\begin{thm}\label{ulimitatoCupini}
    Let $f = f(x, \xi)$ be the integrand defined at \eqref{integrand} and assume that $\omega\in L^t(\Omega)$, with $\left(\frac{\overline{p}^*}{p_n} \right)' < t \le \infty$,
where $\bar{p}^*$ is the Sobolev exponent of $\bar{p}$, with $\bar{p}$ defined at \eqref{definizionePi}.
Then, every local minimizer $u$ of \eqref{functional} is locally bounded in $\Omega$.
\end{thm}

\begin{thm}\label{thmBfinito}
Let $\omega \in  W_{\mathrm{loc}}^{1,\frac{\mathbf{p}+2}{\mathbf{p}+1}}(\Omega) $ and let $u \in {W_{\mathrm{loc}}^{1,\mathbf{p}}}(\Omega)$ be a local minimizer of \eqref{functional} under assumptions \eqref{A2}--\eqref{A4}, with exponents $p_i\geq 2, \forall i=1,\dots,n $, such that
\begin{equation}\label{Ipotesip}
p_n <
\begin{cases}
    \min \Big\{ \dfrac{\overline{p}^*}{t'} ,p_1 +2 \Big\} \, &\text{if } \overline{p} < n,  \\
     p_1 +2 \quad &\text{otherwise},
    \end{cases}
\end{equation}
and with a function $g \in {L^{r}_{\mathrm{loc}}(\Omega)}$, where $r$ satisfies the condition
\begin{equation}\label{Ipotesir}
   r> p_n + 2 .
\end{equation}
Then,
\begin{equation*}
{V_{p_i}(u_{x_i}) \in W^{1,2}_{\mathrm{loc}}(\Omega)} \qquad\forall i=1,\dots,n,
\end{equation*}
and the following estimates
\begin{align*}
    &\sum_{i=1}^{n} \left(  \int_{B_{R/4}} |u_{x_i}|^{p_i +2 } dx \right) \notag\\
    &\leq c  \left( 1+\sum_{i=1}^{n} \left(\Vert u_{x_i} \Vert_{L^{p_i}(B_R)} + {\Vert \omega_{x_i} \Vert_{L^\frac{p_i+2}{p_i+1}(B_{R})}} \right) +\Vert \omega\Vert_{L^t(B_R)}+ \Vert g \Vert_{L^r (B_{R})} \right)^\sigma
\end{align*}
and
\begin{align*}
        &\sum_{i=1}^{n}  \left( \int_{B_{R/4}} |u_{x_i x_j}|^2|u_{x_i}|^{p_i-2}dx \right) \notag\\
         &\leq c  \left( 1+\sum_{i=1}^{n} \left(\Vert u_{x_i} \Vert_{L^{p_i}(B_R)} + {\Vert \omega_{x_i} \Vert_{L^\frac{p_i+2}{p_i+1}(B_{R})}} \right) +\Vert \omega\Vert_{L^t(B_R)}+ \Vert g \Vert_{L^r (B_{R})} \right)^\sigma
\end{align*}
hold for every pair of concentric balls $B_{R/4} \subset B_{R} \Subset \Omega$, where $c = c(n,p_i,\lambda, \Lambda,R, \Vert u \Vert_{\infty})$ and $\sigma= \sigma (n,p_i, r)$ are positive constants.
\end{thm}

\begin{align}\label{Stimacompattauxixjpi+2}
        \sum_{i=1}^{n}   \left( \int_{B_{R/4}} |u_{x_i x_j}|^2|u_{x_i}|^{p_i-2} dx \right) & \leq c  \left(1+ \sum_{i=1}^{n} \left(\Vert u_{x_i} \Vert_{L^{p_i}(B_R)}+{\Vert \omega_{x_i} \Vert_{L^{\frac{p_i+2}{p_i+1}} (B_R) }} \right) + \Vert g \Vert_{L^r (B_{R})} \right)^\sigma,
\end{align}
which concludes the proof.
\end{proof}
\section{Approximation}\label{Appro}

