\documentclass[reqno]{amsart}
\usepackage{a4wide}
\usepackage{hyperref}
\usepackage{amssymb}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2026 (2026), No. 29, pp. 1--22.\newline
ISSN: 1072-6691. URL: https://ejde.math.txstate.edu, DOI: 10.58997/ejde.2026.29}
\thanks{\copyright 2026. This work is licensed under a CC BY 4.0 license.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2026/29\hfil 
Borderline double-phase functional with DMO coefficients]
{Gradient continuity of local minimizers of a borderline double-phase
functional with DMO coefficients}


\author[W. Chang, Y. Tong, S. Zheng \hfil EJDE-2026/29\hfilneg]
{Wenrui Chang, Yuxia Tong, Shenzhou Zheng}

\address{Wenrui Chang \newline
School of Mathematics and Statistics,
Beijing Jiaotong University,
Beijing 100044, China}
\email{wenruichang@bjtu.edu.cn}

\address{Yuxia Tong \newline
School of Mathematics and Statistics,
North China University of Science and Technology,
Tangshan 063210, China}
\email{tongyuxia@126.com}

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

\thanks{Submitted November 28, 2025. Published April 27, 2026.}
\subjclass[2020]{35B65, 35J15, 35J60}
\keywords{Gradient continuity; non-autonomous functional; Dini mean oscillation; \hfill\break\indent borderline double phase}

\begin{abstract}
Let $u \in W^{1,1}(\Omega)$. We study a local gradient continuity of the minimizers  
of functionals with borderline double-phase growth,
$$
\mathcal{F}(u ; \Omega) := \int_{\Omega} \Big(|Du|_\mathbb{A}^{p} + a(x)\, |Du|^{p}_\mathbb{A} \log(e + |Du|_\mathbb{A})\Big) \,dx
$$
for $1 < p < \infty$ and
$|Du|_{\mathbb{A}} :=\langle \mathbb{A}(x) Du, Du \rangle^{1/2}$,
where the matrix $\mathbb{A}(x) = \{A^{\alpha \beta}(x)\}$ is symmetric with
positive upper and lower bounds of its eigenvalues.
We prove that $Du\in C^1_\mathrm{loc}(\Omega)$ based mainly on the
exit-time approach, provided that the moderating factor $a(x)$ and coefficient $\mathbb{A}(x)$ satisfy the so-called Dini-type mean oscillation conditions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks
\newcommand{\fint}{-\hspace{-0.38cm}\int}

\section{Introduction}

Let $\Omega \subset \mathbb{R}^n$ for $n\ge 2$  be a bounded domain and $1 < p < \infty$. We consider the local minimizers $u \in W^{1,1}(\Omega)$ for the 
functional with borderline double phase growth,
\begin{equation}\label{main_funl}
\mathcal{F}(u ; \Omega) := \int_{\Omega} \Big(|Du|_\mathbb{A}^{p} + a(x)\, |Du|^{p}_\mathbb{A} \log(e + |Du|_\mathbb{A})\Big) \,dx,
\end{equation}
where the moderating factor $a: \Omega \to [0,+\infty)$ is bounded, and $|Du|_{\mathbb{A}} := \big\langle \mathbb{A}(x)\,Du, Du \big\rangle^{1/2}$. 
The matrix 
$\big(A^{\alpha\beta}(x)\big)$ for $\alpha,\beta=1,2,\dots,n$
is assumed to be symmetric with $A^{\alpha \beta}(x)=A^{\beta \alpha}(x)$; 
moreover, there exists a constant $K \geq 1$ such that
\begin{equation}\label{cond_A}
K^{-1}|\xi|^2
\leq  \sum_{\alpha,\beta  =1}^{n}A^{\alpha \beta}(x)\, \xi_\alpha \xi_\beta
\leq	K\,|\xi|^2, \quad (x,\xi) \in \Omega\times\mathbb{R}^{n}.
\end{equation}
For a given $\gamma \geq 1$, we define the $\gamma$-moduli of mean oscillation as
\[
\bar{\omega}_{a, \gamma}(\rho) := \sup_{y\in \Omega} \Big(\fint_{B_\rho(y) \cap \Omega} |a(x) - \bar{a}_{\rho}|^\gamma\,dx\Big)^{1/\gamma}
	\quad \text{and} \quad
	\bar{\omega}_{\mathbb{A}, \gamma}(\rho) := \sup_{y\in \Omega} \Big(\fint_{B_\rho(y) \cap \Omega} |\mathbb{A}(x) - \bar{\mathbb{A}}_{\rho}|^\gamma\,dx\Big)^{1/\gamma},
\]
where $\bar{a}_{\rho} = \fint_{B_\rho(y)\cap \Omega} a(x)\,dx$ and $\bar{\mathbb{A}}_{\rho} = \fint_{B_\rho(y) \cap \Omega} \mathbb{A}(x)\,dx$. Moreover, it is supposed that both $a(x)$ and $\mathbb{A}(x)$ satisfy the so-called Dini-type mean oscillation (DMO) introduced first by Spanne \cite{Sp65}, which require
the moduli to satisfy
\begin{equation}\label{DMO}
	\int_{0}^1 \bar{\omega}_{a, \gamma}(\rho) \log \big(\frac{1}{\rho}\big)\, \frac{d\rho}{\rho} < \infty
	\quad \text{and} \quad
	\int_{0}^1 \bar{\omega}_{\mathbb{A}, \gamma}(\rho)\, \frac{d\rho}{\rho} < \infty.
\end{equation}
Now we give the definition of local minimizers to the functional \eqref{main_funl}.

\begin{definition}\label{def-min} \rm
 We say that $u \in W^{1,1}(\Omega)$ is a local minimizer of the functional \eqref{main_funl} if $\mathcal{F}(u ; \Omega) < \infty$ and for any $\Omega' \Subset \Omega$ we have
\[
\mathcal{F}(u ; \Omega')
\leq	\mathcal{F}(u + \phi; \Omega'),\quad   \phi \in W_0^{1,1}(\Omega').
\]
\end{definition}

Such a non-uniform functional \eqref{main_funl} was originated from Zhikov's work \cite{Zh87} on the homogenization of strongly anisotropic materials, and the 
regularity study for elliptic problems with $(p,q)$-growth has drawn extensive attention over the past few decades.
Marcellini proved the Lipschitz regularity for local minimizers of double-phase energy functionals in \cite{Ma89}, and also got the same regularity for non-uniformly elliptic equations in \cite{Ma91}. A typical $(p,q)$-growth functional with
\begin{equation}\label{fun_p_q}
\mathcal{F}_{p,q}(u ; \Omega) := \int_\Omega\Big(|Du|^{p} + a(x)\,|Du|^{q}\Big)\, dx,
\end{equation}
 Colombo-Mingione \cite{CoM15} recently proved the local H\"older continuity of the gradient for local minimizers under the assumptions that
\begin{equation}\label{gap_p_q}
a(x) \in C^{0, \alpha}(\Omega) \quad \text{for } \alpha \in (0,1], \quad \text{and} \quad \frac{q}{p} < 1 + \frac{\alpha}{n};
\end{equation}
while Colombo-Mingione \cite{CoM15B} showed the
$C_{\text{loc}}^{1,\gamma}$-regularity for the bounded minimizers of \eqref{fun_p_q} under a relaxed gap condition $q \le p + \alpha$. Furthermore, Baroni-Colombo-Mingione \cite{BaM18} also got the same gradient H\"older continuity to the local minimizers of functional \eqref{fun_p_q} based on harmonic type approximation by extending this constraint \eqref{gap_p_q} to $q/p \le 1 + \alpha/n$.

On the other hand, as a borderline case of $(p,q)$-growth with functional of the form
\begin{equation}\label{p-plog}
\mathcal{F}_{\log}(u ; \Omega)
:= \int_\Omega \Big( |Du|^p + a(x)\,|Du|^p \log \big(e + |Du|\big) \Big)\, dx,
\end{equation}
Baroni-Colombo-Mingione \cite{BaCM16} proved that the local minimizers of
$\mathcal{F}_{\log}(u ; \Omega)$ can attain an optimal H\"older continuity provided
$\limsup_{r \to 0^+} \omega_a(r)\,\log\big(\frac{1}{r}\big) = 0$ with
$\omega_a(r)$ as the modulus of continuity for $a(\cdot)$, and the gradient H\"older
continuity by adding $\omega_a(r)\lesssim r^{\alpha}$ with $0<\alpha\le 1$. Later, Byun-Youn \cite{ByY18} employed the Riesz potential theory to obtain the pointwise gradient estimate and  $C^1$-regularity to such an elliptic equation with borderline double-phase growth under assumption of the so-called log-Dini continuity with the form $\int_0^1 \omega_a(\rho) \log\big(\frac{1}{\rho}\big)\, \frac{d\rho}{\rho} < \infty$. Recently, Baroni-Coscia \cite{BaC22} also got the gradient continuity of local minimizers to such a functional $\mathcal{F}_{\log}(u ; \Omega)$ under assumption of log-Dini continuity of $a(\cdot)$. More regularity results involving the double-phase problems, we can refer to \cite{ArCW25, Ba25, De23, ZFeng25, LuZF21, RaT24,JZh23} and the references therein.

Associated with $C^1$-regularity of elliptic problems, let us briefly recall some recent progresses. For such  a linear elliptic equation: 
$	-\operatorname{div}(\mathbb{A}(x) Du) = 0$, Lieberman \cite{Li87} first proved  the gradient continuity if $\mathbb{A}(x)$ satisfies the so-called $\alpha$-increasing 
Dini continuity which means that 
$\int_0^1 \omega_{\mathbb{A}}(r)\,\frac{dr}{r} < \infty$
and $\omega_{\mathbb{A}}(r)/r^\alpha$ is non-decreasing for $\alpha> 0$ with $\omega_{\mathbb{A}}(r)$ as a modulus of continuity of $\mathbb{A}(x)$.
Instead of the Dini mean oscillation from Spanne in \cite{Sp65}, Huang in \cite{huang96} proved the $\mathrm{BMO}_\psi$-boundedness and continuity of $Du$ to linear elliptic systems if the leading coefficient $\mathbb{A}(x) \in \mathrm{BMO}_\psi$ with $\psi$ satisfying the Dini continuity and almost increasing of $\psi(r)\big(1+|\log \int_0^r(\psi(t) / t)|\big)$.
Li in \cite{Li17}  gave an open problem for linear elliptic equations: does the $C^1$-regularity hold of solutions if one merely adds DMO condition on 
the coefficients?
Dong-Kim in \cite{DoK17} gave a positive answer to Li's question, and they proved the $C^1$-regularity of solutions to linear elliptic equations under the DMO condition $\int_{0}^1 \bar{\omega}_{\mathbb{A}, 1}(\rho)\, \frac{d\rho}{\rho} < \infty$. However, it is insufficient one only assumes the continuity of coefficients $\mathbb{A}(x)$, since Jin-Maz'ya-Schaftingen \cite{JiMV09} constructed a counterexample to show that the gradient may be discontinuous to the homogeneous linear elliptic equation if $\mathbb{A}(x)$ is merely continuous.
Further, Duzaar-Mingione \cite{DuM10} used  non-linear Wolff potential argument to show the gradient continuity for $p$-Laplace equations under assumption of Dini continuous coefficients. Recently, for $p(x)$-Laplace system
\[
	-\operatorname{div}\big(|Du|^{p(x)-2} Du\big) = f,
\]
Ok \cite{Ok16} obtained a local gradient continuity under assumptions that
$f\in L(n, 1)$ (the so-called Lorentz spaces), and $p(x)$ is log-Dini continuous
with $\int_0^1 \omega_p(r) \log\big(\frac{1}{r}\big)\, \frac{dr}{r} < \infty$,
where $\omega_{p}(\cdot)$ is the modulus of continuity of $p(x)$.
Baroni-Coscia in \cite{BaC22} showed  the $C^1$-regularity of functional \eqref{p-plog} if $a(x)$ satisfies the log-Dini condition. Dong-Lee-Kim \cite{DoLK20} proved a
global $C^1$-regularity of conormal and oblique derivative boundary problem to linear elliptic equations with DMO coefficients, while Dong-Kwon \cite{DoK26} obtained the $C^1$-regularity of solutions to  non-stationary Stokes equations with DMO coefficients.

Motivated by the above-mentioned papers, we are interested in the $C^1$-regularity 
to the local minimizers of  functional \eqref{main_funl}. 
This is an extensive generalization of Baroni-Coscia's work in \cite{BaC22} 
by way of general matrix function $\mathbb{A}(x)$ instead of $Id$, and 
relaxing the regular assumption of $a(x)$ from the log-Dini continuity to the 
Dini-type mean oscillation. To this end, let us write
\[
	\varPsi(x, t) := t^{p} + a(x)\, t^{p} \log(e + t);
\]
and we are now in a position to state our main result.

\begin{theorem}\label{thm_1}
Let $u \in W^{1,1}(\Omega)$  be a local minimizer of the functional 
\eqref{main_funl} with $\varPsi(x, Du) \in L^1(\Omega)$. 
If there exists a constant $\gamma = \gamma\,(n,p,K, \|a\|_{L^\infty(\Omega)}, \|Du\|_{L^p(\Omega)}) \geq 1$ such that  $a(x)$ and $\mathbb{A}(x)$ satisfy \eqref{cond_A} and \eqref{DMO}, 
then we conclude that $Du$ is locally continuous in $\Omega$.
\end{theorem}

The rest of this paper is organized as follows. 
We devote Section 2 to notations, basic inequalities and a few useful lemmas. 
In Section 3, we employ the perturbation argument to perform the comparison and 
decay estimates in the $L^1$ sense based on the freezing coefficients. 
We mainly focus Section 4 on the proofs of local boundedness and continuity of 
gradient to the local minimizers of functional \eqref{main_funl} by way of the stop-time approach.

\section{Preliminaries}

Throughout this paper, we use $c$ to denote a generic positive constant that may 
vary from line to line. Important constants are distinguished by subscripts 
(e.g., $c_1, c_2$). To indicate dependence on specific parameters, we use parentheses; for example, $c(n, p, K)$ indicates that the constant depends only on $n, p, K$.
We use the notation $a \lesssim b$ to explain that there exists a constant $c > 0$ 
such that $a \leq cb$; similarly, $a \gtrsim b$ means $a \geq cb$; $a \approx b$ indicates that there exist two positive constants $c_1, c_2$ satisfying 
$c_1 a \leq b \leq c_2 a$.

