and Phuc as follows; see \cite[Lemma 3.11]{MeP1}.

\begin{lemma}\label{bdd in Lorentz space}
For any $1<\gamma<\infty, 0<q\leq \infty$, there exists a constant 
$C=C(n,\gamma,q)$ such that
\begin{equation*}
 \|\mathcal{M}g\|_{L^{\gamma,q}(\mathbb{R}^n)}
\leq C\|g\|_{L^{\gamma,q}(\mathbb{R}^n)}
\end{equation*}
for all $g\in L^{\gamma,q}(\mathbb{R}^n)$.
\end{lemma}

We will apply the following lemma to prove our global regularity estimates. 
This modified covering lemma accommodates the special needs for the conditions 
of $(\delta,R)$-vanishing and quasiconvex domains; see
 \cite[Lemma 2.5]{ByKSW}.

\begin{lemma}\label{covering lemma}
Assume $E$ and $F$ are measurable sets, 
$E\subset F\subset \Omega$ with $\Omega(\delta,\sigma,1)$-quasiconvex, and 
that there exists an $\varepsilon>0$ such that
\begin{itemize}
\item[(i)] $|E|<\varepsilon |B_{1}|$, and 
\item[(ii)] for every $x\in B_{1}$, and all $r\in (0,1]$,
\begin{equation*}
 |E\cap B_{r}(x)|\geq \varepsilon |B_{r}(x)| \quad implies \quad B_{r}(x)\cap \Omega \subset F.
\end{equation*}
\end{itemize}
Then $|E|\leq (\frac{5}{\sigma})^n\varepsilon |F|$.
\end{lemma}

We also need the following elementary characterization of functions in 
Lorentz spaces, see \cite[Lemma 4.1]{AdP} or 
\cite[Lemma 3.12]{MeP1}.

\begin{lemma}\label{Lorentz norm}
Let $g$ be a nonnegative measurable function in a bounded domain 
$U\subset \mathbb{R}^n$. Let $\theta>0$ and $\lambda>1$ be constants. Then for any
$0<\gamma,q<\infty$, we have
\begin{equation*}
 g\in L^{\gamma,q}(U)\Leftrightarrow 
S:=\sum_{k\geq 1} \lambda^{tk}|\{x\in U: g(x)
> \theta \lambda^{k}\}|^{q/\gamma}<+\infty,
\end{equation*}
and moreover
\begin{equation}
 C^{-1}S\leq \|g\|^{t}_{L^{\gamma,q}(U)}\leq C(|U|^{q/\gamma}+S)
\end{equation}
with constant $C=C(\theta,\lambda,q)>0$. Analogously, for 
$0<\gamma<\infty$ and $q=\infty$ we have
\begin{equation}
 C^{-1}T\leq \|g\|_{L^{\gamma,\infty}(U)}\leq C(|U|^{1/\gamma}+T),
\end{equation}
where $T$ is the quantity
\begin{equation*}
 T:=\sup_{k\geq 1} \lambda^{k}|\{x\in U: |g(x)|
> \theta \lambda^{k}\}|^{1/\gamma}.
\end{equation*}
\end{lemma}


\section{Proof of the main result}

In this section, we are devoted to the proof of our main result based 
on the global estimate in Lorentz spaces for problem 
\eqref{Dirichlet problem} with $b(x,\xi)$ satisfying the assumptions 
(H1) and (H2), see Theorem \ref{Lorentz estimate} below.
To that end, we first introduce the following lemma; cf. 
\cite[Lemma 4.5]{ByKSW}.

\begin{lemma}\label{condition II}
Assume that $u\in W^{1,p}_{0}(\Omega)$ is the weak solution of
 \eqref{Dirichlet problem}. Then there is a constant 
$N_{0}=N_{0}(n,\lambda,\Lambda,p)>1$ so that for any fixed 
$\varepsilon\in (0,1)$, one can find a small constant 
$\delta=\delta(\varepsilon)>0$ such that if $\mathbf{b}$ is 
$(\delta,\frac{48}{\sigma})$-vanishing, $\Omega$ is 
$(\delta,\sigma,\frac{48}{\sigma})$-quasiconvex, and 
$B_{r}(y), 0<r\leq 1, y\in \Omega$, satisfies
\begin{equation*}
 |\{x\in \Omega: \mathcal{M}(|Du|^{p}(x))>N_{0}^{p}\} \cap B_{r}(y)| 
\geq \varepsilon|B_{r}(y)|,
\end{equation*}
then we have
\begin{equation*}
 B_{r}(y)\cap \Omega \subset \{x\in \Omega: \mathcal{M}(|Du|^{p})(x)>1\} 
\cup \{x\in \Omega: \mathcal{M}(|\mathbf{f}|^{p})(x)>\delta^{p}\}.
\end{equation*}
\end{lemma}