In order to perform the approximation argument, let us fix a non-negative smooth kernel $\phi \in \mathcal{C}^\infty_0(B_1(0))$ such that $\int_{B_1(0)} \phi =1$ and consider the corresponding family of mollifiers $(\phi_k)_{k >0}$. We set, for a given ball $B_R \Subset \Omega$ ,
$$a_i^k=a_i * \phi_k, \quad \omega^k= \omega * \phi_k,$$
\begin{equation}\label{gk}
g_k(x)=\underset{i}{\max} \{ |D a_i^k (x)|\}
\end{equation}
and, for $x \in B_R$, we define
\begin{equation}\label{fvarepsilon}
f^k (x,\xi)= \sum_{i=1}^{n} a_i^k(x)|\xi|^{p_i} ,
\end{equation}
 for every $k < \text{dist}(B_R,\Omega)$. We shall use the following $$\mathcal{F}^{k}(v,B_R):=\int_{_{B_R} }\left( f^{k}(x, Dv)-\omega(x)v(x) \right) dx$$ and
$$\mathcal{F}^{k}_{\varepsilon}(v,B_R):= \, \int_{B_R} \left( f^k(x,Dv)+ \varepsilon(1+ |Dv|^2)^{\frac{p_n}{2}}- \omega^k(x)v(x) \right) dx.$$
\begin{flushleft}
One can easily check that $f^k(x, \xi)$ satisfies assumptions \eqref{A2}--\eqref{A3} and \eqref{A'4}, with $K = \sup_{B_R} |g_k(x)| $ that is finite since $a^k_i(x) \in W^{1, \infty}(B_R)$ and then $g_k \in L^{\infty}(B_R)$.\\
\end{flushleft}

We are in position to prove our main result.

\begin{proof}[\textit{{Proof of Theorem~\ref{thmBfinito}}}]
We consider the variational problems
\begin{equation}\label{Pfj}
    \inf \left\{  \int_{B_R} \left( f^k(x,Dv)+ \varepsilon(1+ |Dv|^2)^{\frac{p_n}{2}} -\omega^k(x)v(x)\right) dx \ : \ v \in W^{1,p_n}_0(B_R)+u^\eta \right\},
\end{equation}
where $$u^\eta = u * \varsigma_\eta$$ is the mollification of the local minimizer $u$ of \eqref{functional}, for a sequence of mollifier $\varsigma_n$.\\
It is well known that, by the direct methods of the calculus of variations, there exists a unique solution $u^\eta_{k,\varepsilon} \in  W^{1,p_n}_0(B_R)+u^\eta $ of the problem \eqref{Pfj}. Since the integrand $$f^k(x,Dv)+ \varepsilon(1+ |Dv|^2)^{\frac{p_n}{2}}-\omega^k(x)v(x)$$ satisfies the assumptions of  Theorem \ref{thm2Mascolo}, we have that $$V_{p_i}((u^\eta_{k,\varepsilon})_{x_i}) \in W_{\mathrm{loc}}^{1,2}(B_R) \quad \forall i=1,...,n.$$
Hence we are legitimate to use estimate \eqref{uxistima} of Theorem \ref{AppThm}, to obtain that
\begin{align}
   & \sum_{i=1}^{n}  \int_{B_{R/4}}|(u^\eta_{k,\varepsilon})_{x_i}|^{p_i +2 } dx \notag \\
    & \leq c \left(1+ \sum_{i=1}^{n} \left(\int_{B_R}\, \lvert(u^\eta_{k,\varepsilon})_{x_i} \rvert^{p_i} \ dx+  \Vert \omega^k_{x_i} \Vert_{L^\frac{p_i+2}{p_i+1}(B_{R})} \right)  + \Vert g_k \Vert_{L^r (B_{R})} \right)^\sigma \notag \\
     &  \leq c(\lambda) \left(1+ \sum_{i=1}^{n} \left( \int_{B_R} a_i^k(x)\, \lvert (u^\eta_{k,\varepsilon})_{x_i}\rvert^{p_i} \ dx +  \Vert \omega^k_{x_i} \Vert_{L^\frac{p_i+2}{p_i+1}(B_{R})} \right) + \Vert g_k \Vert_{L^r (B_{R})} \right)^\sigma, \label{der}
\end{align}