Let us first recall related basic notations. 
We write $\varphi_{\log}(t) = t^p \log(e + t)$ for $p \in (1, \infty)$, and  
observe that
\[
\varphi'_{\log}(\cdot) = p t^{p-1} \log(e + t) + \frac{t^p}{e + t},
\quad
\varphi''_{\log}(\cdot) = p(p-1) t^{p-2} \log(e + t) + \frac{2 p t^{p-1}}{e + t} - \frac{t^p}{(e + t)^2}.
\]
A direct calculation yields that
\begin{equation}\label{equiv}
t^2\varphi_{\log}^{\prime\prime}(t)
\approx	t\varphi_{\log}'(t)
\approx	\varphi_{\log}(t) \quad \forall t > 0,
\end{equation}
where the implied constants depend only on $p$.
Since $\log(e + L t) \leq L\log(e + t)$ for  $L \geq 1$ and $ t > 0$, and
$\log(e + \nu t) \geq \nu\log(e + t)$ for  $\nu \in (0, 1)$ and $ t > 0$, then we obtain the following facts:
\begin{equation}\label{est_log_1}
\varphi_{\log}(L t) \leq L^{p+1} \varphi_{\log}(t), \quad \varphi_{\log}(\nu t) \geq \nu^{p+1} \varphi_{\log}(t), \quad \text{for any } L \geq 1,\; \nu \in (0, 1),\; t > 0;
\end{equation}
moreover, it follows from the expression of $\varphi'_{\log}(\cdot)$ that
\begin{equation}\label{est_log_2}
\varphi'_{\log}(L t) \leq L^{p} \varphi'_{\log}(t), \quad \varphi'_{\log}(\nu t) \geq  \nu^{p} \varphi'_{\log}(t), \quad \text{for any } L \geq 1, \nu \in (0, 1), t > 0.
\end{equation}

Let $\mathcal{D} \subset \mathbb{R}^n$  be a measurable set with positive measure. 
For any locally integrable function $f \in L^1(\mathcal{D}, \mathbb{R}^m)$, 
we define the integral average of $f$ over $\mathcal{D}$ by
\[
\langle f \rangle_{\mathcal{D}} =
\fint_{\mathcal{D}} f \,dx =
\frac{1}{|\mathcal{D}|} \int_{\mathcal{D}} f \,dx.
\]
A direct calculation yields that for any $\xi \in \mathbb{R}^n$ and
$f \in L^q(\mathcal{D}, \mathbb{R}^n)$ with $q \geq 1$,
\[
\fint_{\mathcal{D}} |f - \langle f \rangle_\mathcal{D}|^q \,dx
\leq c(n,q) \fint_{\mathcal{D}} |f - \xi|^q \,dx.
\]
In the following context, let $B_r(x_0)$ denote the open ball in
$\mathbb{R}^n$ centered at $x_0$ with radius $r > 0$; we simply write $B_r$
while the center is clear from the context.
The following lemma provides an estimate associated with the logarithmic function,
for details see \cite[Lemma 2.1]{BaC22}.

\begin{lemma}\label{log}
Let $q > 1$ and $\sigma, \beta, \theta \geq 0$ be constants, and let $f \in L^{q}(B_r)$ with $r \leq e^{-1}$. Then there exists a positive constant 
$c = c(n, \beta, \sigma, \theta, q)$ such that
\[
\fint_{B_r}|f| \log^\beta(e+|f|^\sigma) \,dx \leq c\big(1+r^\theta\|f\|_{L^1(B_r)}\big) \Big(\log \big(\frac{1}{r}\big)\Big)^\beta \Big(\fint_{B_r}|f|^{q} \,dx\Big)^{1 / q} .
\]
\end{lemma}

It is important to observe that any function satisfying the Dini mean oscillation condition must be continuous, see \cite[Corollary 1]{Sp65}.

\begin{lemma}\label{DMO_C}
Let $f(x) \in L^1(\Omega)$ satisfy the Dini mean oscillation condition with the form
$$
\int_0^1 \bar{\omega}_f(\rho) \frac{d\rho}{\rho} < \infty \quad \text{with} \quad \bar{\omega}_f(\rho) := \sup_{y \in \Omega} \fint_{B_\rho(y) \cap \Omega} |f(x) - \langle f \rangle_{B_\rho \cap \Omega}| \, dx.
$$
Then $f(x)$ has to be continuous in $\Omega$ with modulus 
$\omega_f(r) = c \int_0^r \bar{\omega}_f(\rho) \, \frac{d\rho}{\rho}$ for some 
$c > 0$.
\end{lemma}

With Lemma \ref{DMO_C} in hand, we readily see that $a(x)$ is log-H\"older 
continuous and $\mathbb{A}(x)$ is continuous provided that $a(x)$ and 
$\mathbb{A}(x)$ meet \eqref{DMO}.  

\begin{lemma}\label{DMO_L}
If both $a(x)$ and $\mathbb{A}(x)$ satisfy \eqref{DMO} for  $\gamma > 1$,
then $a(x)$ is log-H\"older continuous, and $\mathbb{A}(x)$ is continuous in $\Omega$, respectively,  with
\begin{equation}\label{log_Hold}
\sup_{r \in (0,1]} \omega_a(r)\log\big(\frac{1}{r}\big)  < \infty,
\end{equation}
where $\omega_a(r)$ is the continuity modulus of $a(x)$ in $B_r$.
\end{lemma}

\begin{proof}
It follows from the boundedness of $a(x)$ and \eqref{DMO} that
\begin{align*}
\int_{0}^{1} \bar{\omega}_a(\rho) \frac{d\rho}{\rho}
&=  \int_{0}^{1/e} \bar{\omega}_a(\rho) \frac{d\rho}{\rho} + \int_{1/e}^{1} \bar{\omega}_a(\rho) \frac{d\rho}{\rho} \\
&\leq \int_{0}^{1/e} \bar{\omega}_a(\rho)\log \big(\frac{1}{\rho}\big) \frac{d\rho}{\rho} + 2\|a\|_{L^\infty(\Omega)} < \infty.
\end{align*}
By Lemma \ref{DMO_C} we conclude that $a(x)$ is continuous and satisfies 
$\omega_a(r) = c \int_0^r \bar{\omega}_a(\rho)  \frac{d\rho}{\rho}$. Then we use \eqref{DMO} again to obtain that
\[
\sup_{r \in (0,1]} \omega_a(r)\log\big(\frac{1}{r}\big)
\leq \int_{0}^{1} \bar{\omega}_{a, \gamma}(\rho)\log \big(\frac{1}{\rho}\big)\, \frac{d\rho}{\rho} < \infty.
\]
The  continuity of $\mathbb{A}(x)$ is readily obtained due to \eqref{DMO} and Lemma \ref{DMO_C}.
\end{proof}

For the sake of convenience, we collect the parameters in the context and denote them by
\[
\texttt{data} := \Big(n,  p, K,  \|a\|_{L^\infty(\Omega)}, \|Du\|_{L^p(\Omega)}\Big).
\]
In what follows, we present the higher integrability result of the gradient to 
the local minimizers  of functional \eqref{main_funl}, for its proof  see  \cite[Theorem 4.1]{BaCM16}.


\begin{lemma}\label{higher_int}
Let $u \in W^{1,1}(\Omega)$ be a local minimizer of the functional 
\eqref{main_funl}. Assume that $\mathbb{A}(x)$ satisfies \eqref{cond_A} and 
$a(x)$ is log-H\"older continuous with the modulus $\omega_a(r)$ satisfying \eqref{log_Hold}. Then there exist constants 
$\varepsilon = \varepsilon\,(\texttt{data}) > 0$  and 
$c = c (\texttt{data}) \geq 1$ such that
\[
\Big(\fint_{B_{r/2}} \varPsi(x, |Du|)^{1 + \varepsilon} \, dx\Big)^{\frac{1}{1 + \varepsilon}}
\leq c \fint_{B_{r}} \varPsi(x, |Du|)\, dx
\]
for all $B_r \Subset \Omega$.
\end{lemma}

The following lemma describes a self-improving property of the reverse 
H{\"o}lder inequality based on the backward iteration, see \cite[Remark 6.12]{Gi03} 
or \cite[Lemma 3.1]{KuM18}.

\begin{lemma}\label{lem_self_improve}
Let $\mathcal{D}$ be a domain. If for any function 
$g \in L_\mathrm{loc}^{1+\varepsilon}(\mathcal{D})$, there exist $b > 0$ and  
$A \ge 0$ such that for any $B_r \Subset \mathcal{D}$,
\[
\Big(\fint_{B_{r/2}} g^{1+\varepsilon} \, dx\Big)^\frac{1}{1+\varepsilon} \leq b\,\fint_{B_{r}} g \, dx +  A,
\]
then  we have
\[
\Big(\fint_{B_{r/2}} g^{1+\varepsilon} \, dx\Big)^\frac{1}{1+\varepsilon} \leq B\,\Big(\Big(\fint_{B_{r}} g^{\tau} \, dx\Big)^{1/\tau} +  A \Big)
\]
for any \(\tau > 0\), where $B = B\big(n, b, \tau\big) > 0$.
\end{lemma}

Then, the conclusion of Lemma \ref{higher_int} can be  improved by way of Lemma \ref{lem_self_improve}. To this end, let
$$
\bar{\varPsi}_r(t) := t^p + \bar{a}_r\, t^p \log(e + t)
$$
with $\bar{a}_r = \fint_{B_r} a(x)\, dx$.

\begin{lemma}\label{higher_int_p}
Under the same assumptions as Lemma \ref{higher_int},  there exist an 
$R_1 = R_1\,(\|Du\|_{L^p(\Omega)}) > 0$ and $c = c (\texttt{data}) \geq 1$
such that
\[
\Big(\fint_{B_{r/2}} \varPsi(x, |Du|)^{1 + \varepsilon} \, dx\Big)^{\frac{1}{1 + \varepsilon}} \leq c \bar{\varPsi}_r\Big(\fint_{B_{r}} |Du|\, dx\Big)\,,
\]
for any  $r \in (0, R_1)$ with $B_r \Subset \Omega$.
\end{lemma}

\begin{proof}
Employing the self-improving property of Lemma \ref{lem_self_improve}, we obtain
\begin{equation}\label{ineq_rev_s_i}
\Big(\fint_{B_{r/2}} \Big(\varPsi\big(x, |Du|\big)\Big)^{1+\varepsilon} \, dx\Big)^\frac{1}{1+\varepsilon}
\leq c  \Big(\fint_{B_{r}} \Big(\varPsi\big(x, |Du|\big)\Big)^\frac{1}{p+1} \, dx\Big)^{p+1}.
\end{equation}
Note that
\begin{align*}
\Big(\varPsi(x, |Du|)\Big)^\frac{1}{p+1}
&= \Big(|Du|^p + \big(\bar{a}_r - \bar{a}_r + a(x)\big)\,\varphi_{\log}(|Du|) \Big)^\frac{1}{p+1} \\
&\leq |Du|^\frac{p}{p+1} + \big(\bar{a}_r + \omega_a(r)\big)^\frac{1}{p+1}\varphi_{\log}^\frac{1}{p+1}\big(|Du|\big)\big)\,.
\end{align*}
Since both $t^\frac{p}{p+1}$ and $\varphi_{\log}^{\frac{1}{p+1}}(t)$ are 
concave functions,  by the Jensen inequality, we conclude that
\begin{equation}\label{right}
\begin{aligned}
&\Big(\fint_{B_{r}} \Big(\varPsi\big(x, |Du|\big)\Big)^\frac{1}{p+1} \, dx\Big)^{p+1} \\
&\leq c\Big(\fint_{B_{r}} |Du|^\frac{p}{p+1} \, dx\Big)^{p+1}
+ c \Big(\bar{a}_r + \omega_a(r)\Big)\Big(\fint_{B_{r}} \Big(\varphi_{\log}\big(|Du|\big)\Big)^\frac{1}{p+1} \, dx\Big)^{p+1}  \\
&\leq c\Big(\fint_{B_{r}} |Du| \, dx\Big)^{p} + c \Big(\bar{a}_r + \omega_a(r)\Big){\varphi_{\log}}\Big(\fint_{B_{r}} |Du| \, dx\Big) \\
&\leq c\bar{\varPsi}_{r}\Big(\fint_{B_{r}} |Du| \, dx\Big)  +  c\omega_a(r)\,{\varphi_{\log}}\Big(\fint_{B_{r}} |Du| \, dx\Big)\,.
\end{aligned}
\end{equation}
Let us take $R_1 = \frac{1}{e + \omega_n\|Du\|_{L^p(\Omega)}}$ with 
$\omega_n$ as the measure of the unit ball in $\mathbb{R}^n$. 
For $r \in (0, R_1)$, the second term of the right-hand side in \eqref{right} 
is estimated as follows:
\begin{equation}\label{right-2}
\begin{aligned}
\omega_a(r){\varphi_{\log}}\Big(\fint_{B_{r}} |Du| \, dx\Big)
&=   \omega_a(r)\log\Big(e + \fint_{B_{r}} |Du| \, dx\Big)\Big(\fint_{B_{r}} |Du| \, dx\Big)^p \\
&\leq  {\omega_a(r)\,\log\Big(\frac{e+\omega_n\|Du\|_{L^1(B_r)}}{r^n}\Big)\,\bar{\varPsi}_r\Big(\fint_{B_{r}} |Du| \, dx\Big)}    \\
&\leq  {\omega_a(r)\,\log\Big(\frac{1}{r^{n+1}}\Big)\,\bar{\varPsi}_r\Big(\fint_{B_{r}} |Du| \, dx\Big)}    \\
&\leq  c  \bar{\varPsi}_r\Big(\fint_{B_{r}} |Du| \, dx\Big)\,.
\end{aligned}
\end{equation}
Combining \eqref{ineq_rev_s_i}, \eqref{right} and \eqref{right-2}, we obtain the 
desired estimate.
\end{proof}


\section{Comparison estimates}

In this section, we mainly perform the perturbation estimates between the original equation and a local frozen problem over a sequence of concentric balls. 
For $B_{r} = B_r(y) \Subset \Omega$ with $r \in (0, R_1)$ and 
$\theta \in \big(0,  1/4\big)$, we write
\[
r_j = \theta^j r \quad\text{and}\quad
B_j = B_{r_j}(y) \quad\text{for}\quad j = 0, 1, 2, \dots.
\]
Moreover, we denote $\bar{a}_j$ and $\bar{\mathbb{A}}_j $ as the integral averages of $a(x)$ and $\mathbb{A}(x)$ over $B_j$ with
\[
\bar{a}_j = \fint_{B_j} a(x)\, dx \quad \text{and} \quad \bar{\mathbb{A}}_j = \fint_{B_j} \mathbb{A}(x)\, dx,
\]
and introduce the frozen double phase operator $\bar{\varPsi}_j$ defined by
\[
	\bar{\varPsi}_j\,(t) := t^{p} + \bar{a}_j\, t^{p} \log(e + t).
\]
By \eqref{equiv} we readily see that
$t \bar{\varPsi}'_j(t) \approx \bar{\varPsi}_j(t)$ for $t > 0$.
First, we give two inequalities for $\bar{\varPsi}_j$ which is useful for the further analysis.


\begin{lemma}\label{monotone}
For $\bar{\varPsi}_j\,(t)$, we have the following results:
\begin{enumerate}
\item For any $\xi_1, \xi_2 \in \mathbb{R}^{m}\setminus \{\mathbf{0}\}$ and $\mathbb{A}$ satisfying \eqref{cond_A}, we have
\begin{equation}\label{ineq_1}
\big\langle\frac{\bar{\varPsi}_j'(|\xi_1|_\mathbb{A})}{|\xi_1|_\mathbb{A}} \mathbb{A} \xi_1 - \frac{\bar{\varPsi}_j'(|\xi_2|_\mathbb{A})}{|\xi_2|_\mathbb{A}} \mathbb{A} \xi_2, \xi_1 - \xi_2 \big\rangle 
\gtrsim  \frac{\bar{\varPsi}_j'\big(|\xi_1| + |\xi_2|\big)}{|\xi_1| + |\xi_2|} |\xi_1 - \xi_2|^2;
\end{equation}