\begin{theorem}\label{Lorentz estimate}
 Assume $1<p\le \gamma<\infty, 0<q\leq \infty$ and $0<\sigma<1/4$. 
Let $u\in W^{1,p}_{0}(\Omega)$ be the weak solution to the 
Dirichlet problem \eqref{Dirichlet problem} with the vector-valued function 
$\mathbf{b}(x,\xi)$ and $\mathbf{f}\in L^{\gamma,q}(\Omega)$. 
Then there exists small $\delta=\delta(\sigma,n,p,\gamma,q,\lambda,\Lambda)>0$ 
such that if $\mathbf{b}(x,\xi)$ satisfies the assumptions (H1) and (H2), and
$\Omega$ is $(\delta,\sigma,R)$-quasiconvex, then $Du\in L^{\gamma,q}(\Omega)$
 with the estimate
\begin{equation}\label{Lorentz estimate inequality}
 \|Du\|_{L^{\gamma,q}(\Omega)} \leq C\|f\|_{L^{\gamma,q}(\Omega)},
\end{equation}
for some positive constant $C=C(n,\lambda,\Lambda,p,\gamma,q,\theta)$ 
(except in the case $q=\infty$, where it depends only on 
$n,\lambda,\Lambda,p,\gamma,\theta$).
\end{theorem}

\begin{proof} 
Let $\varepsilon>0$ be given, and we take $\delta>0$ and $N_{0}>1$ 
as in Lemma \ref{condition II}. To that end, it suffices to show that for
$\eta=\frac{p+\gamma}{2}$ there holds
\begin{equation}\label{F small}
 \|\mathbf{f}\|_{L^{\eta}(\Omega)}\leq \delta \quad \Rightarrow \quad 
\|Du\|_{L^{\gamma,q}(\Omega)}\leq C.
\end{equation}
In fact, by considering \eqref{F small} and the normalization with
\begin{equation*}
 \tilde{u}=\frac{\delta u}{\|\mathbf{f}\|_{L^{\gamma,q}(\Omega)}+\mu} \quad 
\text{and} \quad 
\tilde{\mathbf{f}}=\frac{\delta \mathbf{f}}{\|\mathbf{f}\|_{L^{\gamma,q}(\Omega)}+\mu}, \quad \mu>0,
\end{equation*}
we derive, after letting $\mu\to 0^{+}$, the desired result. Since 
$p\le \eta\le\gamma$, by Lemma \ref{Lorentz spaces embedding}
we see that the assumption $\|f\|_{L^{\eta}}\leq \delta$ is well defined. 
Therefore, under this assumption we set
\begin{gather*}
 E=\{x\in \Omega: \mathcal{M}(|Du|^{p}(x))>N_{0}^{p}\}, \\
 F=\{x\in \Omega: \mathcal{M}(|Du|^{p})(x)>1\} \cup 
\{x\in \Omega: \mathcal{M}(|\mathbf{f}|^{p})(x)>\delta^{p}\}.
\end{gather*}
Then, using the weak (1-1) estimate of Hardy-Littlewood maximal function,
 $L_{p}$-estimate \eqref{classical Lp estimate}, H\"{o}lder inequality 
and smallness of $\mathbf{f}$ in order,
we can check the first hypothesis of Lemma \ref{covering lemma} as follows:
\begin{align*}
 |E|&\leq  \frac{C}{N_{0}^{p}}\int_{\Omega}|Du|^{p}dx \\
 &\leq  \frac{C}{N_{0}^{p}}\int_{\Omega}|\mathbf{f}|^{p}dx \\
 &\leq  \frac{C}{N_{0}^{p}}\|\mathbf{f}\|^{p}_{L^{\eta}(\Omega)}
|\Omega|^{1-\frac{p}{\eta}} \\
 &\leq  C\delta^{p}|\Omega|^{1-\frac{p}{\eta}} \\
 &\leq  \varepsilon|B_{1}|,
\end{align*}
by choosing a small $\delta=\delta(\varepsilon)>0$, if necessary, 
in order to get the last inequality. Meanwhile, the second hypothesis 
of Lemma \ref{covering lemma}
follows directly from Lemma \ref{condition II}. 
Therefore, by Lemma \ref{covering lemma} we have
\begin{equation} \label{sum}
\begin{aligned}
& |\{x\in \Omega: \mathcal{M}(|Du|^{p})(x)>N_{0}^{p}\}| \\
&\leq  \varepsilon_{1}|\{x\in \Omega: \mathcal{M}(|Du|^{p})(x)>1\}|
+\varepsilon_{1}|\{x\in \Omega: \mathcal{M}(|F|^{p})(x)>\delta^{p}\}|,
\end{aligned}
\end{equation}
for $\varepsilon_{1}=(5/\sigma)^n\varepsilon$.
 Using a simple iteration argument to \eqref{sum}, for any $\tau>0$ we
further have
\begin{align*}
& |\{x\in \Omega: \mathcal{M}(|Du|^{p})(x)>N_{0}^{kp}\}|^{\tau} \\
&\leq  \varepsilon_{2}^{k}|\{x\in \Omega:
\mathcal{M}(|Du|^{p})(x)>1\}|^{\tau}+\sum^{k}_{i=1}\varepsilon_{2}^{i}
|\{x\in \Omega: \mathcal{M}(|F|^{p})(x)>\delta^{p}N_{0}^{(k-i)p}\}|^{\tau},
\end{align*}
where $\varepsilon_{2}=\max\{1,2^{\tau-1}\}\varepsilon_{1}^{\tau}$.
Then it follows that
\begin{align*}
S &:=  \sum^{\infty}_{k=1} N_{0}^{qk} |\{x\in \Omega: \mathcal{M}(|Du|^{p})(x)>N_{0}^{kp}\}|^{q/\gamma} \\
 &\leq  C\sum^{\infty}_{k=1}(N_{0}^{q}\varepsilon_{2})^{k}|\{x\in \Omega: \mathcal{M}(|Du|^{p})(x)>1\}|^{q/\gamma} \\
 &\quad +  C\sum^{\infty}_{k=1}N_{0}^{qk}
 \Big[\sum^{k}_{i=1}\varepsilon_{2}^{i}|\{x\in \Omega:
  \mathcal{M}(|\mathbf{f}|^{p})(x)>\delta^{p}N_{0}^{(k-i)p}\}|^{q/\gamma}\Big] \\
 &\leq  C\sum^{\infty}_{k=1}(N_{0}^{q}\varepsilon_{2})^{k}
 |\Omega|^{q/\gamma}\\
&\quad +C\sum^{\infty}_{i=1}(N_{0}^{q}\varepsilon_{2})^{i}
\Big[\sum^{\infty}_{k=i}N_{0}^{q(k-i)}|\{x\in \Omega:
\mathcal{M}(|\mathbf{f}|^{p})(x)>\delta^{p}N_{0}^{(k-i)p}\}|^{q/\gamma}\Big] \\
 &\leq  C\sum^{\infty}_{k=1}(N_{0}^{q}\varepsilon_{2})^{k}|\Omega|^{q/\gamma}+C\sum^{\infty}_{i=1}(N_{0}^{q}\varepsilon_{2})^{i}\|\mathcal{M}(|\mathbf{f}|^{p})(x)\|^{\frac{q}{p}}_{L^{\frac{\gamma}{p},\frac{q}{p}}(\Omega)} \\
 &\leq  C\sum^{\infty}_{k=1}(N_{0}^{q}\varepsilon_{2})^{k}
\Big(|\Omega|^{q/\gamma}+\||\mathbf{f}|^{p}\|^{\frac{q}{p}}_{L^{\frac{\gamma}{p},
\frac{q}{p}}(\Omega)}\Big).
\end{align*}
Now choosing $\varepsilon$ sufficiently small so that $N_{0}^{q}\varepsilon_{2}<1$,
we obtain
\begin{align*}
 \|Du\|^{q}_{L^{\gamma,q}(\Omega)}
&=\||Du|^{p}\|^{\frac{q}{p}}_{L^{\frac{\gamma}{p},\frac{q}{p}}(\Omega)} \\
&\leq C\|\mathcal{M}(|Du|^{p}(x)\|^{\frac{q}{p}}_{L^{\frac{\gamma}{p},
 \frac{q}{p}}(\Omega)} \\
&\leq C\Big(|\Omega|^{q/\gamma}+\|\mathbf{f}\|^{q}_{L^{\gamma,q}(\Omega)}\Big)
\leq C.
\end{align*}
This completes the proof.
\end{proof}