\begin{lem}\label{LemmaTroisi}
    Let $E \subset \mathbb{R}^n$ be a bounded open set and consider $u \in W^{1, \mathbf{p}}(E)$, with $  p_i \geq 1$, for all $i=1, ..., n$.\\
    If $\overline{p} \le n$, then there exists a positive constant $\gamma_1$ depending on $n, p_i$ and, only in the case $\overline{p}=n$, also on $\overline{p}^*$ and $E$, such that
    \begin{align*}
        \lvert \lvert u \rvert \rvert_{L^{\overline{p}^*}(E)} \leq \gamma_1  \left[ \sum_{i=1}^{n} \lvert \lvert u_{x_i} \rvert \rvert_{L^{p_i}(E)} + \lvert \lvert u \rvert \rvert_{L^1(E)} \right].
    \end{align*}
      If $\overline{p} > n$,  there exists a positive constant $\gamma_2$ depending only on $ n, p_i$ and $E$, such that
    \begin{align*}
        \lvert \lvert u \rvert \rvert_{L^{\infty}(E)} \leq \gamma_2  \left[ \sum_{i=1}^{n} \lvert \lvert u_{x_i} \rvert \rvert_{L^{p_i}(E)} + \lvert \lvert u \rvert \rvert_{L^1(E)} \right].
    \end{align*}
\end{lem}

\section{A regularized problem}\label{PreREG}
In this section we prove a regularity result for more regular problems with respect to \eqref{functional} that will be needed in the approximation procedure. More precisely, for $\varepsilon \geq 0$ we consider
\begin{equation}\label{Eqconepsilon}
 f_\varepsilon(x,u, \xi)= \sum_{i=1}^{n} \, \dfrac{b_i(x)}{p_i} \lvert \xi_i \rvert^{p_i} + \varepsilon(1+|\xi|^\frac{p_n}{2})^2  -\overline{\omega}(x)u
\end{equation}
with
$b_i(x):\Omega \to [0,+\infty), \, i=1, \dots, n$, non-negative, Lipschitz continuous coefficients,
i.e., the following conditions hold
\begin{equation}\label{atilde}
L_i= \sup_{x,y \in \Omega, x \neq y} \frac{|b_i(x)-b_i(y)|}{|x-y|}< \infty , \quad i=1,..,n.
\end{equation}
Setting $\tilde{f}(x, \xi)= \sum_{i=1}^{n} \, b_i(x) \lvert \xi_i \rvert^{p_i},$ one can easily check that
\begin{equation}\label{A'4}
    |D_\xi  \tilde{f}(x,\xi)-D_\xi \tilde{f}(y, \xi)| \leq K |x-y|\, \, \sum_{i=1}^{n} |\xi_i|^{p_i-1} \tag{A3'}
\end{equation}
\noindent for a.e.\ $x,y \in \Omega $ and every $\xi \in \mathbb{R}^{ n}$, where $K=K(L_i, p_i)$
.\\
We recall a Lipschitz regularity Theorem for the minimizers of the functional
\begin{equation}\label{functional2}
    \mathcal{F}_\varepsilon (x, Dv) =\int_\Omega f_\varepsilon(x,v, Dv) \, dx,
\end{equation}
that can be deduced by  \cite[Theorem $8.9$]{giusti} in the case $p=p_n$, where $f_\varepsilon$ was defined at \eqref{Eqconepsilon}.
\begin{thm}\label{thm2Mascolo}
    Let $\tilde{f} $ satisfy \eqref{A2}, \eqref{A3} and \eqref{A'4} and $\overline{\omega} \in L^\infty(\Omega)$.
    Let $u_\varepsilon \in W^{1,p_n}_{\mathrm{loc}}(\Omega)$ be a local minimizer of \eqref{functional2}.
Then, $u_\varepsilon \in W^{1,\infty}_{\mathrm{loc}}(\Omega)$
\end{thm}

We shall use the higher differentiability of the minimizers $u_\varepsilon$ of $\mathcal{F}_\varepsilon$ that, as far  we know, is not available in literature and that is contained in the following
{
\begin{lem}\label{LemmaInduzione}
  Let $\tilde{f}$ satisfy \eqref{A2}, \eqref{A3} and \eqref{A'4} for exponents $ p_i \geq 2,  \forall i=1,\dots,n$, and assume {$\overline{\omega} \in L^\infty(\Omega) \cap W^{1,\frac{\mathbf{p}+2}{\mathbf{p}+1}}_{\mathrm{loc}}(\Omega)$. }
    Let  $u_\varepsilon \in W^{1,p_n}_{\mathrm{loc}}(\Omega) $ be a local minimizer of the functional \eqref{functional2}.
    Then,
    we have  $${V_{p_i}((u_\varepsilon)_{x_i})} \in W^{1,2}_{\mathrm{loc}}(\Omega), \quad \forall i =1,\dots, n.   $$