\item For any $\xi \in \mathbb{R}^{m}\setminus \{\mathbf{0}\}$, both 
$\mathbb{A}_1$ and  $\mathbb{A}_2$ satisfying \eqref{cond_A}, it holds 
\begin{equation}\label{ineq_2}
\big| \frac{\bar{\varPsi}_j'(|\xi|_{\mathbb{A}_1})}{|\xi|_{\mathbb{A}_1}} \mathbb{A}_1 \xi - \frac{\bar{\varPsi}_j'(|\xi|_{\mathbb{A}_2})}{|\xi|_{\mathbb{A}_2}} \mathbb{A}_2 \xi \big|
\lesssim  \bar{\varPsi}_j'(|\xi|) \cdot | \mathbb{A}_1 - \mathbb{A}_2 |;
\end{equation}
\end{enumerate}
where the constants of the inequalities depend only on $p$ and $K$.
\end{lemma}

\begin{proof}
We begin with the proof of (1). Without loss of generality, we assume that $|\xi_1|_\mathbb{A} \leq |\xi_2|_\mathbb{A}$. 
By \eqref{est_log_2} we have that
\[
\Big(\frac{|\xi_1|_\mathbb{A}}{|\xi_2|_\mathbb{A}}\Big)^{p} \varphi_{\log}'(|\xi_2|_\mathbb{A}) \leq \varphi_{\log}'(|\xi_1|_\mathbb{A}) \leq \Big(\frac{|\xi_1|_\mathbb{A}}{|\xi_2|_\mathbb{A}}\Big)^{p - 1} \varphi_{\log}'(|\xi_2|_\mathbb{A}).
\]
Combined with
\[
\Big(\frac{|\xi_1|_\mathbb{A}}{|\xi_2|_\mathbb{A}}\Big)^{p} |\xi_2|_\mathbb{A}^{p-1} \leq |\xi_1|_\mathbb{A}^{p-1} = \Big(\frac{|\xi_1|_\mathbb{A}}{|\xi_2|_\mathbb{A}}\Big)^{p - 1} |\xi_2|_\mathbb{A}^{p-1},
\]
we conclude that
\[
\Big(\frac{|\xi_1|_\mathbb{A}}{|\xi_2|_\mathbb{A}}\Big)^{p} \bar{\varPsi}_j'(|\xi_2|_\mathbb{A}) \leq \bar{\varPsi}_j'(|\xi_1|_\mathbb{A}) \leq \Big(\frac{|\xi_1|_\mathbb{A}}{|\xi_2|_\mathbb{A}}\Big)^{p - 1} \bar{\varPsi}_j'(|\xi_2|_\mathbb{A}).
\]
Thanks to the mean value theorem, there exists a $\theta  \in (p,p+1)$ such that
\[
\bar{\varPsi}_j'(|\xi_1|_\mathbb{A}) = \Big(\frac{|\xi_1|_\mathbb{A}}{|\xi_2|_\mathbb{A}}\Big)^{\theta  - 1} \bar{\varPsi}_j'(|\xi_2|_\mathbb{A}).
\]
Then we have
\begin{equation}\label{ineq_xi}
\begin{aligned}
&\big\langle\frac{\bar{\varPsi}_j'(|\xi_1|_\mathbb{A})}{|\xi_1|_\mathbb{A}} \mathbb{A} \xi_1 - \frac{\bar{\varPsi}_j'(|\xi_2|_\mathbb{A})}{|\xi_2|_\mathbb{A}} \mathbb{A} \xi_2, \xi_1 - \xi_2 \big\rangle \\
&= \big\langle\frac{\bar{\varPsi}_j'(|\xi_2|_\mathbb{A})}{|\xi_2|^{\theta}_\mathbb{A}} |\xi_1|^{\theta -1}_\mathbb{A} \mathbb{A} \xi_1 - \frac{\bar{\varPsi}_j'(|\xi_2|_\mathbb{A})}{|\xi_2|^{\theta}_\mathbb{A}} |\xi_2|^{\theta -1}_\mathbb{A} \mathbb{A} \xi_2, \xi_1 - \xi_2 \big\rangle   \\
&=  \frac{\bar{\varPsi}_j'(|\xi_2|_\mathbb{A})}{|\xi_2|^{\theta}_\mathbb{A}} \big\langle |\xi_1|^{\theta -1}_\mathbb{A} \mathbb{A} \xi_1 - |\xi_2|^{\theta -1}_\mathbb{A} \mathbb{A} \xi_2, \xi_1 - \xi_2 \big\rangle.
\end{aligned}
\end{equation}

On the other hand, it follows from $ |\xi_2|_\mathbb{A} \leq |\xi_2|_\mathbb{A} + |\xi_1|_\mathbb{A} \leq 2|\xi_2|_\mathbb{A}$ and \eqref{cond_A} that
\begin{equation}\label{ineq_xi_2}
\frac{\bar{\varPsi}_j'(|\xi_2|_\mathbb{A})}{|\xi_2|^{\theta}_\mathbb{A}}
\approx \frac{\bar{\varPsi}_j'\big(|\xi_1|_\mathbb{A} + |\xi_2|_\mathbb{A}\big)}{\big(|\xi_1|_\mathbb{A} + |\xi_2|_\mathbb{A}\big)^{\theta}}
\approx \frac{\bar{\varPsi}_j'\big(|\xi_1| + |\xi_2|\big)}{\big(|\xi_1| + |\xi_2|\big)^{\theta}}.
\end{equation}
Moreover,  an algebraic inequality in \cite[(2.27)]{Mi02} gives
\begin{equation}\label{ineq_xi_3}
\big\langle |\xi_1|^{\theta -1}_\mathbb{A} \mathbb{A} \xi_1 - |\xi_2|^{\theta -1}_\mathbb{A} \mathbb{A} \xi_2, \xi_1 - \xi_2 \big\rangle
\gtrsim   \Big(|\xi_1| + |\xi_2|\Big)^{\theta - 1} |\xi_1 - \xi_2|^2
\end{equation}
Combining \eqref{ineq_xi}, \eqref{ineq_xi_2} \eqref{ineq_xi_3} we obtain the desired result (1).

We now turn to the proof of (2). Without loss of generality, we assume  $|\xi|_{\mathbb{A}_1} \leq |\xi|_{\mathbb{A}_2}$. 
A direct calculation yields that
\begin{equation}\label{ineq_A1_2}
\begin{aligned}
\Big| \frac{\bar{\varPsi}_j'(|\xi|_{\mathbb{A}_1})}{|\xi|_{\mathbb{A}_1}} \mathbb{A}_1 \xi - \frac{\bar{\varPsi}_j'(|\xi|_{\mathbb{A}_2})}{|\xi|_{\mathbb{A}_2}} \mathbb{A}_2 \xi \Big|
&\lesssim
\Big| \Big(\frac{\bar{\varPsi}_j'(|\xi|_{\mathbb{A}_1})}{|\xi|_{\mathbb{A}_1}} - \frac{\bar{\varPsi}_j'(|\xi|_{\mathbb{A}_2})}{|\xi|_{\mathbb{A}_2}}\Big) \,\mathbb{A}_1 \xi + \frac{\bar{\varPsi}_j'(|\xi|_{\mathbb{A}_2})}{|\xi|_{\mathbb{A}_2}} (\mathbb{A}_1 - \mathbb{A}_2) \xi \Big| \\
&\lesssim \Big|\frac{\bar{\varPsi}_j'(|\xi|_{\mathbb{A}_1})}{|\xi|_{\mathbb{A}_1}} - \frac{\bar{\varPsi}_j'(|\xi|_{\mathbb{A}_2})}{|\xi|_{\mathbb{A}_2}}\Big| \cdot |\xi| + \bar{\varPsi}_j'(|\xi|) \cdot | \mathbb{A}_1 - \mathbb{A}_2 |.
\end{aligned}
\end{equation}
For the first term of the right-hand side, we employ the mean value theorem and
$t\bar{\varPsi}_j''(t) \thickapprox \bar{\varPsi}_j'(t)$ to obtain that there 
exists a $z \in (|\xi|_{\mathbb{A}_1}, |\xi|_{\mathbb{A}_2})$ such that
\begin{equation}\label{eq_1}
\Big|\frac{\bar{\varPsi}_j'(|\xi|_{\mathbb{A}_1})}{|\xi|_{\mathbb{A}_1}} - \frac{\bar{\varPsi}_j'(|\xi|_{\mathbb{A}_2})}{|\xi|_{\mathbb{A}_2}}\Big|
=  \Big|\frac{z\bar{\varPsi}_j''(z) - \bar{\varPsi}_j'(z)}{z^2}\Big| \cdot \big||\xi|_{\mathbb{A}_1} - |\xi|_{\mathbb{A}_2}\big|  
\lesssim  \frac{\bar{\varPsi}_j'(z)}{z^2} \big||\xi|_{\mathbb{A}_1} - |\xi|_{\mathbb{A}_2}\big|.
\end{equation}
It follows from \eqref{cond_A} that 
$|\xi|^2 \approx |\xi|^2_{\mathbb{A}_1} \approx |\xi|^2_{\mathbb{A}_2}$, by 
$z \in (|\xi|_{\mathbb{A}_1}, |\xi|_{\mathbb{A}_2})$ which implies that
\begin{equation}\label{eq_2}
\begin{aligned}
\frac{\bar{\varPsi}_j'(z)}{z^2}\big||\xi|_{\mathbb{A}_1} - |\xi|_{\mathbb{A}_2}\big|
&\approx \frac{\bar{\varPsi}_j'(|\xi|)}{|\xi|^2}\frac{\big| |\xi|^2_{\mathbb{A}_1} - |\xi|^2_{\mathbb{A}_2}\big|}{|\xi|_{\mathbb{A}_1} + |\xi|_{\mathbb{A}_2}} \\
&\lesssim \frac{\bar{\varPsi}_j'(|\xi|)}{|\xi|^2}\frac{\big|\langle \mathbb{A}_1 \xi, \xi \rangle -  \langle \mathbb{A}_2 \xi, \xi \rangle\big|}{|\xi|} \\
&\lesssim \frac{\bar{\varPsi}_j'(|\xi|)}{|\xi|} \cdot \big| \mathbb{A}_1 - \mathbb{A}_2 \big|.
\end{aligned}
\end{equation}
Let us put \eqref{eq_1} and \eqref{eq_2} together, to obtain
\begin{equation}\label{ineq_A1_2_2}
\Big|\frac{\bar{\varPsi}_j'(|\xi|_{\mathbb{A}_1})}{|\xi|_{\mathbb{A}_1}} - \frac{\bar{\varPsi}_j'(|\xi|_{\mathbb{A}_2})}{|\xi|_{\mathbb{A}_2}}\Big| \cdot |\xi|
\lesssim \bar{\varPsi}_j'(|\xi|) \cdot \big| \mathbb{A}_1 - \mathbb{A}_2 \big|.
\end{equation}
We insert  \eqref{ineq_A1_2_2} into \eqref{ineq_A1_2} to conclude the desired result (2).
\end{proof}

Let $v_j \in u + W_0^{1,1}\big(\frac{1}{2}B_j\big)$ be a minimizer of the local frozen functional
\begin{equation}\label{fun_v}
	\mathcal{F}\big(v_j ; \tfrac{1}{2}B_j\big): = \int_{\frac{1}{2}B_j} \bar{\varPsi}_j\big(|Dv_j|_{\mathbb{A}}\big) \,dx.
\end{equation}
Note that $Dv_j$ as a minimizer of the local frozen functional, which enjoys the 
higher integrability governed by the boundary data, see \cite[Lemma 4.3]{De20} or \cite[Theorem 3.4]{BaC22}. More precisely, there exists a constant $c = c (n,  p, K) \geq 1$ such that
\begin{equation}\label{high_bound}
\Big(\fint_{\frac{1}{2}B_j} \Big(\bar{\varPsi}_j\big(|Dv_j|\big)\Big)^{1 + \varepsilon} \, dx\Big)^{\frac{1}{1 + \varepsilon}} \leq c \Big(\fint_{\frac{1}{2}B_j} \Big(\bar{\varPsi}_j\big(|Du|\big)\Big)^{1 + \varepsilon} \, dx\Big)^{\frac{1}{1 + \varepsilon}}
\end{equation}
with $\varepsilon$ as the same in Lemma \ref{higher_int_p}.

Next we give a comparison estimate between $Du$ and $Dv_j$. 
To this end, we introduce an auxiliary vector field 
$V_{\bar{\varPsi}_j}(\xi):\mathbb{R}^{n}\to \mathbb{R}^{n}$ defined by
\[
V_{\bar{\varPsi}_j}(\xi)
:=	\Big(\frac{\bar{\varPsi}_j'(|\xi|)}{|\xi|}
\Big)^{1/2} \xi.
\]
We immediately know that
$|V_{\bar{\varPsi}_j}(\xi)|^2 \thickapprox  \bar{\varPsi}_j(|\xi|)$  for
$\xi \in \mathbb{R}^{n}\setminus \{\mathbf{0}\}$; moreover, for any
$\xi_1, \xi_2 \in \mathbb{R}^{n}\setminus \{\mathbf{0}\}$ it holds
\begin{equation}\label{est_V}
\big|V_{\bar{\varPsi}_j}(\xi_1) - V_{\bar{\varPsi}_j}(\xi_2)\big|^2 \thickapprox \frac{\bar{\varPsi}_j'(|\xi_1|+ |\xi_2|)}{|\xi_1| + |\xi_2|} |\xi_1 - \xi_2|^2,
\end{equation}
where the implied constant depends only on $p$, see \cite[Lemma 2.4]{Di09}.


\begin{lemma}\label{comp_1}
Let $u \in W^{1,1}(\Omega)$ be a local minimizer of the functional \eqref{main_funl},  $v_j \in u + W_0^{1,1}\big(\frac{1}{2}B_j\big)$ be a minimizer of the local frozen functional \eqref{fun_v}, and $\gamma = \frac{2(2+\varepsilon)}{\varepsilon}$. 
If $a(x)$ is log-H\"older continuous as shown in \eqref{log_Hold} and 
$\mathbb{A}(x)$ satisfies \eqref{cond_A}, then there exists a constant 
$c = c(\texttt{data}) \geq 1$ with
\[
\fint_{\frac{1}{2}B_j} \big|V_{\bar{\varPsi}_j}(Du) - V_{\bar{\varPsi}_j}(Dv_j)\big|^2 \, dx \leq c \Big(\bar{\omega}_{a, \gamma}(r_j)\log(\tfrac{1}{r_j})\Big)^2\,\bar{\varPsi}_j\, \Big(\fint_{B_j} |Du|\, dx \Big)\,.
\]
\end{lemma}