The main ingredient to prove Theorem \ref{main result} is to use  Poisson's 
formula to construct a regular Dirichlet problem whose nonlinearity satisfies 
the assumptions (H1) and (H2). Here, we first define a vector-valued 
function $\mathbf{c}(x,\xi):\mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}^n$ by
\begin{equation}\label{c-defi}
 |\xi|^{p-1}\mathbf{c}(x,\xi)=\mathbf{a}(x,\xi)-\mathbf{b}(x,\xi).
\end{equation}
Then, from \eqref{a-small-regu} it yields that for any sufficiently small 
$\delta >0$ there exists a positive constant $M=M(\delta)$ such that
\begin{equation}\label{c-range}
 |\xi|\geq M\Rightarrow |\mathbf{c}(x,\xi)|\leq 2\delta,
\end{equation}
uniformly in $x\in \mathbb{R}^n$. For any fixed point $x\in \mathbb{R}^n$,
 we consider the  Poisson integral
\begin{equation*}
 P[\mathbf{c}(x,\cdot)](\xi):=\int_{\partial_{B_{M}}}\mathbf{c}(x,\eta)
K(\xi,\eta)d\sigma(\eta) \quad \text{for } \xi\in B_{M},
\end{equation*}
where
\begin{equation*}
 K(\xi,\eta)=\frac{M^{2}-|\xi|^{2}}{M\omega_{n-1}|\xi-\eta|^n} \quad 
\text {for all } \xi\in B_{M}  \text{ and } \eta\in \partial B_{M}
\end{equation*}
is the Poisson kernel for the ball $B_{M}\subset \mathbb{R}^n$ with  radius $M$, 
and $\omega_{n-1}$ is the surface area of the unit sphere $\partial B_{1}$
in $\mathbb{R}^n$. Let us denote a new vector-valued function 
$\tilde{\mathbf{c}}(x,\xi)$ by
\begin{equation}\label{new-c-defi}
 \tilde{\mathbf{c}}(x,\xi)=
\begin{cases}
\mathbf{c}(x,\xi), & \text {if $|\xi|\geq M$},\\
P[\mathbf{c}(x,\cdot)](\xi), & \text {if $|\xi|< M$}.
\end{cases}
\end{equation}
Then we see that $\tilde{\mathbf{c}}(x,\xi)$ is a vector-valued 
function defined in $\mathbb{R}^n\times \mathbb{R}^n$. 
By the maximum principle and \eqref{c-range}, it follows
that for any $\xi\in \mathbb{R}^n$ there holds
\begin{equation}\label{new-c-range}
 |\tilde{\mathbf{c}}(x,\xi)|\leq 2\delta,
\end{equation}
uniformly in $x\in \mathbb{R}^n$.