\end{lem}

\begin{lem}\label{D1}
Let $1<p<+\infty$. There exist positive constants $c_1$ and $c_2$, depending only on $n$ and $p$, such that
\begin{center}
{$c_1(|\xi|^{2}+|\eta|^{2})^{\frac{p-2}{2}} \leq \dfrac{|V_{p}(\xi)-V_{p}(\eta)|^{2}}{|\xi-\eta|^{2}} \leq c_2(|\xi|^{2}+|\eta|^{2})^{\frac{p-2}{2}} $},
\end{center}
for any $\xi, \eta \in \mathbb{R}^{n}$, with $\xi \neq \eta$.
\end{lem}

The aim of this paper is to investigate the differentiability properties of locally bounded minimizers of convex functionals, exhibiting an orthotropic structure, depending also on the $x$-variable and on the minimizer itself. More precisely, we consider
\begin{equation}\label{functional}
 \mathcal{F}(u,\Omega):= \, \int_\Omega \left[f(x, Du)-\omega(x)u(x) \right] dx,
\end{equation}
\noindent where $\Omega \subset \mathbb{R}^n$ is an open subset, $n\geq 2$, {$u : \Omega \to \mathbb{R}$}, and $f:\Omega\times\mathbb{R}^{ n}\to [0,+\infty)$ has the following form
\begin{equation}
    f= f(x, \xi)= \sum_{i=1}^{n} \,     \dfrac{a_i(x)}{p_i} \lvert \xi_i \rvert^{p_i}, \label{integrand}
\end{equation}
with $\xi=(\xi_1,\xi_2,...,\xi_n) \in \mathbb{R}^n$.
Here, we assume that $2\leq p_1  \leq \cdots \leq p_n$  and $a_i(x):\Omega \to \mathbb{R}$ are bounded, measurable coefficients such that, for every $i = 1,...,n$,   $$0< \lambda \leq a_i (x) \leq \Lambda  ,$$ for some positive constants $\lambda$, $ \Lambda$ and for a.e.\ $ x \in \Omega$.

One can easily check  that there exist positive constants $ L, l$ depending on $\lambda, \Lambda, p_i$ such that
\begin{equation}\label{A2}
    \langle D_\xi f(x, \xi)-D_\xi f(x, \eta),\xi-\eta\rangle  \geq  l \sum_{i=1}^{n}(|\xi_i|^2+|\eta_i |^2)^\frac{p_i -2}{2}|\xi_i - \eta_i |^2 \tag{A1}
\end{equation}
\begin{equation}\label{A3}
     |D_\xi f(x, \xi)-D_\xi f(x, \eta)| \leq  L \sum_{i=1}^{n}(|\xi_i |^2+|\eta_i |^2)^\frac{p_i -2}{2}|\xi_i - \eta_i | ,\tag{A2}
\end{equation}

\noindent for a.e.\ $x \in \Omega$ and all $\xi, \eta \in \mathbb{R}^{ n}$.
Concerning the dependence on the $x$-variable, we assume that $$a_i(x) \in W^{1,r}_{\mathrm{loc}}(\Omega) \qquad \forall i=1, \dots,n ,$$ so that,
denoting by $g(x)=  \underset{i}{\max} \{|Da_i(x)| \}$, we have that $g(x) \in L_{\mathrm{loc}}^r (\Omega)$ and the following inequality
\begin{equation}\label{A4}
    |D_\xi f(x,\xi)-D_\xi f(y, \xi)| \leq |x-y|\,(g(x)+g(y)) \, \sum_{i=1}^{n} |\xi_i|^{p_i-1} \tag{A3}
\end{equation}
\noindent holds for a.e.\ $x,y \in \Omega$ and every $\xi \in \mathbb{R}^{n}$.

For the exponents $p_i$, we shall denote by $\mathbf{p} = (p_1, p_2, \ldots, p_n)$ and by
\begin{equation*}
    W^{1,\mathbf{p}}_{\mathrm{loc}}(\Omega) = \left\{ u \in W^{1,1}_{\mathrm{loc}}(\Omega) : u_{x_i} \in L^{p_i}_{\mathrm{loc}}(\Omega), \, i = 1, \ldots, n \right\},
    \end{equation*}
the  corresponding anisotropic Sobolev space.\\
We denote by $\overline{p}$ the harmonic average of $p_i$ and by $\overline{p}^*$ the Sobolev conjugate exponent of $\overline{p}$, i.e.
 \begin{align}\label{definizionePi}
        \frac{1}{\overline{p}} = \frac{1}{n} \sum_{i=1}^{n} \frac{1}{p_i}, \qquad \overline{p}^* = \begin{cases}\frac{n \overline{p}}{n- \overline{p}} & \quad \text{if} \ \overline{p} <n,\\
        \text{any exponent in } [1,\infty) & \quad \text{if} \  \overline{p} \ge n.
        \end{cases}
    \end{align}
   For an exponent $1 \le s \le \infty$, we will denote by $s'$ the H\"older conjugate of $s$.\\
In the following, we will assume that the function $\omega(x) $ belongs to the Sobolev space  $W^{1,{\frac{\mathbf{p}+2}{\mathbf{p}+1}}}_{\mathrm{loc}}(\Omega)$, with ${\frac{\mathbf{p}+2}{\mathbf{p}+1}} = (\frac{p_1+2}{p_1+1}, \frac{p_2+2}{p_2+1}, \ldots, \frac{p_n+2}{p_n+1})$.\\