\begin{proof}
 From the Euler-Lagrange equations of functionals \eqref{main_funl} and \eqref{fun_v},
 respectively, we use  $\phi = u - v_j \in W_0^{1,1}\big(\frac{1}{2}B_j\big)$ as 
 the test function to have that
\begin{gather*}
\int_{\frac{1}{2}B_j}\big\langle \frac{\varPsi'(x, |Du|_{\mathbb{A}})}{|Du|_{\mathbb{A}}} \mathbb{A} Du, Du - Dv_j \big\rangle\, dx 
= 0,\\
\int_{\frac{1}{2}B_j}\big\langle \frac{\bar{\varPsi}_j'(|Dv_j|_{\mathbb{A}})}{|Dv_j|_{\mathbb{A}}} \mathbb{A} Dv_j, Du - Dv_j \big\rangle\, dx = 0.
\end{gather*}
It follows from \eqref{ineq_1} of Lemma \ref{monotone} that
\begin{align*}
&\fint_{\frac{1}{2}B_j} \frac{\bar{\varPsi}_j'(|Du|+|Dv_j|)}{|Du| + |Dv_j|} \, |Du - Dv_j|^2\, dx   \\
&\leq  c \fint_{\frac{1}{2}B_j}\big\langle \frac{\bar{\varPsi}_j'(|Du|_{\mathbb{A}})}{|Du|_{\mathbb{A}}} {\mathbb{A}} Du    - \frac{\bar{\varPsi}_j'(|Dv_j|_{{\mathbb{A}}})}{|Dv_j|_{{\mathbb{A}}}} {\mathbb{A}} Dv_j, Du - Dv_j \big\rangle\, dx   \\
&=  c\fint_{\frac{1}{2}B_j}\big\langle \frac{\bar{\varPsi}_j'(|Du|_{{\mathbb{A}}})}{|Du|_{{\mathbb{A}}}} {\mathbb{A}} Du    - \frac{\varPsi'(x, |Du|_{\mathbb{A}})}{|Du|_{\mathbb{A}}} \mathbb{A} Du, Du - Dv_j \big\rangle\, dx   \\
&\leq   {c \fint_{\frac{1}{2}B_j} \big|a(x) - \bar{a}_j\big|\, \big|Du\big|^{p-1}\,\log\big(e + |Du|\big)\,\big|Du - Dv_j\big| \, dx}.
\end{align*}
By the Young inequality with {$\epsilon  > 0$} to be chosen later, we deduce 
that
\begin{equation}\label{Du_Dv_j}
\begin{aligned}
&\fint_{\frac{1}{2}B_j} \frac{\bar{\varPsi}_j'(|Du|+|Dv_j|)}{|Du| + |Dv_j|} \, |Du - Dv_j|^2\, dx  \\
&\leq \fint_{\frac{1}{2}B_j} |a(x) - \bar{a}_j|\, |Du|^{p-1}\,\log(e + |Du|)\,|Du - Dv_j| \, dx   \\
&\leq  c \fint_{\frac{1}{2}B_j} |a(x) - \bar{a}_j|\, \Big(|Du|+|Dv_j|\Big)^{\frac{p}{2} + \frac{p-2}{2}} \log(e + |Du|)\,|Du - Dv_j| \, dx \\
&\leq  c_\epsilon \fint_{\frac{1}{2}B_j} |a(x) - \bar{a}_j|^2\Big(|Du|+|Dv_j|\Big)^{p} \Big(\log(e + |Du|)\Big)^2 \, dx   \\
&\quad +\, \epsilon \fint_{\frac{1}{2}B_j} \Big({|Du| + |Dv_j|}\Big)^{p-2} \, |Du - Dv_j|^2\, dx  \\
&=  c_\epsilon \, I_1 + \epsilon I_2.
\end{aligned}
\end{equation}
To estimate $I_1$, we employ the H\"older inequality and Lemma \ref{log} to obtain that
\begin{align*}
I_1 &\leq  \Big(\fint_{\frac{1}{2}B_j} |a(x) - \bar{a}_j|^\frac{2(2+\varepsilon)}{\varepsilon}\,dx
 \Big)^\frac{\varepsilon}{2+\varepsilon}  
 \Big(\fint_{\frac{1}{2}B_j} \Big(|Du|+|Dv_j|\Big)^{\frac{p(2+\varepsilon)}{2}} \Big(\log(e + |Du|)\Big)^{2+\varepsilon} \, dx\Big)^\frac{2}{2+\varepsilon}   \\
&\leq  c \big(\bar{\omega}_{a, \gamma}(r_j)\big)^2 \Big(\Big(\log\Big(\tfrac{1}{r_j}\Big)\Big)^{2+\varepsilon}
\Big(\fint_{\frac{1}{2}B_j} \Big(|Du|+|Dv_j|\Big)^{p(1+\varepsilon)} \, dx\Big)^\frac{2+\varepsilon}{2(1+\varepsilon)}\Big)^\frac{2}{2+\varepsilon}   \\
&\leq  c \Big(\bar{\omega}_{a, \gamma}(r_j)\log(\tfrac{1}{r_j})\Big)^2\Big(\fint_{\frac{1}{2}B_j} \Big(\bar{\varPsi}_j\big(|Du|\big)\Big)^{1 + \varepsilon} \, dx\Big)^{\frac{1}{1 + \varepsilon}}  \\
&\leq  c \Big(\bar{\omega}_{a, \gamma}(r_j)\log(\tfrac{1}{r_j})\Big)^2\,\bar{\varPsi}_j\, \Big(\fint_{B_j} |Du|\, dx \Big)\,,
\end{align*}
where we used  Lemma \ref{higher_int_p} in the last inequality.
For $I_2$, it is obvious that
$$
I_2 \leq c \fint_{\frac{1}{2}B_j} \frac{\bar{\varPsi}_j'(|Du|+|Dv_j|)}{|Du| + |Dv_j|} \, |Du - Dv_j|^2\, dx.
$$
Let us now insert the estimates of $I_1$ and $I_2$ into \eqref{Du_Dv_j}, 
and  pick $\epsilon$ small enough so as to absorb the term $\epsilon I_2$ 
into the left-hand side of \eqref{Du_Dv_j}, then we conclude the desired 
estimate from \eqref{est_V}.
\end{proof}

In what follows, we introduce the local limiting frozen functional.
Let $w_j \in v_j + W_0^{1,1}\big(\frac{1}{4}B_j\big)$ be a minimizer of the 
local functional
\begin{equation}\label{fun_w}
	\mathcal{F}(w_j; \tfrac{1}{4}B_j): = \int_{\frac{1}{4}B_j} \bar{\varPsi}_j\,\big(|Dw_j|_{\bar{\mathbb{A}}_j}\big) \,dx.
\end{equation}
It is obvious that the minimizer $w_j$ also has the higher integrability 
governed by the boundary value. To be concrete, we have
\begin{equation}\label{high_bound_w}
\begin{aligned}
\Big(\fint_{\frac{1}{4}B_j} \Big(\bar{\varPsi}_j\big(|Dw_j|\big)\Big)^{1 + \varepsilon} \, dx\Big)^{\frac{1}{1 + \varepsilon}}
&\lesssim \Big(\fint_{\frac{1}{4}B_j} \Big(\bar{\varPsi}_j\big(|Dv_j|\big)\Big)^{1 + \varepsilon} \, dx\Big)^{\frac{1}{1 + \varepsilon}}   \\
&\lesssim \Big(\fint_{\frac{1}{2}B_j} \Big(\bar{\varPsi}_j\big(|Du|\big)\Big)^{1 + \varepsilon} \, dx\Big)^{\frac{1}{1 + \varepsilon}}
\end{aligned}
\end{equation}
where $\varepsilon$ is the same as in Lemma \ref{higher_int_p}. Moreover, by considering $w_j$ as the minimizer of frozen functional with constant coefficients, 
it enjoys the local gradient H\"older continuity, for detail see 
\cite[Lemma 4.2]{ChKW26}.

\begin{lemma}\label{holder_w}
Let  $w_j \in W^{1, \bar{\varPsi}_j}(B_j)$ be a minimizer of the limiting 
frozen functional \eqref{fun_w} with $\bar{\mathbb{A}}_j$ satisfying 
\eqref{cond_A}. Then there are positive constants $\alpha = \alpha(n,  p) \in (0, 1)$, $c_1 = c_1(n,  p)$ and $\tau_1 = \tau_1(n,  p)$ such that
\begin{equation}\label{holder_Dw}
\operatorname{osc}_{\tau B_j}  Dw_j  \leq c_1 \tau^{\alpha} \fint_{B_j} |Dw_j - \langle Dw_j \rangle _{B_j}|\, dx \quad \text{for any } \tau \in (0, \tau_1).
\end{equation}
\end{lemma}


We are now in a position to present the  comparison estimate between $Dv_j$ and $Dw_j$. More precisely, we have the following lemma.

\begin{lemma}\label{comp_2}
Let $v_j \in u + W_0^{1,1}\big(\frac{1}{2}B_j\big)$ be a minimizer of local 
functional \eqref{fun_v}, $w_j \in v_j + W_0^{1,1}\big(\frac{1}{4}B_j\big)$ 
be a minimizer of limiting functional\eqref{fun_w}, and  
$\gamma = \frac{2(2+\varepsilon)}{\varepsilon}$. 
If $\mathbb{A}(x)$ satisfies \eqref{cond_A}, then there exists a constant  
$c = c(\texttt{data}) \geq 1$  with
\[
\fint_{\frac{1}{4}B_j} \big|V_{\bar{\varPsi}_j}(Dv_j) - V_{\bar{\varPsi}_j}(Dw_j)\big|^2\, dx \leq c \big(\bar{\omega}_{\mathbb{A}, \gamma}(r_j)\big)^2\, \bar{\varPsi}_j\Big(\fint_{B_j} |Du|\, dx \Big).
\]
\end{lemma}

\begin{proof} We use $\phi = v_j - w_j \in W_0^{1,1}\big(\frac{1}{4}B_j\big)$ as the test function to
the Euler-Lagrange equations of functionals \eqref{fun_v} and \eqref{fun_w}, respectively, to conclude that
\begin{gather*}
\int_{\frac{1}{4}B_j} \big\langle \frac{\bar{\varPsi}_j'(|Dv_j|_{\mathbb{A}})}{|Dv_j|_{\mathbb{A}}} \mathbb{A} Dv_j, Dv_j - Dw_j \big\rangle\, dx = 0, \\
\int_{\frac{1}{4}B_j}\big\langle \frac{\bar{\varPsi}_j'(|Dw_j|_{\bar{\mathbb{A}}_j})}{|Dw_j|_{\bar{\mathbb{A}}_j}} \bar{\mathbb{A}}_j Dw_j, Dv_j - Dw_j \big\rangle\, dx = 0.
\end{gather*}
By \eqref{ineq_1} we have that
\begin{align*}
&\fint_{\frac{1}{4}B_j}
 \frac{\bar{\varPsi}_j'(|Dv_j|+|Dw_j|)}{|Dv_j| + |Dw_j|} \, |Dv_j - Dw_j|^2\, dx   \\
&\leq  c \fint_{\frac{1}{4}B_j}\big\langle \frac{\bar{\varPsi}_j'(|Dv_j|_{\bar{\mathbb{A}}_j})}{|Dv_j|_{\bar{\mathbb{A}}_j}} \bar{\mathbb{A}}_j Dv_j    - \frac{\bar{\varPsi}_j'(|Dw_j|_{\bar{\mathbb{A}}_j})}{|Dw_j|_{\bar{\mathbb{A}}_j}} \bar{\mathbb{A}}_j Dw_j, Dv_j - Dw_j \big\rangle\, dx   \\
&=  c\fint_{\frac{1}{4}B_j}\big\langle \frac{\bar{\varPsi}_j'(|Dv_j|_{\bar{\mathbb{A}}_j})}{|Dv_j|_{\bar{\mathbb{A}}_j}} \bar{\mathbb{A}}_j Dv_j    - \frac{\bar{\varPsi}_j'(|Dv_j|_{\mathbb{A}})}{|Dv_j|_{\mathbb{A}}} \mathbb{A} Dv_j, Dv_j - Dw_j \big\rangle\, dx.
\end{align*}
By \eqref{ineq_2},  $t\bar{\varPsi}_j'(t) \approx \bar{\varPsi}_j(t)$ and the 
Young inequality, we obtain that
\begin{equation}\label{Dv_Dw_j}
\begin{aligned}
&\fint_{\frac{1}{4}B_j} \frac{\bar{\varPsi}_j'(|Dv_j|+|Dw_j|)}{|Dv_j| + |Dw_j|} \, |Dv_j - Dw_j|^2\, dx   \\
&\leq  c \fint_{\frac{1}{4}B_j} \, \bar{\varPsi}_j'\big(|Dv_j| + |Dw_j|\big)\, | \bar{\mathbb{A}}_j - \mathbb{A} |\,|Dv_j - Dw_j| \,dx   \\
&\leq  c \fint_{\frac{1}{4}B_j} |\bar{\mathbb{A}}_j - \mathbb{A}|\Big(\bar{\varPsi}_j\big(|Dv_j| + |Dw_j|\big)\Big)^{1/2}\Big(\frac{\bar{\varPsi}_j'\big(|Dv_j| + |Dw_j|\big)}{|Dv_j| + |Dw_j|}\Big)^{1/2}\, |Dv_j - Dw_j| \, dx   \\
&\leq  c \fint_{\frac{1}{4}B_j} |\bar{\mathbb{A}}_j - \mathbb{A}|^2\, \Big(\bar{\varPsi}_j(|Dv_j|) + \bar{\varPsi}_j(|Dw_j|)\Big) \, dx   \\
&\quad +\, \frac{1}{4} \fint_{\frac{1}{4}B_j} \frac{\bar{\varPsi}_j'\big(|Dv_j| + |Dw_j|\big)}{|Dv_j| + |Dw_j|}\, |Dv_j - Dw_j|^2 \, dx.
\end{aligned}
\end{equation}
To estimate the first term of the right-hand side, we employ the H\"older inequality, \eqref{high_bound_w} and \eqref{high_bound} to show that
\begin{equation}\label{Dv_Dw_j_1}
\begin{aligned}
&\fint_{\frac{1}{4}B_j} |\bar{\mathbb{A}}_j - \mathbb{A}|^2\,  \Big(\bar{\varPsi}_j(|Dv_j|) + \bar{\varPsi}_j(|Dw_j|)\Big) \, dx   \\
&\leq  c \Big(\fint_{\frac{1}{4}B_j} |\bar{\mathbb{A}}_j - \mathbb{A}|^{\frac{2(2+\varepsilon)}{\varepsilon}}\,dx \Big)^\frac{\varepsilon}{2+\varepsilon}\, \Big(\fint_{\frac{1}{4}B_j} \Big(\bar{\varPsi}_j(|Dv_j|)\Big)^\frac{2+\varepsilon}{2} \, dx \Big)^\frac{2}{2+\varepsilon }  \\
&\quad + c \Big(\fint_{\frac{1}{4}B_j} |\bar{\mathbb{A}}_j - \mathbb{A}|^{\frac{2(2+\varepsilon)}{\varepsilon}}\,dx \Big)^\frac{\varepsilon}{2+\varepsilon}\, \Big(\fint_{\frac{1}{4}B_j} \Big(\bar{\varPsi}_j(|Dw_j|)\Big)^\frac{2+\varepsilon}{2} \, dx \Big)^\frac{2}{2+\varepsilon }  \\
&\leq  c \big(\bar{\omega}_{\mathbb{A}, \gamma}(r_j)\big)^2\, \Big(\fint_{\frac{1}{4}B_j} \Big(\bar{\varPsi}_j(|Du|)\Big)^\frac{2+\varepsilon}{2} \, dx \Big)^\frac{2}{2+\varepsilon }  \\
&\leq  c \big(\bar{\omega}_{\mathbb{A}, \gamma}(r_j)\big)^2\, \bar{\varPsi}_j\Big(\fint_{B j} |Du|\, dx \Big),
\end{aligned}
\end{equation}
where we used  Lemma \ref{higher_int_p} in the last step. 
Putting \eqref{Dv_Dw_j_1} into \eqref{Dv_Dw_j}, we obtain the desired estimate in accordance with \eqref{est_V}.
\end{proof}

With Lemmas \ref{comp_1} and \ref{comp_2} in hand, we devote the following to the comparison estimate between $Du$ and $Dw_j$ in the $L^1$ senses. Let
\[
\varUpsilon(\rho) = \bar{\omega}_{\mathbb{A}, \gamma}(\rho) + \bar{\omega}_{a, \gamma}(\rho)\log(\tfrac{1}{\rho}).
\]


\begin{lemma}\label{comp_L_1}
Let $u \in W^{1,1}(\Omega)$ be a local minimizer of the functional \eqref{main_funl}, and $w_j \in v_j + W_0^{1,1}\big(\frac{1}{4}B_j\big)$ be a minimizer of the limit functional \eqref{fun_w}. If $a(x)$ is log-H\"older continuous as \eqref{log_Hold} and $\mathbb{A}(x)$ satisfies \eqref{cond_A}, then there exist a constant  
$c_2 = c_2\,(\texttt{data}) \geq 1$ such that
\[
\fint_{\frac{1}{4}B_j} |Du - Dw_j|\, dx
\leq c_2 \big(\varUpsilon(r_j)\big)^\frac{1}{p+1}\,\fint_{B_j} |Du|\, dx.
\]
\end{lemma}