Now, by combining \eqref{c-defi} with \eqref{new-c-defi} we derive
\begin{equation}  \label{a-equ-1}
\begin{aligned}
 \mathbf{a}(x,\xi)
&= \mathbf{b}(x,\xi)+|\xi|^{p-1}\mathbf{c}(x,\xi) \\
&= \mathbf{b}(x,\xi)+|\xi|^{p-1}\tilde{\mathbf{c}}(x,\xi)
 +|\xi|^{p-1}\chi_{\{|\xi|<M\}}(\mathbf{c}(x,\xi)-\tilde{\mathbf{c}}(x,\xi)),
\end{aligned}
\end{equation}
where $\chi_{\{|\xi|<M\}}$ is the characteristic function on the set
 $\{\xi\in \mathbb{R}^n: |\xi|< M\}$.


Here, we introduce a new nonlinearity $\tilde{\mathbf{a}}(x,\xi)$, which 
is regular problem transferred from the asymptotically regular
one. More precisley, for a given weak solution $u\in W^{1,p}_{0}(\Omega)$ 
of the Dirichlet problem \eqref{Dirichlet problem} we define
$\tilde{\mathbf{a}}(x,\xi):\mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}^n$
by
\begin{equation}\label{new-a-defi}
 \tilde{\mathbf{a}}(x,\xi):=\mathbf{b}(x,\xi)
+|\xi|^{p-1}\tilde{\mathbf{c}}(x,Du(x)).
\end{equation}

The following lemma play an important role in the proof of our main 
Theorem \ref{main Lorentz estimate}.

\begin{lemma}\label{new-a-regular}
 Let $u\in W^{1,p}_{0}(\Omega)$ be a weak solution of the Dirichlet problem 
\eqref{Dirichlet problem}. Assume that $\mathbf{a}(x,\xi)$ is asymptotically 
$\delta$-regular with $\mathbf{b}(x,\xi)$ satisfying the assumptions 
(H1) and (H2). Then we have the following conclusions:

(i)  If $0<\delta<\min\left\{\frac{\lambda}{4(p-1)},1\right\}$, 
then $\tilde{\mathbf{a}}(x,\xi)$ satisfies the ellipticity and growth conditions:
\begin{equation}\label{ellipticity and growth conditions-new-a}
 \begin{gathered}
 \langle \partial_{\xi}\tilde{\mathbf{a}}(x,\xi)\eta,\eta\rangle
 \geq \frac{\lambda}{2}|\xi|^{p-2}|\eta|^{2}, \\
 |\tilde{\mathbf{a}}(x,\xi)|+|\xi||\partial_{\xi}\tilde{\mathbf{a}}(x,\xi)|
 \leq \widetilde{\Lambda}|\xi|^{p-1},
 \end{gathered}
\end{equation}
for almost every $x\in\Omega$ and all $\xi,\eta\in \mathbb{R}^n$, where 
$\widetilde{\Lambda}=\Lambda+p$.

(ii) $\tilde{\mathbf{a}}(x,\xi)$ satisfies the $(5\delta,R)$-vanishing condition.
\end{lemma}