\noindent Let us recall the following definition of local minimizer.
\begin{dfn}
{ \em A function {$u\in W_{\rm loc}^{1,\mathbf{p}}(\Omega)$} is a  local minimizer of
\eqref{functional} if, for every open subset $\tilde{\Omega} \Subset \Omega$, we have
$\mathcal{F}(u;\tilde{\Omega})\le  \mathcal{F}(\varphi;\tilde{\Omega})$
for all $\varphi\in \mathcal{C}_0^{\infty}(\tilde{\Omega})$.}
\end{dfn}
The energy densities $f$ that satisfy anisotropic growth condition fit into the wider context of those with $(p,q)$-growth condition as follows
$$|\xi|^p \leq f(x,\xi) \leq (1 + |\xi|^2)^{\frac{q}{2}}, \qquad1<p<q, $$
where $p= c(p_i)$ and $q= \max \{p_i : i=1, \dots, n \}.$\\
The study of regularity properties of local minimizers to functionals with non-standard growth started with the pioneering papers by Marcellini \cite{MarcelliniEs1, MarcelliniEs2}.
 It is now well understood that, in order to obtain some regularity for the local minimizers of functionals with $(p,q)$-growth, $p<q$,  a restriction between the exponents $p$ and $q$ must be imposed (see the counterexamples in \cite{Giaquinta,MarcelliniEs3}).  Indeed, the gap $\frac{q}{p}$ cannot differ to much from $1$ and sufficient conditions to the regularity can be expressed as
$$\frac{q}{p}\le c(n)\displaystyle \to 1\,\,\textnormal{as}\,\,\,\,n\to \infty,$$
according to \cite{MarcelliniEs1,MarcelliniEs2}.

In recent years there has been a considerable of interest in variational problems with $(p,q)$-growth conditions (see, for example, \cite{Adi,Barone,6,12,SuggeritoRefery,Ming1,Ming2,Ming3,Ming4,Ele1,Ele2,Ele3,25,Hasto,Koch,16} and references therein). It is impossible to give a complete overview on the subject, however we refer the interested reader to the recent survey \cite{surveyMar1,surveyMing}.\\
Compared to the above mentioned papers, the difference with our work lies in the fact that the behavior of our functional at \eqref{functional} depends on the components of the gradient and, although they exhibit $(p,q)$-growth, they need to be treated with appropriate arguments.\\
Actually, the regularity of local minimizers of functionals with orthotropic structure as in \eqref{functional} is a well-studied problem: some results can be found in \cite{6,5,LemmaBrasco,Carozza,17,53,russo}, for the homogeneous case \( \omega = 0 \), and for the non-homogeneous one, we refer to  \cite{BraTau,CMM}.
\\
In the autonomous case, i.e.\ $a_i(x) \equiv 1$,  for all $i=1,...,n$, it has been proved in \cite{Brascop12}, for the sub-quadratic case  with $\omega \in L^{1+ \frac{2}{p_1}}_{\mathrm{loc}}(\Omega)$,
that any local minimizer \( u \in W^{1,\mathbf{p}}_{\text{loc}}(\Omega) \cap L^\infty_{\text{loc}}(\Omega) \)  satisfies
\begin{equation}
    |u_{x_i}|^{\frac{p_i - 2}{2}}  \, u_{x_i} \in W^{1,2}_{\text{loc}}(\Omega), \quad i = 1, \dots,n. \label{diffproperties}
\end{equation}
For the super-quadratic case, Brasco et al.\ \cite{BraTau} consider the following widely degenerate orthotropic functional
$$F(u; \Omega) = \sum_{i=1}^{n} \int_{\Omega} \frac{1}{p_i} \left( |u_{x_i}| - \delta_i \right)^{p_i} \, dx - \int_{\Omega} \omega \,u \, dx, $$
with $\delta_i \ge 0$, $p_i \ge 2$, $ u \in W^{1,\mathbf{p}}_{\text{loc}}(\Omega)\cap L^\infty_{\text{loc}}(\Omega)$ and
$ \omega \in W^{1,\mathbf{p'}}_{\text{loc}}(\Omega), $ $\mathbf{p'}=(p_1', \cdots, p_n')$, where $p_i'$ is the H\"older conjugate of $p_i$.
The authors proved the higher differentiability of the local minimizers, that is
\[
\left( |u_{x_i}| - \delta_i \right)^\frac{p_i}{2}_+ \frac{u_{x_i}}{|u_{x_i}|}  \in W^{1,2}_{\text{loc}}(\Omega), \quad i = 1, \dots, n.
\]
In particular, for local weak solutions of the anisotropic orthotropic $p$-Laplace equation (i.e. the case $\delta_i = 0$), they get the regularity at \eqref{diffproperties}.