\begin{proof}
Note that the Young inequality yields
\begin{align*}
|Du - Dw_j| 
&=  \Big(\varUpsilon(r_j)\,\big(|Du| + |Dw_j|\big)\Big)^{1/2}\cdot\Big(\varUpsilon(r_j)\,\big(|Du| + |Dw_j|\big)\Big)^{-1/2} |Du - Dw_j| \\
&\leq  \frac{1}{2}\Big(\varUpsilon(r_j)\,\big(|Du| + |Dw_j|\big) + \Big(\varUpsilon(r_j)\,\big(|Du| + |Dw_j|\big)\Big)^{-1}\,|Du - Dw_j|^2\Big)\,.
\end{align*}
By $\bar{\varPsi}_j(t) \approx t\,\bar{\varPsi}'_j(t)$, then we conclude that
\begin{equation}\label{Du-Dw}
\begin{aligned}
&\bar{\varPsi}_j\big(|Du - Dw_j|\big) \\[6pt]
&\leq  c \bar{\varPsi}'_j\big(|Du| + |Dw_j|\big)\,|Du - Dw_j|  \\
&\leq   c \bar{\varPsi}'_j\big(|Du| + |Dw_j|\big)\Big(\varUpsilon(r_j)\,\big(|Du| + |Dw_j|\big)  +  \Big(\varUpsilon(r_j)\,\big({|Du| + |Dw_j|}\big)\Big)^{-1}|Du - Dw_j|^2\Big) \\
&=   c \Big(\varUpsilon(r_j)\bar{\varPsi}_j\big(|Du| + |Dw_j|\big)  +  \big(\varUpsilon(r_j)\big)^{-1}\frac{\bar{\varPsi}'_j\big(|Du| + |Dw_j|\big)}{|Du| + |Dw_j|}|Du - Dw_j|^2\Big)\,.
\end{aligned}
\end{equation}
 For the second term of the right-hand side, by \eqref{est_V}, Lemmas \ref{comp_1} 
 and \ref{comp_2} we deduce that
\begin{equation}\label{Du_Dw}
\begin{aligned}
&\fint_{\frac{1}{4}B_j} \frac{\bar{\varPsi}_j'(|Du|+|Dw_j|)}{|Du| + |Dw_j|} \, |Du - Dw_j|^2\, dx   \\
&\leq  c \fint_{\frac{1}{4}B_j} \big|V_{\bar{\varPsi}_j}(Du) - V_{\bar{\varPsi}_j}(Dw_j)\big|^2\, dx  \\
&\leq  c \fint_{\frac{1}{4}B_j} \big|V_{\bar{\varPsi}_j}(Du) - V_{\bar{\varPsi}_j}(Dv_j)\big|^2\, dx + c \fint_{\frac{1}{4}B_j} \big|V_{\bar{\varPsi}_j}(Dv_j) - V_{\bar{\varPsi}_j}(Dw_j)\big|^2\, dx   \\
&\leq  c \big(\varUpsilon(r_j)\big)^2\,\bar{\varPsi}_j\Big(\fint_{B_j} |Du|\, dx \Big).
\end{aligned}
\end{equation}
We integrate \eqref{Du-Dw} over $\frac{1}{4}B_j$, and use \eqref{high_bound_w} \eqref{Du_Dw} to obtain that
\begin{equation}\label{phi_u_w}
\begin{aligned}
\fint_{\frac{1}{4}B_j} \bar{\varPsi}_j\big(|Du - Dw_j|\big)\, dx
&\leq   c  \varUpsilon(r_j)\fint_{\frac{1}{4}B_j} \bar{\varPsi}_j\big(|Du|+ |Dw_j|\big) \, dx  \\
&\quad + c \big(\varUpsilon(r_j)\big)^{-1}\fint_{\frac{1}{4}B_j} \frac{\bar{\varPsi}'_j\big(|Du| + |Dw_j|\big)}{|Du| + |Dw_j|}|Du - Dw_j|^2 \, dx \\
&\leq   c  \varUpsilon(r_j)\fint_{\frac{1}{2}B_j} \bar{\varPsi}_j\big(|Du|\big) \, dx + c \varUpsilon(r_j)\, \bar{\varPsi}_j\, \Big(\fint_{B_j} |Du|\, dx\Big) \\
&\leq   c  \varUpsilon(r_j)\, \bar{\varPsi}_j\, \Big(\fint_{B_j} |Du|\, dx\Big),
\end{aligned}
\end{equation}
We used Lemma \ref{higher_int_p} in the last inequality. 
Since $\varUpsilon(r_j) \leq c$ due to the boundness of $\mathbb{A}$ and 
\eqref{DMO_L}, we employ the Jensen inequality, \eqref{phi_u_w} and 
\eqref{est_log_1} to have 
\begin{align*}
\bar{\varPsi}_j\Big(\fint_{\frac{1}{4}B_j} |Du-Dw_j|\, dx\Big)
&\leq   \fint_{\frac{1}{4}B_j} \bar{\varPsi}_j\big(|Du - Dw_j|\big)\, dx  \\
&\leq   c  \varUpsilon(r_j)\, \bar{\varPsi}_j\, \Big(\fint_{B_j} |Du|\, dx\Big)  \\
&\leq   \bar{\varPsi}_j\, \Big(c \big(\varUpsilon(r_j)\big)^\frac{1}{p+1}  \fint_{B_j} |Du|\, dx\Big)\,,
\end{align*}
which yields the desired estimate in accordance with the non-decreasing property 
of $\bar{\varPsi}_j$.
\end{proof}

On the basis of Lemma \ref{comp_L_1}, we  give a decay estimate of the energy functional.

\begin{lemma}\label{lem_decay}
Let $u \in W^{1,1}(\Omega)$ be a local minimizer of the functional \eqref{main_funl}.
If $a(x)$ is log-H\"older continuous as shown in \eqref{log_Hold} and 
$\mathbb{A}(x)$ satisfies \eqref{cond_A}, then for any 
$\theta \in (0, \min\{\frac{1}{4}, \tau_1\})$ there exists 
$c_3 = c_3\,(\texttt{data}) \geq 1$ such that
\[
\fint_{B_{j+1}} |Du - \langle Du \rangle _{B_{j+1}}|\, dx
\leq c_3\Big(\theta^{\,\alpha} \fint_{B_j}
|Du - \langle Du \rangle _{B_j}|\, dx + \theta^{-n}\big(\varUpsilon(r_j)\big)^\frac{1}{p+1} \fint_{B_j} |Du|\, dx\Big)\,.
\]
\end{lemma}

\begin{proof} Note that
\begin{equation}\label{dacy_1}
\begin{aligned}
&\fint_{B_{j+1}} |Du - \langle Du \rangle _{B_{j+1}}|\, dx \\
&\leq  2\fint_{B_{j+1}} |Du - \langle Dw_j \rangle _{B_{j+1}}|\, dx  \\
&\leq  2\fint_{B_{j+1}} |Du - Dw_j|\, dx + 2\fint_{B_{j+1}} |Dw_j - \langle Dw_j \rangle _{B_{j+1}}|\, dx  \\
&\leq  c \theta^{-n}\fint_{\frac{1}{4}B_j} |Du - Dw_j|\, dx + c \theta^{\,\alpha} \fint_{\frac{1}{4}B_j} |Dw_j - \langle Dw_j \rangle _{\frac{1}{4}B_j}|\, dx  \\
&\leq  \theta^{-n}\fint_{\frac{1}{4}B_j} |Du - Dw_j|\, dx + c \theta^{\,\alpha}\Big(
 \fint_{\frac{1}{4}B_j} |Dw_j - Du|\, dx + \fint_{\frac{1}{4}B_j} |Du - \langle Du \rangle _{\frac{1}{4}B_j}|\, dx\Big),
\end{aligned}
\end{equation}
where we used Lemma \ref{holder_w} in the third inequality. 
By confining $0<\theta <\frac 14$ we obtain that
\begin{align*}
\fint_{B_{j+1}} |Du - \langle Du \rangle _{B_{j+1}}|\, dx
&\leq  c \theta^{-n} \fint_{\frac{1}{4}B_j} |Du - Dw_j|\, dx +   c \theta^{\,\alpha} \fint_{\frac{1}{4}B_j} |Du - \langle Du \rangle _{\frac{1}{4}B_j}|\, dx \\
&\leq  c \theta^{-n}\big(\varUpsilon(r_j)\big)^\frac{1}{p+1} \fint_{B_j} |Du|\, dx + c \theta^{\,\alpha} \fint_{B_j} |Du - \langle Du \rangle _{B_j}|\, dx,
\end{align*}
where we used Lemma  \ref{comp_L_1} in the last inequality, and we complete 
the proof.
\end{proof}

In the sequel, we  show a comparison estimate between $Du$ and $Dw_j$ 
so as to be used to prove the iterating decay of $Du$.

\begin{lemma}\label{lem_comp_ul}
Let $u \in W^{1, 1}(\Omega)$ be a local minimizer of the functional 
\eqref{main_funl}, and $w_j \in v_j + W_0^{1,1}\big(\frac{1}{4}B_j\big)$ be a 
minimizer of the limit functional \eqref{fun_w}. Assume that $a(x)$ is 
log-H\"older continuous as shown in  \eqref{log_Hold} and $\mathbb{A}(x)$ satisfies 
\eqref{cond_A}. If for $\lambda > 0$ and $\Lambda \geq 1$ one has
\begin{equation}\label{comp_cond}
\frac{\lambda}{\Lambda} \leq \inf_{\frac{1}{4}B_{j+1}} |Dw_j|
\quad \text{and} \quad
\fint_{B_j} |Du|\, dx \leq \lambda,
\end{equation}
then there exists $c = c (\texttt{data}) \geq 1$ such that
\[
\fint_{\frac{1}{4}B_{j+1}} |Du - Dw_j|\, dx \leq {c  \theta^{-n} \Lambda^{p}}\,\varUpsilon(r_j)\,\lambda.
\]
\end{lemma}


\begin{proof}
By \eqref{high_bound_w}  and Lemma \ref{higher_int_p}, we have
\[
\bar{\varPsi}_j\Big(\fint_{\frac{1}{4}B_j} |Dw_j|\, dx\Big)
\leq \fint_{\frac{1}{4}B_j} \bar{\varPsi}_j\, \big(|Dw_j|\big)\, dx
\leq c \bar{\varPsi}_j\Big(\fint_{B_j} |Du|\, dx\Big)\,,
\]
which follows from  \eqref{est_log_1} that
\begin{equation}\label{ener_est_L_1}
\fint_{\frac{1}{4}B_j} |Dw_j|\, dx \leq c \fint_{B_j} |Du|\, dx\,.
\end{equation}
Therefore, by the Young inequality and $0<\theta <\frac 14$  we obtain that
\begin{equation}\label{comp_ul_1}
\begin{aligned}
\fint_{\frac{1}{4}B_{j+1}} |Du - Dw_j|\, dx
&\leq   \varUpsilon(r_j) \fint_{\frac{1}{4}B_{j+1}} \Big(|Du| + |Dw_j|\Big)\, dx  +  \frac{1}{\varUpsilon(r_j)} \fint_{\frac{1}{4}B_{j+1}} \frac{|Du - Dw_j|^2}{|Du| + |Dw_j|}\, dx  \\
&\leq   c  \theta^{-n} \varUpsilon(r_j) \fint_{B_j} |Du|\, dx   +  \frac{1}{\varUpsilon(r_j)} \fint_{\frac{1}{4}B_{j+1}} \frac{|Du - Dw_j|^2}{|Du| + |Dw_j|}\, dx   \\
&\leq  c  \theta^{-n} \varUpsilon(r_j) \lambda  +  \frac{1}{\varUpsilon(r_j)} \fint_{\frac{1}{4}B_{j+1}} \frac{|Du - Dw_j|^2}{|Du| + |Dw_j|}\, dx.
\end{aligned}
\end{equation}
To estimate the second term of the right-hand side, we notice that
\[
1 \leq \frac{\bar{\varPsi}'_j(|Dw_j|)}{\bar{\varPsi}'_j(\lambda/\Lambda)} \leq \frac{\bar{\varPsi}'_j(|Du| + |Dw_j|)}{\Lambda^{-p}\bar{\varPsi}'_j(\lambda)}
 \]
in $\frac{1}{4}B_{j+1}$ because of \eqref{comp_cond} and \eqref{est_log_2}.
Then we employ \eqref{Du_Dw}  to obtain that
\begin{equation}\label{comp_ul_3}
\begin{aligned}
\fint_{\frac{1}{4}B_{j+1}} \frac{|Du - Dw_j|^2}{|Du| + |Dw_j|}\, dx
&\leq  \fint_{\frac{1}{4}B_{j+1}} \frac{\bar{\varPsi}'_j(|Du| + |Dw_j|)}{\Lambda^{-p}\bar{\varPsi}'_j(\lambda)}\frac{|Du - Dw_j|^2}{|Du| + |Dw_j|}\, dx  \\
&\leq  \frac{\theta^{-n}\Lambda^p}{\bar{\varPsi}'_j(\lambda)} \fint_{\frac{1}{4}B_j} \frac{\bar{\varPsi}'_j(|Du| + |Dw_j|)}{|Du| + |Dw_j|} |Du - Dw_j|^2\, dx  \\
&\leq  \frac{c \theta^{-n} \Lambda^{p}}{\bar{\varPsi}'_j(\lambda)}\big( \varUpsilon(r_j)\big)^2 \bar{\varPsi}_j\,  \Big(\fint_{B_j} |Du|\, dx\Big)  \\
&\leq  \frac{c \theta^{-n} \Lambda^{p}}{\bar{\varPsi}'_j(\lambda)}\big( \varUpsilon(r_j)\big)^2 \bar{\varPsi}_j(\lambda)  \\
&\leq  {c  \theta^{-n} \Lambda^{p}}(\varUpsilon(r_j))^2 \lambda,
\end{aligned}
\end{equation}
where we used $\bar{\varPsi}_j(\lambda) \approx \lambda \bar{\varPsi}'_j(\lambda)$ in the last step.
By inserting \eqref{comp_ul_3} into \eqref{comp_ul_1} we complete the proof.
\end{proof}

With Lemma \ref{lem_comp_ul}, we are able to show the iterating decay 
associated to $Du$

\begin{lemma}\label{dacy_u}
Let $u \in W^{1, 1}(\Omega)$ be a local minimizer of the functional 
\eqref{main_funl}. Assume that $a(x)$ is log-H\"older continuous as 
\eqref{log_Hold} and $\mathbb{A}(x)$ satisfies \eqref{cond_A}. 
For a given $\Lambda \geq 1$, if there is a positive constant 
$\theta_1 = \theta_1(\texttt{data}, \Lambda)$, such that for any 
$\theta \in (0, \theta_1)$ and $\lambda > 0$ it holds
\begin{equation}\label{cond_dacy_u}
\frac{\lambda}{\Lambda} \leq \fint_{B_{j+2}} |Du|\, dx\,,
\quad
\fint_{B_j} |Du|\, dx \leq \lambda
\quad \text{and} \quad
\varUpsilon(r_j) \leq \Big(\frac{\theta^{3n}}{c_2 \Lambda}\Big)^{p+1},
\end{equation}
then there exists a constant 
$c^*_1 = c^*_1(\texttt{data}, \theta, \Lambda) \geq 1$ such that
\[
\fint_{B_{j+2}} |Du - \langle Du \rangle _{B_{j+2}}|\, dx \leq c_4 \ \theta^\alpha \fint_{B_{j+1}} |Du - \langle Du \rangle _{B_{j+1}}|\, dx + c^*_1\varUpsilon(r_j)\,\lambda
\]
with $c_4$ being independent of $\theta$.
\end{lemma}

\begin{proof}
Let us use Lemma \ref{comp_L_1} to obtain that
\begin{align*}
\fint_{B_{j+2}} |Du - Dw_j|\, dx
&\leq   \theta^{-2n} \fint_{\frac{1}{4}B_j} |Du - Dw_j|\, dx  \\
&\leq   c_2\theta^{-2n} \big(\varUpsilon(r_j)\big)^\frac{1}{p+1}  \fint_{B_j} |Du|\, dx  \\
&\leq   \frac{\lambda}{2\Lambda}
\end{align*}
with the last inequality because of \eqref{cond_dacy_u}. 
Therefore, by $\fint_{B_{j+2}} |Du|\, dx \geq \frac{\lambda}{\Lambda}$ as in \eqref{cond_dacy_u} we conclude that
\begin{equation}\label{sup_w_1}
\begin{aligned}
\sup_{\frac{1}{4}B_{j+1}}|Dw_j|
&\geq \fint_{B_{j+2}} |Dw_j|\, dx   \\
&\geq \fint_{B_{j+2}} |Du|\, dx - \fint_{B_{j+2}} |Du - Dw_j|\, dx   \\
&\geq \frac{\lambda}{\Lambda} - \fint_{B_{j+2}} |Du - Dw_j|\, dx \\
&\geq \frac{\lambda}{2\Lambda}.
\end{aligned}
\end{equation}
On the other hand, let us confine $0<\theta_1 \leq \tau_1$ and 
$c \theta_1^{\,\alpha} \leq \frac{1}{4\Lambda}$  with $\tau_1$ as in Lemma \ref{holder_w}. For any $0<\theta\le \theta_1$, then by \eqref{holder_Dw} and \eqref{ener_est_L_1} we obtain that
\begin{equation}\label{osc_w_1}
\big|\underset{\frac{1}{4}B_{j+1}}{\mathrm{osc}}Dw_j \big|\leq c  \theta^{\,\alpha} \fint_{\frac{1}{4}B_j} |Dw_j|\, dx \leq c \theta^{\,\alpha} \fint_{B_j} |Du|\, dx \le c \theta^{\,\alpha} \lambda \le  \frac{\lambda}{4\Lambda},
\end{equation}
where we used  $\fint_{B_j} |Du|\, dx \leq \lambda$ in \eqref{cond_dacy_u} in the last second inequality.