\begin{proof} (i) For any given 
$0<\delta<\min\left\{\frac{\lambda}{4(p-1)},1\right\}$, by \eqref{new-a-defi} 
and \eqref{new-c-range} it follows that
\begin{equation}\label{new-a-range}
 |\tilde{\mathbf{a}}(x,\xi)|\leq |\mathbf{b}(x,\xi)|+2|\xi|^{p-1}.
\end{equation}
Since $\mathbf{b}(x,\xi)$ and $|\xi|^{p-1}$ are differentiable in $\xi$ 
it implies
\begin{equation} \label{new-a-diff}
\begin{aligned}
 \partial_{\xi}\tilde{\mathbf{a}}(x,\xi)
&= \partial_{\xi}\mathbf{b}(x,\xi)
 +\tilde{\mathbf{c}}(x,Du(x))D_{\xi}(|\xi|^{p-1})^{T} \\
&= \partial_{\xi}\mathbf{b}(x,\xi)+\tilde{\mathbf{c}}(x,Du(x))
[(p-1)|\xi|^{p-3}\xi]^{T},
\end{aligned}
\end{equation}
and further using \eqref{new-c-range} and $\delta\leq1$, we obtain
\begin{equation}\label{new-a-diff-range}
 |\partial_{\xi}\tilde{\mathbf{a}}(x,\xi)|
\leq |\partial_{\xi}\mathbf{b}(x,\xi)|+2(p-1)|\xi|^{p-2}.
\end{equation}
Then, by \eqref{new-a-range}, \eqref{new-a-diff-range} and
\eqref{ellipticity and growth conditions} it follows that
\begin{align*}
|\tilde{\mathbf{a}}(x,\xi)|+|\xi||\partial_{\xi}\tilde{\mathbf{a}}(x,\xi)|
 &\leq |\mathbf{b}(x,\xi)|+2|\xi|^{p-1}+|\xi||\partial_{\xi}\mathbf{b}(x,\xi)|+2(p-1)|\xi|^{p-1} \\
 &\leq \Lambda|\xi|^{p-1}+2|\xi|^{p-1}+2(p-1)|\xi|^{p-1} \\
 &=  \tilde{\Lambda}|\xi|^{p-1},
\end{align*}
where $\tilde{\Lambda}=\Lambda+2p$.
On the other hand, by \eqref{new-a-diff},
\eqref{ellipticity and growth conditions} and \eqref{new-c-range} we conclude that
\begin{align*}
 \langle \partial_{\xi}\tilde{\mathbf{a}}(x,\xi)\eta,\eta\rangle
&= \langle \partial_{\xi}\mathbf{b}(x,\xi)\eta,
 \eta\rangle+(p-1)|\xi|^{p-3}\tilde{\mathbf{c}}(x,Du(x))\xi^{T}\eta\cdot \eta \\
&\geq \lambda|\xi|^{p-2}|\eta|^{2}-2\delta(p-1)|\xi|^{p-2}|\eta|^{2} \\
&= (\lambda-2\delta(p-1))|\xi|^{p-2}|\eta|^{2} \\
&\geq \frac{\lambda}{2}|\xi|^{p-2}|\eta|^{2}.
\end{align*}
Considering $0<\delta\leq\frac{\lambda}{4(p-1)}$ we notice that
$\lambda-2\delta(p-1)\geq \frac{\lambda}{2}$. So (i) is proved.

(ii) Let $0<r\leq R$ and $y\in \mathbb{R}^n$. Then, for any 
$\xi\in \mathbb{R}^n$ and any $\varepsilon>0$ it follows from 
\eqref{new-a-defi} and \eqref{new-c-range} that
\begin{align*}
 |\tilde{\mathbf{a}}(x,\xi)-\overline{\tilde{\mathbf{a}}}_{B_{r}(y)}(\xi)|
 &\leq |\mathbf{b}(x,\xi)-\overline{\mathbf{b}}_{B_{r}(y)}(\xi)|
 +2\varepsilon|\xi|^{p-1}+2\varepsilon|\xi|^{p-1} \\
 &= |\mathbf{b}(x,\xi)-\overline{\mathbf{b}}_{B_{r}(y)}(\xi)|
 +4\varepsilon|\xi|^{p-1}.
\end{align*}
So
\begin{align*}
\omega_{\tilde{\mathbf{a}}}(R)
&:=\sup_{0<r\leq R}  \sup_{\xi \in \mathbb{R}^n} 
-\hspace{-3.5mm}\int_{B_{r}(y)}\frac{\tilde{\mathbf{a}}(x,\xi)
-\overline{\tilde{\mathbf{a}}}_{B_{r}(y)}(\xi)}{(1+|\xi|)^{p-1}}dx \\
&\leq \sup_{0<r\leq R}  \sup_{\xi \in \mathbb{R}^n} 
-\hspace{-3.5mm}\int_{B_{r}(y)}\frac{\mathbf{b}(x,\xi)
-\overline{\mathbf{b}}_{B_{r}(y)}(\xi)}{(1+|\xi|)^{p-1}}dx+4\varepsilon.
\end{align*}
Since $\mathbf{b}(x,\xi)$ is $(\delta,R)$-vanishing, we know that there exists 
$R_{0}>0$ such that for any $0<R\leq R_{0}$ we have
\begin{equation*}
 \omega_{\tilde{\mathbf{a}}}(R)\leq \varepsilon +4\varepsilon=5\varepsilon
\end{equation*}
namely, $\tilde{\mathbf{a}}(x,\xi)$ satisfies the $(5\delta,R)$-vanishing 
only if we choose $\delta=\varepsilon$. So $(ii)$ is proved.
\end{proof}

We are now ready to prove our main result.