 In the paper \cite{16} by Marcellini, it is clearly stated that currently there are few known regularity results for problems of this type, in relation to the properties of the coefficients $a_i(x)$. Indeed, as far as we know, few regularity results are available for the local minimizers of functionals with an orthotropic structure in the so-called non-autonomous case (see \cite{feopass,russo}).
More precisely, in the super-quadratic case it has been recently proved in \cite{russo} that, assuming the condition $p_n < p_1+2$ and Sobolev regularity on the coefficients $a_i(x)$ of the type $W^{1,r}$, with $r > p_n+2$, locally bounded minimizers $u$ of the functional \eqref{functional}, with $\omega  \equiv 0$, satisfy the higher differentiability properties at \eqref{diffproperties}.\\
\noindent Here, the main novelty with respect to the paper \cite{russo} is that we allow the presence of a forcing term $\omega$. We are interested in the conditions that must be imposed on the function $\omega$ in order to achieve analogous higher differentiability results for the solutions. \\
 Actually, our main result consists in proving that a suitable Sobolev regularity of the coefficients $a_i(x)$ and of the forcing term $\omega$ transfers into a higher differentiability of integer order for the local minimizers of \eqref{functional}. Moreover, we will require a weaker differentiability assumption on $\omega$ with respect to the one made in \cite{BraTau}. Indeed,  since $\frac{p_i+2}{p_i+1} < p_i'$, we have the embedding $W^{1, \mathbf{p'}}_{\mathrm{loc}}(\Omega) \hookrightarrow W^{1,{\frac{\mathbf{p}+2} {\mathbf{p}+1}}}_{\mathrm{loc}}(\Omega)$.

We recall that recent studies have shown that the weak differentiability of the map \( D_\xi f(x,\xi) \), whether of integer or fractional order with respect to the \( x \)-variable, is a sufficient condition to achieve higher differentiability (see \cite{cup, Torricelli, 25,40} for the case of Sobolev spaces with integer order and \cite{ 2,15, Grimaldi1, Ipocoana1, Russo} for the fractional one) both in case of standard and non standard growth. \\
In particular, the general requirement for obtaining higher differentiability for local minimizers is a Sobolev-type regularity for the coefficients with an exponent
$r$ that is greater than or equal to the dimension
$n$. When dealing with bounded minimizers, the situation changes, and sufficient assumptions on the Sobolev regularity of the coefficients can be made independently of the dimension (see \cite{CKP, Colombo, cup,25}). Our result follows this approach. Even though the order of summability of the coefficients depends on the dimension $n$, it does so only through the bound that allows us to handle locally bounded minimizers.

{In the statement our main result, we shall use the  auxiliary function $V_{p}(\xi)$  defined by
\begin{equation}\label{DefVp}
    V_{p}(\xi):= |\xi|^{\frac{p-2}{2}} \xi,
    \end{equation}
for all $\xi\in \mathbb{R}^{n}$.