Let us put \eqref{sup_w_1} and \eqref{osc_w_1} together, to obtain
\begin{equation}\label{inf_w_1}
\inf_{\frac{1}{4}B_{j+1}} |Dw_j|
\geq \sup_{\frac{1}{4}B_{j+1}}|Dw_j| - \big|\underset{\frac{1}{4}B_{j+1}}{\mathrm{osc}}Dw_j\big|
\geq \frac{\lambda}{2\Lambda}- \frac{\lambda}{4\Lambda} = \frac{\lambda}{4\Lambda}.
\end{equation}
According to the validity of \eqref{cond_dacy_u} and \eqref{inf_w_1}, we use 
Lemma \ref{lem_comp_ul} to obtain
\[
\fint_{\frac{1}{4}B_{j+1}} |Du - Dw_j|\, dx \leq {c  \theta^{-n} \Lambda^{p}}\,\varUpsilon(r_j)\,\lambda.
\]
Finally, we employ the same approach as in \eqref{dacy_1}, to conclude that
\begin{align*}
\fint_{B_{j+2}} |Du - \langle Du \rangle _{B_{j+2}}|\, dx
&\leq   c \theta^{-n}\fint_{\frac{1}{4}B_{j+1}} |Du - Dw_j|\, dx  +c \theta^{\,\alpha} \fint_{\frac{1}{4}B_{j+1}} |Du - \langle Du \rangle _{\frac{1}{4}B_{j+1}}|\, dx \\
&\leq   {c  \theta^{-2n} \Lambda^{p}}\,\varUpsilon(r_j)\,\lambda +c \theta^{\,\alpha} \fint_{\frac{1}{4}B_{j+1}} |Du - \langle Du \rangle _{\frac{1}{4}B_{j+1}}|\, dx,
\end{align*}
which yields the desired estimate.
\end{proof}

\section{Proof of main result}

In this section, we prove the continuity of gradient to the local minimizers of the functional \eqref{main_funl} by imposing DMO condition on $a(x)$ and 
$\mathbb{A}(x)$ as in \eqref{DMO}. First, we show  a control relation by an integral 
to the DMO assumption on $a(x)$ and $\mathbb{A}(x)$ in \eqref{DMO}.

\begin{lemma}\label{lem_gap} 
Let $r_j = \theta^j r$ for $\theta \in \big(0,  1/4\big) $ and 
$ j = 0, 1, 2, \dots$. If both $a(x)$ and $\mathbb{A}(x)$ are imposed on 
the DMO assumption as \eqref{DMO}, then there exists 
$c^*_2 = c^*_2(\texttt{data},\theta) \geq 1$ such that
\begin{equation}
\sum_{j=0}^{\infty} \varUpsilon(r_j) \leq c^*_2\int_0^{2r} \varUpsilon(\rho) \, \frac{d\rho}{\rho}.
\end{equation}
\end{lemma}

\begin{proof}
A direct calculation yields that
\begin{equation}\label{claim}
\begin{aligned}
\int_0^{2r} \varUpsilon(\rho) \, \frac{d\rho}{\rho}
&=  \sum_{j=0}^{\infty} \int_{r_{j+1}}^{r_j} \bar{\omega}_{a,\gamma}(\rho) 
 \log\big(\frac{1}{\rho}\big) \, \frac{d\rho}{\rho}  +  \int_r^{2r}
 \bar{\omega}_{a,\gamma}(\rho) \log\big(\frac{1}{\rho}\big) \, \frac{d\rho}{\rho} \\
&\quad + \sum_{j=0}^{\infty} \int_{r_{j+1}}^{r_j} \bar{\omega}_{\mathbb{A},\gamma}(\rho) \, \frac{d\rho}{\rho}  +  \int_r^{2r} \bar{\omega}_{\mathbb{A},\gamma}(\rho) \, \frac{d\rho}{\rho} \\
&:= I_1+I_2+I_3+I_4.
\end{aligned}
\end{equation}
For $I_1$, let $\rho \in (r_{j+1}, r_j)$, and for any $y \in \Omega$ we obtain that
\begin{align*}
\Big(\fint_{B_{j+1}(y)\cap \Omega} |a - \langle a \rangle_{B_{j+1}\cap \Omega}|^\gamma \, dx\Big)^{1/\gamma}
&\leq   c \Big(\fint_{B_{j+1}(y) \cap \Omega} |a - \langle a \rangle_{B_{\rho}\cap \Omega}|^\gamma \, dx\Big)^{1/\gamma} \\
&\leq   c \Big(\frac{1}{r^{\,n}_{j+1}}\int_{B_{\rho}(y)\cap \Omega} |a - \langle a \rangle_{B_{\rho}\cap \Omega}|^\gamma \, dx\Big)^{1/\gamma} \\
&\leq  c \theta^{\frac{-n}{\gamma}} \Big(\frac{1}{\rho^{\,n}}\int_{B_{\rho}(y) \cap \Omega} |a - \langle a \rangle_{B_{\rho}\cap \Omega}|^\gamma \, dx\Big)^{1/\gamma},
\end{align*}
which follows from Definition of $\bar{\omega}_{a, \gamma}$ that
\begin{equation}\label{omega_a}
\bar{\omega}_{a, \gamma}(r_{j+1})
\leq   c \theta^{\frac{-n}{\gamma}} \bar{\omega}_{a, \gamma}(\rho).
\end{equation}
Then we obtain that
\begin{equation}\label{sum_1}
\begin{aligned}
\sum_{j=0}^{\infty} \int_{r_{j+1}}^{r_j} \bar{\omega}_{a,\gamma}(\rho) \log\big(\frac{1}{\rho}\big) \, \frac{d\rho}{\rho}
&\geq c \theta^{\frac{n}{\gamma}} \sum_{j=0}^{\infty} \bar{\omega}_{a, \gamma}(r_{j+1}) \log\big(\frac{1}{r_j}\big) \int_{r_{j+1}}^{r_j} \frac{d\rho}{\rho} \\
&=  c \theta^{\frac{n}{\gamma}} \sum_{j=0}^{\infty} \bar{\omega}_{a, \gamma}(r_{j+1}) \log\big(\frac{1}{r_j}\big) \log\big(\frac{1}{\theta}\big).
\end{aligned}
\end{equation}
Since $0< \theta < 1/4$, a simple calculation yields that
\begin{equation}\label{ineq_log}
\log\big(\frac{1}{r_{j+1}}\big)
= \log\big(\frac{1}{\theta}\big) + \log\big(\frac{1}{r_j}\big)
\leq 2\log\big(\frac{1}{\theta}\big) \log\big(\frac{1}{r_j}\big).
\end{equation}
Together with \eqref{sum_1} and \eqref{ineq_log}, we arrive at
our first claim
\begin{equation}\label{claim_1}
\sum_{j=0}^{\infty} \int_{r_{j+1}}^{r_j} \bar{\omega}_{a,\gamma}(\rho) \log\big(\frac{1}{\rho}\big) \, \frac{d\rho}{\rho} \geq c  \theta^{\frac{n}{\gamma}} \sum_{j=0}^{\infty} \bar{\omega}_{a, \gamma}(r_{j+1}) \log\big(\frac{1}{r_{j+1}}\big)\,.
\end{equation}

For $I_2$, by the same argument as \eqref{omega_a}, we  see
$\bar{\omega}_{a, \gamma}(r) \leq c  \bar{\omega}_{a, \gamma}(\rho)$ for 
$\rho \in (r, 2r)$, which implies that
\begin{equation}\label{claim_2}
\begin{aligned}
\int_r^{2r} \bar{\omega}_{a,\gamma}(\rho) \log\big(\frac{1}{\rho}\big) \, \frac{d\rho}{\rho}
&\geq c \bar{\omega}_{a, \gamma}(r) \log\Big(\frac{1}{2r}\Big) \int_r^{2r} \frac{d\rho}{\rho} \\
&\geq c \bar{\omega}_{a, \gamma}(r) \log\big(\frac{1}{r}\big).
\end{aligned}
\end{equation}

For $I_3$, we also have 
$\bar{\omega}_{\mathbb{A},\gamma}(r_{j+1}) \leq c \theta^{\frac{-n}{\gamma}} \bar{\omega}_{\mathbb{A},\gamma}(\rho)$ 
for $\rho \in (r_{j+1}, r_j)$, which implies that
\begin{equation}\label{claim_3}
\begin{aligned}
\sum_{j=0}^{\infty} \int_{r_{j+1}}^{r_j} \bar{\omega}_{\mathbb{A},\gamma}(\rho) \, \frac{d\rho}{\rho}
&\geq c \theta^{\frac{n}{\gamma}} \sum_{j=0}^{\infty} \bar{\omega}_{\mathbb{A}, \gamma}(r_{j+1}) \int_{r_{j+1}}^{r_j} \frac{d\rho}{\rho} \\
&=  c \theta^{\frac{n}{\gamma}} \sum_{j=0}^{\infty} \bar{\omega}_{\mathbb{A}, \gamma}(r_{j+1}) \log\big(\frac{1}{\theta}\big) \\
&\geq c \theta^{\frac{n}{\gamma}} \sum_{j=0}^{\infty} \bar{\omega}_{\mathbb{A}, \gamma}(r_{j+1}).
\end{aligned}
\end{equation}

For $I_4$, we obtain that $\bar{\omega}_{\mathbb{A},\gamma}(r) \leq c \bar{\omega}_{\mathbb{A},\gamma}(\rho)$ 
for $\rho \in (r, 2r)$, and it yields
\begin{equation}\label{claim_4}
\int_r^{2r} \bar{\omega}_{\mathbb{A},\gamma}(\rho) \, \frac{d\rho}{\rho} \geq c \bar{\omega}_{\mathbb{A}, \gamma}(r).
\end{equation}
Finally, let us put \eqref{claim_1} \eqref{claim_2} \eqref{claim_3} 
and \eqref{claim_4} into \eqref{claim}, to attain that
\begin{align*}
&\int_0^{2r} \varUpsilon(\rho)  \frac{d\rho}{\rho}\\
&\geq c  \theta^{\frac{n}{\gamma}} \sum_{j=0}^{\infty} \bar{\omega}_{a, \gamma}(r_{j+1}) \log\big(\frac{1}{r_{j+1}}\big) + c \bar{\omega}_{a, \gamma}(r) \log\big(\frac{1}{r}\big)  
 + c \theta^{\frac{n}{\gamma}} \sum_{j=0}^{\infty} \bar{\omega}_{\mathbb{A}, \gamma}(r_{j+1})  + c \bar{\omega}_{\mathbb{A}, \gamma}(r) \\
&\geq c  \theta^{\frac{n}{\gamma}} \sum_{j=0}^{\infty} \varUpsilon(r_j),
\end{align*}
which completes the proof.
\end{proof}

We now show the local gradient boundedness to the local minimizer of the functional \eqref{main_funl}.

\begin{theorem}\label{theo_bound_Du}
Let $u \in W^{1, 1}(\Omega)$ be a local minimizer of the functional \eqref{main_funl}. Assume that $a(x)$ and $\mathbb{A}(x)$ satisfy \eqref{DMO} and \eqref{cond_A}. 
Then there exist $c = c(\texttt{data}) \geq 1$ and $R = R(\texttt{data}) > 0$ such 
that for $r \in (0, R)$ with $B_{2r} \Subset \Omega$ one has
\begin{equation}\label{sup_u}
		\|Du\|_{L^\infty(B_{r})} \leq c\fint_{B_{2r}} |Du|\, dx.
\end{equation}
\end{theorem}

\begin{proof}  
Let $y \in B_r$ be the Lebesgue point of $Du$. To specify some constants, we 
let $\Lambda = 100$, and
\begin{equation}\label{set_theta}
		\theta = \min\Big\{\theta_1, \Big(\frac{1}{200c_3^2}\Big)^\frac{1}{2\alpha}, \Big(\frac{1}{4c_4}\Big)^{1/\alpha}\Big\},
\end{equation}
where  $\theta_1 = \theta_1(\texttt{data}, \Lambda)$ as in Lemma \ref{dacy_u}.
By Lemma \ref{lem_gap} we choose $R>0$ sufficiently small such that for any 
$r \in (0, R)$ one has
\begin{equation}\label{sum_G}
\sum_{j=0}^{\infty} \varUpsilon(r_j) 
\leq c^*_2\, \int_0^{2R} \varUpsilon(\rho) \, \frac{d\rho}{\rho}  
\leq \min\Big\{\Big(\frac{\theta^{3n}}{100 c_2}\Big)^{p+1}, \Big(\frac{\theta^{4n}}{400c_3}\Big)^{p+1}, \frac{\theta^{\,2n}}{20c^*_1} \Big\}.
\end{equation}
Let
\[
\lambda := 200\theta^{-2n} \fint_{B_{r}} |Du|\, dx, 
\]
and
\begin{equation}\label{C_j_u}
C_j := \fint_{B_j} |Du|\, dx + \theta^{-2n} \fint_{B_j} |Du - \langle Du \rangle _{B_j}|\, dx.
\end{equation}

We  prove our main result using the exit-time method. To this end, let us define
\[
\mathbb{J} := \big\{j \in \mathbb{N} \cup \{0\} : C_j \leq \frac{\lambda}{50} \big\},
\]
We see that $\mathbb{J} \neq \emptyset$ because $\{0\} \in \mathbb{J}$. First, if $\mathbb{J}$ is infinite, then we can find an infinite subsequence of $\{C_j\}$, denoted by $\{C_i\}$, such that $C_i \leq \frac{\lambda}{50}$ for all
 $i \in \mathbb{N} \cup \{0\}$. Let $i \to  \infty$, by the Lebesgue differentiation theorem we obtain that
\begin{equation}
|Du(y)| \leq \liminf_{i\to  \infty} \fint_{B_i(y)} |Du|\,dx
\leq \liminf_{i\to  \infty} C_i \leq \frac{\lambda}{50}
\leq c \fint_{B_r(y)} |Du| \,dx,
\end{equation}
which makes \eqref{sup_u} true.