\begin{proof}[Proof of Theorem \ref{main result}]
 From \eqref{a-equ-1} and \eqref{new-a-defi}, for any given $0<\delta<1$ there 
exists a positive constant $M=M(\delta)>1$ and a vector-valued function 
$\tilde{\mathbf{c}}(x,Du)$ such that $\tilde{\mathbf{c}}(x,Du)\leq 2\delta$ and
\begin{align*}
&\mathbf{a}(x,Du)\\
&= \mathbf{b}(x,Du)+|Du|^{p-1}\tilde{\mathbf{c}}(x,Du)
 +|Du|^{p-1}\chi_{\{|Du|<M\}}(\mathbf{c}(x,Du)-\tilde{\mathbf{c}}(x,Du)) \\
&= \tilde{\mathbf{a}}(x,Du)+|Du|^{p-1}\chi_{\{|Du|<M\}}(\mathbf{c}
 (x,Du)-\tilde{\mathbf{c}}(x,Du)),
\end{align*}
which implies
\begin{equation*}
 div \mathbf{a}(x,Du)=div \tilde{\mathbf{a}}(x,Du)+\operatorname{div}(|Du|^{p-1}\chi_{\{|Du|<M\}}(\mathbf{c}(x,Du)-\tilde{\mathbf{c}}(x,Du))).
\end{equation*}
Thus from \eqref{Dirichlet problem} and the above equality, we see that $u\in W^{1,p}_{0}(\Omega)$ is a weak solution of
\begin{equation} \label{main result proof}
\begin{aligned}
 \operatorname{div} \mathbf{\tilde{a}}(x,Du)
&=\operatorname{div}(|\mathbf{f}|^{p-2}\mathbf{f})
 -\operatorname{div}(|Du|^{p-1}\chi_{\{|Du|<M\}}
(\mathbf{c}(x,Du)-\tilde{\mathbf{c}}(x,Du))) \\
 &=\operatorname{div}(|\mathbf{f}|^{p-2}\mathbf{f}+|Du|^{p-1}
\chi_{\{|Du|<M\}}(\tilde{\mathbf{c}}(x,Du)-\mathbf{c}(x,Du))) \\
 &=\operatorname{div}(|\mathbf{g}|^{p-2}\mathbf{g}),
\end{aligned}
\end{equation}
where
\begin{equation*}
 \mathbf{g}=\frac{|\mathbf{f}|^{p-2}\mathbf{f}+|Du|^{p-1}
\chi_{\{|Du|<M\}}(\tilde{\mathbf{c}}(x,Du)
-\mathbf{c}(x,Du))}{||\mathbf{f}|^{p-2}\mathbf{f}+|Du|^{p-1}
\chi_{\{|Du|<M\}}(\tilde{\mathbf{c}}(x,Du)-\mathbf{c}(x,Du))|^{\frac{p-2}{p-1}}}
\end{equation*}
if
\begin{equation*}
 \big||\mathbf{f}|^{p-2}\mathbf{f}+|Du|^{p-1}
\chi_{\{|Du|<M\}}(\tilde{\mathbf{c}}(x,Du)-\mathbf{c}(x,Du))\big|\neq 0,
\end{equation*}
while $\mathbf{g}=0$ if
\begin{equation*}
 \big||\mathbf{f}|^{p-2}\mathbf{f}+|Du|^{p-1}\chi_{\{|Du|<M\}}
(\tilde{\mathbf{c}}(x,Du)-\mathbf{c}(x,Du))\big|=0.
\end{equation*}
Then it is clear that $|\mathbf{g}|^{p-1}$ belongs to $L^{\gamma,q}$
locally in $\Omega$ with
\begin{equation*}
 \|\mathbf{g}\|_{L^{\gamma,q}(\Omega)}
=q\int^{\infty}_{0}(\mu^{\gamma}|\{z\in \Omega:
|\mathbf{g}(z)|>\mu\}|)^{q/\gamma}\frac{d\mu}{\mu}.
\end{equation*}
Let
\begin{equation}\label{h-1}
 \mathbf{h}=|\mathbf{f}|^{p-2}\mathbf{f}+|Du|^{p-1}
\chi_{\{|Du|<M\}}(\tilde{\mathbf{c}}(x,Du)-\mathbf{c}(x,Du)),
\end{equation}
this yields
\begin{equation}\label{h-2}
 |\mathbf{g}|=|\mathbf{h}|^{\frac{1}{p-1}} \Rightarrow |\mathbf{g}|^{p-1}=|\mathbf{h}|.
\end{equation}
Then we obtain
\begin{equation*}
 \mu^{p-1}<|\mathbf{g}(z)|^{p-1}=|\mathbf{h}(z)|\leq |\mathbf{f}(z)|^{p-1}+4|Du|^{p-1}\chi_{\{|Du|<M\}},
\end{equation*}
and
\begin{align*}
 &|\{z\in \Omega:|\mathbf{g}(z)|>\mu\}| \\
 &\leq |\{z\in \Omega:|\mathbf{f}(z)|>\frac{\mu}{2^{\frac{1}{p-1}}}\}|+|\{z\in \Omega:4|Du(z)|^{p-1}\chi_{\{|Du|<M\}}>\frac{\mu}{2^{\frac{1}{p-1}}}\}|.
\end{align*}
Therefore,
\begin{align*}
\|\mathbf{g}\|_{L^{\gamma,q}(\Omega)}
&\leq  q\int^{\infty}_{0}\Big(\mu^{\gamma}|\{z\in \Omega:|\mathbf{f}(z)|
 >\frac{\mu}{2^{\frac{1}{p-1}}}\}|\Big)^{q/\gamma}\frac{d\mu}{\mu} \\
 &\quad + q\int^{\infty}_{0}\Big(\mu^{\gamma}|\{z\in \Omega:2|Du(z)|^{p-1}
\chi_{\{|Du|<M\}}>\frac{\mu}{2^{\frac{1}{p-1}}}\}|\Big)^{q/\gamma}\frac{d\mu}{\mu} \\
 &= 2^{\frac{q}{p-1}}q\int^{\infty}_{0}\Big(\mu^{\gamma}|\{z\in \Omega
 :|\mathbf{f}(z)|>\mu\}|\Big)^{q/\gamma}\frac{d\mu}{\mu} \\
 &\quad + 2^{\frac{q}{p-1}}q\int^{\infty}_{0}\Big(\mu^{\gamma}|
\{z\in \Omega:4|Du(z)|^{p-1}\chi_{\{|Du|<M\}}>\mu\}|\Big)^{q/\gamma}\frac{d\mu}{\mu}.
\end{align*}
Note that
\begin{equation*}
 |\{z\in \Omega:4|Du(z)|^{p-1}\chi_{\{|Du|<M\}}>\mu\}|
\leq |\{z\in \Omega:4M^{p-1}>\mu\}|,
\end{equation*}
it follows that
\begin{align}
&\|\mathbf{g}\|_{L^{\gamma,q}(\Omega)}  \nonumber \\
 &\leq 2^{\frac{q}{p-1}}\|\mathbf{f}\|_{L^{\gamma,q}(\Omega)}
 +2^{\frac{q}{p-1}}q\int^{\infty}_{0}\left(\mu^{\gamma}|\{z\in
 \Omega: 4M^{p-1}>\mu\}|\right)^{q/\gamma}\frac{d\mu}{\mu} \nonumber \\
 &\leq 2^{\frac{q}{p-1}}\|\mathbf{f}\|_{L^{\gamma,q}(\Omega)}
 +2^{\frac{q}{p-1}}q\int^{4M^{p-1}}_{0}\left(\mu^{\gamma}|\{z\in
 \Omega: 4M^{p-1}>\mu\}|\right)^{q/\gamma}\frac{d\mu}{\mu} \nonumber\\
 &\quad + 2^{\frac{q}{p-1}}q\int^{\infty}_{4M^{p-1}}\left(\mu^{\gamma}|\{z\in
 \Omega: 4M^{p-1}>\mu\}|\right)^{q/\gamma}\frac{d\mu}{\mu} \nonumber\\
 &= 2^{\frac{q}{p-1}}\|\mathbf{f}\|_{L^{\gamma,q}(\Omega)}
 +2^{\frac{q}{p-1}}q\int^{4M^{p-1}}_{0}
 \left(\mu^{\gamma}|\Omega|\right)^{q/\gamma}\frac{d\mu}{\mu}+0 \nonumber\\
 &= 2^{\frac{q}{p-1}}\|\mathbf{f}\|_{L^{\gamma,q}(\Omega)}
 +2^{\frac{q}{p-1}}q|\Omega|^{q/\gamma}\int^{4M^{p-1}}_{0}\mu^{q-1}d\mu \nonumber\\
 &= 2^{\frac{q}{p-1}}\|\mathbf{f}\|_{L^{\gamma,q}(\Omega)}+2^{\frac{q}{p-1}}
 |\Omega|^{q/\gamma}(4M^{p-1})^{q} \nonumber\\
 &\leq C(\|\mathbf{f}\|_{L^{\gamma,q}(\Omega)}+1), \label{G-Lorentz-esti}
\end{align}
where $C=C(n,\delta,p,\gamma,q,\theta,|\Omega|)$ is a positive constant.