We will denote by
 \begin{align}\label{definizionet}
        \frac{1}{\overline{q}} = \frac{1}{n} \sum_{i=1}^{n} \frac{p_i+1}{p_i+2}, \qquad t: = \begin{cases}\frac{n \overline{q}}{n- \overline{q}} & \quad \text{if} \ \overline{q} <n,\\
        \text{any exponent in } \left(\left(\frac{\overline{p}^*}{p_n}\right)',\infty \right) & \quad \text{if} \  \overline{q} \ge n.
        \end{cases}
    \end{align}

Our aim is to prove the following

For the exponents $p_i$, we shall denote by $\mathbf{p} = (p_1, p_2, \ldots, p_n)$ and by
\begin{equation*}
 W^{1,\mathbf{p}}_{\mathrm{loc}}(\Omega) = \left\{ u \in W^{1,1}_{\mathrm{loc}}(\Omega) : u_{x_i} \in L^{p_i}_{\mathrm{loc}}(\Omega), \, i = 1, \ldots, n \right\},
 \end{equation*}
the corresponding anisotropic Sobolev space.\\
We denote by $\overline{p}$ the harmonic average of $p_i$ and by $\overline{p}^*$ the Sobolev conjugate exponent of $\overline{p}$, i.e.
 \begin{align}\label{definizionePi}
 \frac{1}{\overline{p}} = \frac{1}{n} \sum_{i=1}^{n} \frac{1}{p_i}, \qquad \overline{p}^* = \begin{cases}\frac{n \overline{p}}{n- \overline{p}} & \quad \text{if} \ \overline{p} <n,\\
 \text{any exponent in } [1,\infty) & \quad \text{if} \ \overline{p} \ge n.
 \end{cases}
 \end{align}
 For an exponent $1 \le s \le \infty$, we will denote by $s'$ the H\"older conjugate of $s$.\\
In the following, we will assume that the function $\omega(x) $ belongs to the Sobolev space $W^{1,{\frac{\mathbf{p}+2}{\mathbf{p}+1}}}_{\mathrm{loc}}(\Omega)$, with ${\frac{\mathbf{p}+2}{\mathbf{p}+1}} = (\frac{p_1+2}{p_1+1}, \frac{p_2+2}{p_2+1}, \ldots, \frac{p_n+2}{p_n+1})$.\\

\noindent Let us recall the following definition of local minimizer.
\begin{dfn}
{ \em A function {$u\in W_{\rm loc}^{1,\mathbf{p}}(\Omega)$} is a local minimizer of
\eqref{functional} if, for every open subset $\tilde{\Omega} \Subset \Omega$, we have
$\mathcal{F}(u;\tilde{\Omega})\le \mathcal{F}(\varphi;\tilde{\Omega})$
for all $\varphi\in \mathcal{C}_0^{\infty}(\tilde{\Omega})$.}
\end{dfn}
The energy densities $f$ that satisfy anisotropic growth condition fit into the wider context of those with $(p,q)$-growth condition as follows
$$|\xi|^p \leq f(x,\xi) \leq (1 + |\xi|^2)^{\frac{q}{2}}, \qquad1<p<q, $$
where $p= c(p_i)$ and $q= \max \{p_i : i=1, \dots, n \}.$\\
The study of regularity properties of local minimizers to functionals with non-standard growth started with the pioneering papers by Marcellini \cite{MarcelliniEs1, MarcelliniEs2}.
 It is now well understood that, in order to obtain some regularity for the local minimizers of functionals with $(p,q)$-growth, $p<q$, a restriction between the exponents $p$ and $q$ must be imposed (see the counterexamples in \cite{Giaquinta,MarcelliniEs3}). Indeed, the gap $\frac{q}{p}$ cannot differ to much from $1$ and sufficient conditions to the regularity can be expressed as
$$\frac{q}{p}\le c(n)\displaystyle \to 1\,\,\textnormal{as}\,\,\,\,n\to \infty,$$
according to \cite{MarcelliniEs1,MarcelliniEs2}.