Otherwise, we  next consider that $\mathbb{J}$ is finite. 
Let $j_m = \max \mathbb{J}$ be the exit time, by the definition of $\mathbb{J}$ 
we see that
\begin{equation}\label{C_j_inf}
C_{j_m} \leq  \frac{\lambda}{50}
\quad \text{and} \quad
C_j > \frac{\lambda}{50} \quad \text{for } j \geq j_m + 1.
\end{equation}
We set
\[
a_j = |\langle Du \rangle_{B_j}|
\quad \text{and} \quad
E_j = \fint_{B_j} |Du - \langle Du \rangle_{B_j}|\, dx.
\]
In what follows, we use the method of induction to show that
\begin{equation}\label{result_induct}
a_j + E_j  \leq \lambda \quad \text{for } j \geq j_m.
\end{equation}
In the case of $j = j_m$, it is obvious that $a_j + E_j \leq C_{j_m} \leq \frac{\lambda}{50} \leq \lambda$.
Assume $a_j + E_j\leq \lambda$ for all $j \in \{j_m, \ldots, k\}$,
 we  prove that $a_{k+1} + E_{k+1} \leq \lambda$. To this end, by the elementary calculations we have that
\begin{equation}\label{a_k+1}
\begin{aligned}
a_{k+1}
&=  \big|\langle Du \rangle_{B_{k+1}} - \langle Du \rangle_{B_{k}} + \langle Du \rangle_{B_{k}}-  \dots - \langle Du \rangle_{B_{j_m}} + \langle Du \rangle_{B_{j_m}}\big| \\
&\leq  \fint_{B_{k+1}} |Du - \langle Du \rangle_{B_{k}}|\, dx  + \dots + \fint_{B_{j_m+1}} |Du - \langle Du \rangle_{B_{j_m}}|\, dx  + \fint_{B_{j_m}} |Du|\, dx  \\
&\leq  \theta^{-n}E_k + \dots + \theta^{-n}E_{j_m + 1}  + C_{j_m}  \\
&=  \theta^{-n} \sum_{j = j_m + 1}^{k} E_j +  C_{j_m}.
\end{aligned}
\end{equation}
To estimate $\theta^{-n}\sum_{j = j_m+1}^{k} E_j$, let us verify Condition \eqref{cond_dacy_u} of Lemma \ref{dacy_u}.
Applying Lemma \ref{lem_decay} to $E_{j+2}$ repeatedly, together with
$c_3^2\theta^{\,2\alpha} \leq \frac{1}{200}$ in \eqref{set_theta} we obtain that
\begin{equation}\label{E_j+2}
\begin{aligned}
&E_{j+2}\\
&\leq  (c_3\theta)^{\alpha(j+2-j_m)} E_{j_m} + c_3\theta^{-n}
\sum_{i = j_m}^{j+1} \big(\varUpsilon(r_{i})\big)^\frac{1}{p+1}  \fint_{B_i} |Du|\, dx \\
&\leq  (c_3\theta)^{2\alpha}{E_{j_m}} +  c_3\theta^{-n} \Big(\sum_{i = j_m}^{j} \big(\varUpsilon(r_{i})\big)^\frac{1}{p+1} \fint_{B_i} |Du|\, dx\Big)
+ c_3\theta^{-n} \big(\varUpsilon(r_{j+1})\big)^\frac{1}{p+1} \fint_{B_{j+1}} |Du|\, dx \\
&\leq  \frac{E_{j_m}}{200} + c_3\theta^{-n} \Big(\sum_{i = j_m}^{j} \big(\varUpsilon(r_{i})\big)^\frac{1}{p+1}  \fint_{B_i} |Du|\, dx\Big) + c_3\theta^{-2n} \big(\varUpsilon(r_{j+1})\big)^\frac{1}{p+1} \fint_{B_j} |Du|\, dx \\
&\leq  \frac{E_{j_m}}{200} + c_3\theta^{-2n} \sum_{i = j_m}^{j} \Big(\big(\varUpsilon(r_{i})\big)^\frac{1}{p+1} + \big(\varUpsilon(r_{i+1})\big)^\frac{1}{p+1}\Big) \fint_{B_i} |Du|\, dx.
\end{aligned}
\end{equation}
Since
\begin{equation}\label{E_j+2_1}
E_{j_m} \leq \theta^{2n} C_{j_m} \leq \theta^{2n}\frac{\lambda}{50}
\end{equation}
from \eqref{C_j_u} and \eqref{C_j_inf}. Moreover,
\begin{equation}\label{E_j+2_2}
\begin{aligned}
&c_3\theta^{-2n}\sum_{i = j_m}^{j} \Big(\big(\varUpsilon(r_{i})\big)^\frac{1}{p+1} + \big(\varUpsilon(r_{i+1})\big)^\frac{1}{p+1}\Big)  \\
&\leq  2 c_3\theta^{-2n} \sum_{i = 0}^{\infty}\big(\varUpsilon(r_{i})\big)^\frac{1}{p+1}
\leq 2 c_3\theta^{-2n} \Big(\sum_{i = 0}^{\infty}\varUpsilon(r_{i})\Big)^\frac{1}{p+1}
\leq \frac{\theta^{2n}}{200}
\end{aligned}
\end{equation}
because of \eqref{sum_G};
\begin{equation}\label{E_j+2_3}
\fint_{B_{i}} |Du|\, dx \leq a_i + E_i \leq \lambda
\quad \text{for } i \in \{j_m, \ldots,k\}
\end{equation}
from the assumption of induction.
Then we put \eqref{E_j+2_1} \eqref{E_j+2_2} \eqref{E_j+2_3} into \eqref{E_j+2} to derive that
\[
E_{j+2} \leq \frac{\theta^{2n}\lambda}{200} + \frac{\theta^{2n}\lambda}{200} \leq \frac{\theta^{2n}\lambda}{100} \quad \text{for } j \in \{j_m, \ldots, k\},
\]
which follows from $C_{j+2} > \frac{\lambda}{50}$ that
\begin{equation}\label{dacy_u_cond_1_final}
\fint_{B_{j+2}} |Du|\, dx
= C_{j+2} - \theta^{-2n} E_{j+2}
\geq \frac{\lambda}{50} -  \frac{\lambda}{100}
\geq \frac{\lambda}{100} \quad \text{for } j \in \{j_m, \ldots, k\}.
\end{equation}

Inequalities \eqref{dacy_u_cond_1_final} \eqref{E_j+2_3} and \eqref{sum_G} ensure the validity of Condition \eqref{cond_dacy_u}, 
then we employ Lemma \ref{dacy_u} to obtain
\[
E_{j+2} \leq \frac{1}{4}E_{j+1}+ c^*_1\,\varUpsilon(r_j)\,\lambda \quad \text{for } j \in \{j_m, \ldots, k\},
\]
where we used $c_4\theta^{\,\alpha} \leq 1/4$ in \eqref{set_theta}.
This implies that
\[
\sum_{j = j_m+2}^{k+2}E_j
\leq  \frac{1}{4}\sum_{j = j_m + 1}^{k + 1}E_j + c^*_1 \,\lambda  \sum_{j = j_m}^{k}\varUpsilon(r_j).
\]
We add $E_{j_m+1}$ to both sides, and use $C_{j_m} \leq \frac{\lambda}{50}$
in \eqref{C_j_inf} and $c^*_1\sum_{j = 0}^{\infty} \varUpsilon(r_j) \leq \frac{\theta^n}{20}$
in \eqref{sum_G} to obtain that
\begin{align*}
\sum_{j = j_m+1}^{k+2}E_j
&\leq  \frac{1}{4}\sum_{j = j_m+1}^{k+1}E_j + E_{j_m+1} + c^*_1 \,\lambda  \sum_{j = j_m}^{k}\varUpsilon(r_j)  \\
&\leq  \frac{1}{4}\sum_{j = j_m+1}^{k+2}E_j + 2\theta^{\,n}\,C_{j_m} + c^*_1 \,\lambda  \sum_{j = j_m}^{k}\varUpsilon(r_j)  \\
&\leq  \frac{1}{4}\sum_{j = j_m+1}^{k+2}E_j + \frac{\lambda}{20}\theta^{\,n} + \frac{\lambda}{20}\theta^{\,n}
\end{align*}
Thus,
\begin{equation}\label{sum_E}
\sum_{j = j_m+1}^{k+2}E_j \leq \frac{\lambda}{4}\theta^{\,n},
\end{equation}
which implies that $E_{k+1} \leq \frac{\lambda}{2}$.
Let us now put \eqref{sum_E} into \eqref{a_k+1} to deduce that
\[
a_{k+1}
\leq \frac{\lambda}{4} +  C_{j_m}
\leq \frac{\lambda}{4} +  \frac{\lambda}{50} \leq \frac{\lambda}{2}.
\]
This makes \eqref{result_induct} true for any $j \geq j_m$, and it follows from  the Lebesgue differentiation theorem that
\begin{equation}
|Du(y)| \leq \liminf_{j\to  \infty} \fint_{B_j(y)} |Du|\,dx
\leq \liminf_{j\to  \infty} \Big(a_j + E_j\Big) \leq \lambda
\leq c \fint_{B_r(y)} |Du| \,dx,
\end{equation}
which completes the proof.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm_1}]

Let $\Omega_0 \Subset \Omega' \Subset \Omega$ and
$\mathrm{dist}(\Omega_0, \Omega') = \mathrm{dist}(\Omega', \Omega) = \tfrac{1}{2}\mathrm{dist}(\Omega_0, \Omega)$.
By Theorem \ref{theo_bound_Du} we see that $Du$ is locally bounded in $\Omega$, 
and denote
\begin{equation}\label{Du_lambda}
\lambda = \|Du\|_{L^\infty(\Omega')} + 1.
\end{equation}
It suffices to prove that $\langle Du \rangle_{B_r(y)}$  for fixed point 
$ y \in \Omega_0$ is a Cauchy sequence in $r$, which means that for any 
$\epsilon > 0$ there is a radius $R_\epsilon \leq \mathrm{dist}\{\Omega_0, \Omega\}/8$ such that
for any  $\rho_1, \rho_2 \in (0, R_\epsilon]$ one has
\begin{equation}\label{kx}
| \langle Du \rangle_{B_{\rho_1}(y)} - \langle Du \rangle_{B_{\rho_2}(y)} | \leq \frac{\epsilon \lambda}{3}.
\end{equation}
Let $\rho_1 = R_\epsilon, \rho_2 \to 0$  we obtain
\[
| \langle Du \rangle_{B_{R_\epsilon}(y)} - Du(y) | \leq \frac{\epsilon \lambda}{3}.
\]
In accordance with the absolute continuity of the integral in $B_r(y)$,
we see that $y \mapsto \langle Du \rangle_{B_{R_\epsilon}(y)}$ is  continuous,
and find a small $\tilde{R}_\epsilon \in (0, R_\epsilon)$ such that for
$|x-y| < \tilde{R}_\epsilon$ one has
\[
| \langle Du \rangle_{B_{R_\epsilon}(x)} - \langle Du \rangle_{B_{R_\epsilon}(y)} | \leq \frac{\epsilon \lambda}{3},
\]
which leads to
\begin{align*}
|Du(x) - Du(y)|
&\leq  |Du(x) - \langle Du \rangle_{B_{R_\epsilon}(x)}| + |\langle Du \rangle_{B_{R_\epsilon}(x)} - \langle Du \rangle_{B_{R_\epsilon}(y)}| + |\langle Du \rangle_{B_{R_\epsilon}(y)} - Du(y)|\\
&\leq   \epsilon \lambda.
\end{align*}

To prove $\langle Du \rangle_{B_r(y)}$  is a Cauchy sequence, we divide this
 into two steps as follows.
\smallskip

\noindent\textbf{Step 1.} VMO type estimate of $Du$. 
It suffices to prove  that for any $\epsilon \in (0,1)$, there exists an 
$r_\epsilon \in (0,1/4)$  such that for $i \geq 2$  one has
\begin{equation}\label{vmo_est}
\fint_{B_{\theta^i r_\epsilon}} |Du - \langle Du \rangle _{B_{\theta^i r_\epsilon}}|\, dx \leq \epsilon \lambda.
\end{equation}
Let $\Lambda = \frac{200}{\epsilon}$ and 
$\theta_1 = \theta_1(\texttt{data}, \Lambda)$ as in Lemma \ref{dacy_u}. 
We choose
\begin{equation}\label{theta_2}
\theta = \min\Big\{\theta_1, \Big(\frac{\epsilon}{4c_4}\Big)^{1/\alpha}\Big\}.
\end{equation}
By Lemma \ref{lem_gap} we can take $r_\epsilon$ sufficient small such that
\begin{equation}\label{sum_G_2}
\sum_{i=0}^{\infty} \varUpsilon(r_i) \leq c^*_2\, \int_0^{2r_\epsilon} \varUpsilon(\rho) \, \frac{d\rho}{\rho}  \leq \min\Big\{\Big(\frac{\epsilon\theta^{3n}}{200 c_2}\Big)^{p+1}, \frac{\theta^{\,2n}\epsilon}{200c^*_1} \Big\}.
\end{equation}
We set
$r_i = \theta^i r_\epsilon$ and $B_i = B_{r_i}$ for $i = 0, 1, 2, \dots$,
and
\begin{equation}\label{set_L}
\mathbb{I} := \big\{2 \leq i \in \mathbb{N}: \fint_{B_i} |Du|\, dx < \frac{\epsilon\lambda}{200}\big\}.
\end{equation}
If $i \in \mathbb{I}$, then we obtain that
\[
\fint_{B_{i}} |Du - \langle Du \rangle _{B_{i}}|\, dx \leq 2 \fint_{B_{i}} |Du|\, dx \leq \frac{2\epsilon\lambda}{200} \leq \epsilon \lambda.
\]
Otherwise, if $i \notin  \mathbb{I}$, by \eqref{set_L} and \eqref{Du_lambda} we see that
\begin{align*}
\frac{\epsilon \lambda}{200} \leq \fint_{B_{i}}  |Du|\, dx   \quad \text{and} \quad \fint_{B_{i-2}} |Du|\, dx \leq \lambda,
\end{align*}
which combined with \eqref{sum_G_2} just meets Condition \eqref{cond_dacy_u},
then we employ Lemma \ref{dacy_u} to obtain
\begin{align*}
\fint_{B_{i}} |Du - \langle Du \rangle _{B_{i}}|\, dx
&\leq  c_4 \theta^{\,\alpha} \fint_{B_{i-1}} |Du - \langle Du \rangle _{B_{i-1}}|\, dx + c^*_1\varUpsilon(r_{i-2})\lambda \\
&\le  2c_4\, \theta^{\,\alpha} \lambda + c^*_1\varUpsilon(r_{i-2})\,\lambda.
\end{align*}
Since $2c_4 \theta^{\,\alpha} \leq \frac{\epsilon}{2}$ is confined  in \eqref{theta_2} and $c^*_1 \varUpsilon(r_{i-2}) \leq \epsilon$  in \eqref{sum_G_2}, we obtain that \eqref{vmo_est}.
\smallskip

\noindent\textbf{Step 2.} Cauchy sequence of $\langle Du \rangle _{B_k}$ for $k\ge 3$.
More precisely, we  prove that
\begin{equation}\label{Du_k_h}
\big|\langle Du \rangle _{B_k} -  \langle Du \rangle _{B_h}\big| \leq \frac{\epsilon \lambda}{10} \quad \text{for  } 3 \leq k < h \text{ with }k,h \in \mathbb{N}.
\end{equation}
By the VMO type estimate of $Du$, it yields that  there exists a new 
$r_\epsilon \in (0,1/4)$ such that for any $i \geq 2$ it holds
\begin{equation}\label{vmo_est_2}
\fint_{B_{i}} |Du - \langle Du \rangle _{B_{i}}|\, dx \leq \frac{\theta^{n}\epsilon}{200} \lambda .
\end{equation}
We denote 
\[
i_m := \begin{cases}
\min \mathbb{I}, &\text{for } \mathbb{I} \neq \emptyset,\\
\infty, &\text{for } \mathbb{I} = \emptyset,
\end{cases}
\]
with  $\mathbb{I}$ as in \eqref{set_L}. Let us consider the proof of \eqref{Du_k_h}
in three cases.
\smallskip