Recalling Lemma \ref{new-a-regular} and using \eqref{main result proof} 
and \eqref{G-Lorentz-esti}, we employ Theorem \ref{Lorentz estimate}
with $\mathbf{b}(x,\xi)$ replaced by $\tilde{\mathbf{a}}(x,\xi)$ and 
$\mathbf{f}$ replaced by $\mathbf{g}$, respectively, which completes 
the proof. 
\end{proof}

\subsection*{Acknowledgments}
The work is supported by: the NSF of China (11371050),
Fundamental Research Funds for the Central Universities (S16JB00040),
NSF of China under grants 2013AA013702 (863) and 2013CB834205 (973).


\begin{thebibliography}{00}

\bibitem{AcM1} Acerbi, E.; Mingione, G.; 
\emph{Gradient estimates for a class of parabolic systems.}
 Duke Math. J. \textbf{136}(2) (2007), 285-320.

\bibitem{AdP} Adimurthil, K.; Phuc, N. C.; 
\emph{Global Lorentz and Lorentz-Morrey estimates below the natural 
exponent for quasilinear equations.} Calc. Var. \textbf{54}(2015), 3107-3139.

\bibitem{Bar1} Baroni, P.; 
\emph{Lorentz estimates for obstacle parabolic problems.} 
Nonlinear Anal. \textbf{96} (2014), 167-188.

\bibitem{Bar2} Baroni, P.; 
\emph{Lorentz estimates for degenerate and singular evolutionary systems.} 
J. Differ. Equ. \textbf{255} (2013), 2927-2951.