\noindent\textbf{Case 1.} $k < h \leq i_m$. By \eqref{set_L} and \eqref{Du_lambda} we see that
\[
\frac{\epsilon \lambda}{200} 
\leq \fint_{B_i}  |Du|\, dx   \quad \text{and} \quad \fint_{B_{i-2}} |Du|\, dx \leq \lambda \quad \text{for } i \in \{k, \ldots, h-1\}.
\]
Employing Lemma \ref{dacy_u}, together with $2c_4 \theta^{\,\alpha} \leq \frac{1}{2}$ in \eqref{theta_2}, we obtain that
\begin{align*}
&\sum_{i=k}^{h-1}  \fint_{B_{i}} |Du - \langle Du \rangle _{B_{i}}|\, dx  \\
&\leq  c_4 \theta^{\,\alpha} \sum_{i=k-1}^{h-2}\fint_{B_{i}} |Du - \langle Du \rangle _{B_{i}}|\, dx + c^*_1\,\lambda\sum_{i=k-2}^{h-3}\varUpsilon(r_{i}) \\
&\leq  \frac{1}{2} \sum_{i=k}^{h-1}  \fint_{B_{i}} |Du - \langle Du \rangle _{B_{i}}|\, dx + \frac{1}{2}\fint_{B_{k-1}} |Du - \langle Du \rangle _{B_{k-1}}|\, dx + c^*_1\,\lambda\sum_{i=k-2}^{h-3}\varUpsilon(r_{i}),
\end{align*}
which yields
$$
\sum_{i=k}^{h-1}  \fint_{B_{i}} |Du - \langle Du \rangle _{B_{i}}|\, dx
\leq \fint_{B_{k-1}} |Du - \langle Du \rangle _{B_{k-1}}|\, dx + 2 c^*_1\,\lambda\sum_{i=k-2}^{h-3}\varUpsilon(r_{i}).
$$
 We use \eqref{vmo_est_2} and \eqref{sum_G_2} to obtain
\[
\sum_{i=k}^{h-1}  \fint_{B_{i}} |Du - \langle Du \rangle _{B_{i}}|\, dx
\leq \frac{\theta^{n}\epsilon}{200} \lambda + \frac{\theta^{n}\epsilon}{80} \lambda \leq \frac{\theta^n \epsilon }{40}\lambda.
\]
Therefore,
\begin{align*}
\big|\langle Du \rangle _{B_k} -  \langle Du \rangle _{B_h}\big|
&\leq  \sum_{i=k}^{h-1} \big|\langle Du \rangle _{B_i} -  \langle Du \rangle _{B_{i+1} }\big|  \\
&\leq  \theta^{-n} \sum_{i=k}^{h-1} \fint_{B_{i}} |Du - \langle Du \rangle _{B_{i}}|\, dx \leq \frac{\epsilon \lambda}{40}.
\end{align*}

\noindent\textbf{Case 2.} $i_m \leq k < h$ for $i_m < \infty$, 
 we claim that
\begin{equation}\label{k_h_lambda}
\big|\langle Du \rangle _{B_k} \big| \leq \frac{\epsilon \lambda}{20} \quad \text{and} \quad \big|\langle Du \rangle _{B_{h}}\big| \leq \frac{\epsilon \lambda}{20},
\end{equation}
which obviously leads to  \eqref{Du_k_h}.
It suffices to prove the first inequality of \eqref{k_h_lambda} in the following, 
since the second one can be obtained by the same way.
In fact, if $k \in \mathbb{I}$, it is true obviously. 
If $k \notin \mathbb{I}$, we can find $k_0 \in \mathbb{I}$ satisfying 
$i_m \leq k_0 < k$ and $\{k_0+1, \dots, k\} \cap \mathbb{I} = \emptyset$. In particular, we have 
\[
\frac{\epsilon \lambda}{200} 
\leq \fint_{B_i}  |Du|\, dx   \quad \text{and} \quad \fint_{B_{i-2}} |Du|\, dx \leq \lambda \quad \text{for } i \in \{k_0 + 1, \ldots, k\}.
\]
By Lemma \ref{dacy_u} with $2c_4 \theta^{\,\alpha} \leq \frac{1}{2}$ as in \eqref{theta_2} we obtain that
\begin{align*}
&\sum_{i=k_0}^{k}  \fint_{B_{i}} |Du - \langle Du \rangle _{B_{i}}|\, dx\\
&=   \sum_{i=k_0 + 1}^{k}  \fint_{B_{i}} |Du - \langle Du \rangle _{B_{i}}|\, dx  + \fint_{B_{k_0}} |Du - \langle Du \rangle _{B_{k_0}}|\, dx  \\
&\leq  c_4 \theta^{\,\alpha} \sum_{i=k_0}^{k-1}\fint_{B_{i}} |Du - \langle Du \rangle _{B_{i}}|\, dx + c^*_1\,\lambda\sum_{i=k_0 - 1}^{k-2}\varUpsilon(r_{i})
 +  \fint_{B_{k_0}} |Du - \langle Du \rangle _{B_{k_0}}|\, dx \\
&\leq  \frac{1}{2}\sum_{i=k_0}^{k}  \fint_{B_{i}} |Du - \langle Du \rangle _{B_{i}}|\, dx  + \frac{\theta^{n}\epsilon}{200} \lambda + \frac{\theta^{n}\epsilon}{200} \lambda,
\end{align*}
where we used \eqref{sum_G_2} and \eqref{vmo_est_2} in the last inequality. Hence
\[
\sum_{i=k_0}^{k}  \fint_{B_{i}} |Du - \langle Du \rangle _{B_{i}}|\, dx \leq \frac{\theta^{n}\epsilon}{40} \lambda,
\]
and
\begin{align*}
\big|\langle Du \rangle _{B_k} \big|
&\leq  \big|\langle Du \rangle _{B_{k_0}} - \langle Du \rangle _{B_k}\big| + \big|\langle Du \rangle _{B_{k_0}} \big|  \\
&\leq  \sum_{i=k_0}^{k-1} \big|\langle Du \rangle _{B_i} -  \langle Du \rangle _{B_{i+1} }\big| + \fint_{B_{k_0}} |Du|\, dx  \\
&\leq  \theta^{-n} \sum_{i=k_0}^{k}  \fint_{B_{i}} |Du - \langle Du \rangle _{B_{i}}|\, dx + \frac{\epsilon \lambda}{200} \leq \frac{\epsilon \lambda}{20}.
\end{align*}

\noindent\textbf{Case 3.} $ k < i_m < h$ for $i_m < \infty$. 
From  Case 2 with $h > i_m$ we obtain
\[
\big|\langle Du \rangle _{B_{h}}\big| \leq \frac{\epsilon \lambda}{20},
\]
and by  Case 1 with $k < i_m$ one has
\[
\big|\langle Du \rangle _{B_k} -  \langle Du \rangle _{B_{i_m}}\big| \leq \frac{\epsilon \lambda}{40}.
\]
Then we conclude that
\[
\big|\langle Du \rangle _{B_k} \big|
\leq  \big|\langle Du \rangle _{B_k} -  \langle Du \rangle _{B_{i_m}}\big	| + \big|\langle Du \rangle _{B_{i_m}}\big|
\leq  \frac{\epsilon \lambda}{40} + \frac{\epsilon \lambda}{200} \leq \frac{\epsilon \lambda}{20},
\]
which yields \eqref{Du_k_h} as Case 2. By combining \textbf{Cases 1-3}, we obtain that $\{ \langle Du \rangle _{B_k}\}$ is a Cauchy sequence.


Finally, we  prove \eqref{kx} in the continuous way with respect to the radius  
$\rho$. For any $0 < \rho_1 < \rho_2 < \theta^3 r_\epsilon$, there exist two 
integers $k,h \geq 3$ such that 
$\theta^{k+1}r_\epsilon \leq \rho_1 < \theta^{k}r_\epsilon$
and $\theta^{h+1}r_\epsilon \leq \rho_2 < \theta^{h}r_\epsilon$. By \eqref{vmo_est_2} it implies that
\begin{align*}
\big|\langle Du \rangle _{B_{k}} -  \langle Du \rangle _{B_{\rho_1}}\big|
&\leq  \fint_{B_{\rho_1}} \big|Du - \langle Du \rangle _{B_{k}}\big|\, dx  \\
&\leq  \Big(\frac{\theta^{k}r_\epsilon}{\rho_1}\Big)^n\fint_{B_{k}} |Du - \langle Du \rangle _{B_{k}}|\, dx \\
&\leq  \theta^{-n} \cdot \frac{\theta^{n}\epsilon}{200} \lambda = \frac{\epsilon \lambda}{200}.
\end{align*}
By the same reasoning, we also have
\[
\big|\langle Du \rangle _{B_{h}} -  \langle Du \rangle _{B_{\rho_2}}\big| \leq	 \frac{\epsilon \lambda}{200}.
\]
It follows from  \eqref{Du_k_h} that
\begin{align*}
\big|\langle Du \rangle _{B_{\rho_1}} -  \langle Du \rangle _{B_{\rho_2}}\big|
&\leq  \big|\langle Du \rangle _{B_{k}} -  \langle Du \rangle _{B_{h}}\big| + \big|\langle Du \rangle _{B_{\rho_1}} -  \langle Du \rangle _{B_{k}}\big| + \big|\langle Du \rangle _{B_{h}} -  \langle Du \rangle _{B_{\rho_2}}\big|  \\
&\leq  \frac{\epsilon \lambda}{10} + \frac{\epsilon \lambda}{200} + \frac{\epsilon \lambda}{200} \leq \frac{\epsilon \lambda}{5}.
\end{align*}
Let $R_\epsilon = \theta^3 r_\epsilon$, then \eqref{kx} holds for $\rho_1, \rho_2 \in (0, R_\epsilon)$ and $y \in \Omega_0$. This completes the proof.
\end{proof}


\subsection*{Acknowledgements}
We would like to thank the anonymous referees for their valuable comments and suggestions. 
This work was supported by the National Natural Science Foundation of
China (No. 12071021).



\begin{thebibliography}{10}

\bibitem{ArCW25} R. Arora, {\'A}. Crespo-Blanco, P. Winkert;
On logarithmic double phase problems, 
\emph{J. Differ. Equ.}, \textbf{433} (2025), 113247.

\bibitem{Ba25} P. Baroni;
A new condition ensuring gradient continuity for minimizers of non-autonomous functionals with mild phase transition, 
\emph{Calc. Var. Partial Differ. Equ.}, \textbf{64} (2025), 112.

\bibitem{BaCM16} P. Baroni, M. Colombo, G. Mingione;
Non-autonomous functionals, borderline cases and related function classes, 
\emph{St. Petersburg Math. J.}, \textbf{27} (2016), 347--379.

\bibitem{BaM18} P. Baroni, M. Colombo, G. Mingione;
Regularity for general functionals with double phase, 
\emph{Calc. Var. Partial Differ. Equ.}, \textbf{57} (2018), 1--48.

\bibitem{BaC22} P. Baroni, A. Coscia;
Gradient regularity for non-autonomous functionals with Dini or non-Dini continuous coefficients, 
\emph{Electron. J. Differ. Equ.}, \textbf{2022} (2022), 80.

\bibitem{ByY18} S. S. Byun, Y. Youn;
Riesz potential estimates for a class of double phase problems, 
\emph{J. Differ. Equ.}, \textbf{264} (2018), 1263--1316.

\bibitem{ChKW26} I. Chlebicka, M. Kim, M. Weidner;
Gradient Riesz potential estimates for a general class of measure data quasilinear systems, 
\emph{Adv. Calc. Var.}, 2026. doi:10.1515/acv-2025-008.

\bibitem{CoM15} M. Colombo, G. Mingione;
Regularity for double phase variational problems, 
\emph{Arch. Ration. Mech. Anal}., \textbf{215} (2015), 443--496.

\bibitem{CoM15B} M. Colombo, G. Mingione;
Bounded minimisers of double phase variational integrals, 
\emph{Arch. Ration. Mech. Anal.}, \textbf{218} (2015), 219--273.

\bibitem{De20} C. De Filippis;
On the regularity of the $\omega$-minima of $\varphi$-functionals, 
\emph{Nonlinear Anal.}, \textbf{194} (2020), 111464.

\bibitem{De23} C. De Filippis, G. Mingione;
Regularity for double phase problems at nearly linear growth, 
\emph{Arch. Ration. Mech. Anal.}, \textbf{247} (2023), 85.

\bibitem{DuM10} F. Duzaar, G. Mingione;
Gradient continuity estimates, 
\emph{Calc. Var. Partial Differ. Equ.}, \textbf{39} (2010), 379--418.

\bibitem{Di09} L. Diening, B. Stroffolini, A. Verde;
Everywhere regularity of functionals with $\varphi$-growth, 
\emph{Manuscr. Math.}, \textbf{129} (2009), 449--481.

\bibitem{DoK17} H. Dong, S. Kim;
On $C^1$, $C^2$, and weak type-$(1, 1)$ estimates for linear elliptic operators, \emph{Commun. Partial Differ. Equ.}, \textbf{42} (2017), 417--435.

\bibitem{DoLK20} H. Dong, J. Lee, S. Kim;
On conormal and oblique derivative problem for elliptic equations with Dini mean oscillation coefficients, 
\emph{Indiana Univ. Math. J.}, \textbf{69} (2020), 1815--1853.

\bibitem{DoK26} H. Dong, H. Kwon;
Spatial $C^1$, $C^2$, and Schauder estimates for nonstationary Stokes equations with Dini mean oscillation coefficients, 
\emph{Calc. Var. Partial Differ. Equ.}, \textbf{65} (2026), 43.

\bibitem{ZFeng25}Z. Feng, J. Zhang, S. Zheng, Weighted gradient estimates to nonlinearelliptic equations of  p(x) -growth with measure data,
\emph{J. Math. Pures Appl.}, \textbf{203} (2025), Paper No. 103756.

\bibitem{Gi03} E. Giusti;
\emph{Direct methods in the calculus of variations}, 
World Scientific Publishing Co. Pte. Ltd., Singapore, 2003.


\bibitem{huang96} Q. Huang;
Estimates on the generalized {Morrey} spaces {$L_{\phi}^{2, \lambda} $} and {$BMO_{\phi}$} for linear elliptic systems, 
\emph{Indiana Univ. Math. J.}, \textbf{45} (1996), 397--439.

\bibitem{JiMV09} T. Jin, V. Maz'ya, J. Van Schaftingen;
Pathological solutions to elliptic problems in divergence form with continuous coefficients, \emph{C. R. Math.}, \textbf{347} (2009), 773--778.

\bibitem{KuM18} T. Kuusi, G. Mingione;
Vectorial nonlinear potential theory, 
\emph{J. Eur. Math. Soc.}, \textbf{20} (2018), 929--964.

\bibitem{Li17} Y. Li;
On the $C^1$ regularity of solutions to divergence form elliptic systems with Dini-continuous coefficients, 
\emph{Chin. Ann. Math. Ser. B}, \textbf{38} (2017), 489--496.

\bibitem{Li87} G. M. Lieberman;
H{\"o}lder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions, 
\emph{Ann. Mat. Pura Appl.}, \textbf{148} (1987), 77--99.

\bibitem{LuZF21} H. Lv,  S. Zheng, Z. Feng;
Existence results for nonlinear Schr\"{o}dinger equations involving the fractional $(p,q)$-Laplacian and critical nonlinearities, 
\emph{Electron. J. Differ. Equ.} \textbf{2021} (2021), 1--24.

\bibitem{Ma89} P. Marcellini;
Regularity of minimizers of integrals of the calculus of variations with non-standard growth conditions, 
\emph{Arch. Ration. Mech. Anal.}, \textbf{105} (1989), 267--284.

\bibitem{Ma91} P. Marcellini;
Regularity and existence of solutions of elliptic equations with $p, q$-growth conditions, 
\emph{J. Differ. Equ.}, \textbf{90} (1991), 1--30.

\bibitem{Mi02} M. Misawa;
Local H{\"o}lder regularity of gradients for evolutional $p$-Laplacian systems,
\emph{Annali di Matematica Pura ed Applicata}, \textbf{181} (2002), 389--405.

\bibitem{Ok16} J. Ok;
Gradient continuity for $p(\cdot)$-Laplace systems, 
\emph{Nonlinear Anal.}, \textbf{141} (2016), 139--166.

\bibitem{RaT24} M. Ragusa, A. Tachikawa;
Regularity of minimizers for double phase functionals of borderline case with variable exponents, 
\emph{Adv. Nonlinear Anal.}, \textbf{13} (2024), 20240017.

\bibitem{Sp65} S. Spanne;
Some function spaces defined using the mean oscillation over cubes,
\emph{Ann. Scuola Norm. Sup. Pisa Cl. Sci.}, \textbf{19} (1965), 593--608.

\bibitem{JZh23}
J. Zhang, S. Zheng, Z. Feng, Gradient estimates of general nonlinear singular elliptic equations with measure data,
\emph{J. Differ. Equ.}, \textbf{372} (2023), 402-457.

\bibitem{Zh87} V. V. Zhikov;
Averaging of functionals of the calculus of variations and elasticity theory, \emph{Izv. Akad. Nauk SSSR Ser. Mat.}, \textbf{29} (1987), 33.

\end{thebibliography}

\end{document}