\bibitem{Bar3} Baroni, P.; 
\emph{Marcinkiewicz estimates for degenerate parabolic equations with measure data.} 
J. Funct. Anal. \textbf{267} (2014), 3397-3426.

\bibitem{ByCO} Byun, S. S.; Cho, Y.; Oh, J.; 
\emph{Global Calder\'{o}n-Zygmund theory for nonlinear elliptic obstacle problems 
with asymptotically regular nonlinearities.} Nonlinear Anal. 123-124 (2015), 150-157.

\bibitem{ByKSW} Byun, S. S.; Kwon, H.; So, H.; Wang, L.; 
\emph{Nonlinear gradient estimates for elliptic equations in quasiconvex domains.} 
Calc. Var. \textbf{54} (2015), 1425-1453.

\bibitem{ByW1} Byun, S. S.; Wang, L.; 
\emph{Elliptic equations with BMO coefficients in Reifenberg domains.} 
Commu. Pure Appl. Math. \textbf{57}(10) (2004), 1283-310.

\bibitem{ByW2} Byun, S. S.; Wang, L.; 
\emph{Elliptic equations with BMO nonlinearity in Reifenberg domains.} 
Advances in Mathematics \textbf{219}(6) (2008), 1937-1971.

\bibitem{ByW3} Byun, S.; Wang, L.; 
\emph{Elliptic equations with measurable coefficients in Reifenberg domains.} 
Advances in Mathematics \textbf{225}(5) (2010), 2648-2673.

\bibitem{ByW4} Byun, S.; Wang, L.; 
\emph{Nonlinear gradient estimates for elliptic equations of general type.} 
Cal. Var. \textbf{45}(3-4) (2012), 403-419.


\bibitem{ByOW} Byun, S. S.; Oh, J.; Wang, L.;
 \emph{Global Calder\'{o}n-Zygmund theory for asymptotically regular nonlinear 
elliptic and parabolic equations.}
 Int. Math. Res. Not. IMRN \textbf{2015}(17) (2015), 8289-8308.

\bibitem{ChE} Chipot, M.; Evans, L. C.; 
\emph{Linearisation at infinity and Lipschitz estimates for certain problems 
in the calculus of variations.} Proc. Roy. Soc. Edinburgh Sect. A. 
\textbf{102}(3-4) (1986), 291-303.

\bibitem{DaT} David, G.; Toro, T.; 
\emph{Reifenberg flat metric spaces, snowballs, and embeddings.} 
Math. Ann. \textbf{315}(4) (1999), 641-710.

\bibitem{Fos} Foss, M.; \emph{Global regularity for almost minimizers 
of nonconvex variational problems.} Ann. Mat. Pura Appl. (4) 
\textbf{187}(2) (2008), 263-321.

\bibitem{HajM} Haj{\l}asz, P.; Martio, O.; 
\emph{Traces of Sobolev functions on fractal type sets and characterization 
of extension domains.} J Funct. Anal. \textbf{143}(1) (1997), 221-246.

\bibitem{JiaLW1} Jia, H.; Li, D.; Wang, L.; 
\emph{Global regularity for divergence form elliptic equations on quasiconvex
 domains.} J. Differ. Equ. \textbf{249}(12) (2010), 3132-3147.

\bibitem{JiaLW2} Jia, H.; Li, D.; Wang, L.; 
\emph{Global regularity for divergence form elliptic equations in Orlicz 
spaces on quasiconvex domains.} Nonlinear Anal. \textbf{74}(4) (2011), 1336-1344.

\bibitem{MeP1} Mengesha, T.; Phuc, N. C.; 
\emph{Global estimates for quasilinear elliptic equations on Reifenberg 
flat domains.} Arch. Rational Mech. Anal. \textbf{203} (2012), 189-216.

\bibitem{Ray} Raymond, J. P.; 
\emph{Lipschitz regularity of solutions of some asymptotically convex problems.}
 Proceedings of the Royal Society of Edinburgh,
Section A \textbf{117}(1-2)(1991), 59-73.

\bibitem{ScS1} Scheven, C.; Schmidt, T.; 
\emph{Asymptotically regular problems I: Higher integrability.} 
J. Differential Equations \textbf{248} (2010), 745-791.

\bibitem{ScS2} Scheven, C.; Schmidt, T.; 
\emph{Asymptotically regular problems II: Partial Lipschitz continuity 
and a singular set of positive measure.} Ann. Sc. Norm.
Super. Pisa Cl. Sci. (5) \textbf{8}(3) (2009), 469-507.

\bibitem{Tal} Talenti G; 
\emph{Elliptic equations and rearrangements.} 
Ann Scuola Norm. Sup. Pisa Cl. Sci. (IV) \textbf{3} (1976), 697-718.

\bibitem{ZhZ} Zhang C.; Zhou S. L.; 
\emph{Global weighted estimates for quasilinear elliptic equations 
with non-standard growth.} J. Funct. Anal. \textbf{267}(2014), 605-642.

\bibitem{ZhZhe} Zhang J.; Zheng S.; 
\emph{Lorentz estimate for nonlinear parabolic obstacle problems with 
asymptotically regular nonlinearities.} Nonlinear Anal. \textbf{134} (2016),
189-203.

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