In recent years there has been a considerable of interest in variational problems with $(p,q)$-growth conditions (see, for example, \cite{Adi,Barone,6,12,SuggeritoRefery,Ming1,Ming2,Ming3,Ming4,Ele1,Ele2,Ele3,25,Hasto,Koch,16} and references therein). It is impossible to give a complete overview on the subject, however we refer the interested reader to the recent survey \cite{surveyMar1,surveyMing}.\\
Compared to the above mentioned papers, the difference with our work lies in the fact that the behavior of our functional at \eqref{functional} depends on the components of the gradient and, although they exhibit $(p,q)$-growth, they need to be treated with appropriate arguments.\\
Actually, the regularity of local minimizers of functionals with orthotropic structure as in \eqref{functional} is a well-studied problem: some results can be found in \cite{6,5,LemmaBrasco,Carozza,17,53,russo}, for the homogeneous case \( \omega = 0 \), and for the non-homogeneous one, we refer to \cite{BraTau,CMM}.
\\
In the autonomous case, i.e.\ $a_i(x) \equiv 1$, for all $i=1,...,n$, it has been proved in \cite{Brascop12}, for the sub-quadratic case with $\omega \in L^{1+ \frac{2}{p_1}}_{\mathrm{loc}}(\Omega)$,
that any local minimizer \( u \in W^{1,\mathbf{p}}_{\text{loc}}(\Omega) \cap L^\infty_{\text{loc}}(\Omega) \) satisfies
\begin{equation}
 |u_{x_i}|^{\frac{p_i - 2}{2}} \, u_{x_i} \in W^{1,2}_{\text{loc}}(\Omega), \quad i = 1, \dots,n. \label{diffproperties}
\end{equation}
For the super-quadratic case, Brasco et al.\ \cite{BraTau} consider the following widely degenerate orthotropic functional
$$F(u; \Omega) = \sum_{i=1}^{n} \int_{\Omega} \frac{1}{p_i} \left( |u_{x_i}| - \delta_i \right)^{p_i} \, dx - \int_{\Omega} \omega \,u \, dx, $$
with $\delta_i \ge 0$, $p_i \ge 2$, $ u \in W^{1,\mathbf{p}}_{\text{loc}}(\Omega)\cap L^\infty_{\text{loc}}(\Omega)$ and
$ \omega \in W^{1,\mathbf{p'}}_{\text{loc}}(\Omega), $ $\mathbf{p'}=(p_1', \cdots, p_n')$, where $p_i'$ is the H\"older conjugate of $p_i$.
The authors proved the higher differentiability of the local minimizers, that is
\[
\left( |u_{x_i}| - \delta_i \right)^\frac{p_i}{2}_+ \frac{u_{x_i}}{|u_{x_i}|} \in W^{1,2}_{\text{loc}}(\Omega), \quad i = 1, \dots, n.
\]
In particular, for local weak solutions of the anisotropic orthotropic $p$-Laplace equation (i.e. the case $\delta_i = 0$), they get the regularity at \eqref{diffproperties}.

We will denote by
 \begin{align}\label{definizionet}
 \frac{1}{\overline{q}} = \frac{1}{n} \sum_{i=1}^{n} \frac{p_i+1}{p_i+2}, \qquad t: = \begin{cases}\frac{n \overline{q}}{n- \overline{q}} & \quad \text{if} \ \overline{q} <n,\\
 \text{any exponent in } \left(\left(\frac{\overline{p}^*}{p_n}\right)',\infty \right) & \quad \text{if} \ \overline{q} \ge n.
 \end{cases}
 \end{align}

Our aim is to prove the following

\begin{rmk}
We observe that condition \eqref{Ipotesip} in Theorem \ref{thmBfinito} ensures that $u \in L^\infty_{\mathrm{loc}}(\Omega)$. Indeed, by Lemma \ref{LemmaTroisi}, the assumption $\omega \in W_{\mathrm{loc}}^{1,\frac{\mathbf{p}+2}{\mathbf{p}+1}}(\Omega)$ implies $\omega \in L^t_{\mathrm{loc}}(\Omega)$, where $t$ was defined at \eqref{D1}. Thus, assuming that the exponents $t$ and $p_i$ satisfy the relation $p_n < \frac{\overline{p}^*}{t'}$, we obtain that local minimizers of \eqref{functional} are locally bounded in $\Omega$ (see Theorem \ref{ulimitatoCupini} below).
If $\omega \in L^\infty_{\mathrm{loc}}(\Omega)$, then this relation becomes $p_n < \overline{p}^*$, which is the usual one when dealing with bounded minimizers (see \cite{CMMjota}).
If $\overline{p} \ge n$, this condition disappears, since according to \eqref{definizionePi} we can choose $\overline{p}^*$ large enough so that $p_1+2 < \frac{\overline{p}^*}{t'}$. \\
Moreover, since $p_i \ge 2$, for every $i=1,...,n$, it holds $\overline{q} < \overline{p}$. Therefore,
 if $\overline{p}<n$, the exponent $t$ appearing in \eqref{Ipotesip} is equal to $\frac{n \overline{q}}{n-\overline{q}}$.\\
 Eventually, we note that the exponents $\overline{p}^*$ and $t$ depend on $p_i$, hence the condition $p_n < \frac{\overline{p}^*}{t'}$ is necessary, since is not satisfied for any choice of exponents $p_i$.
\end{rmk}

\section{A regularized problem}\label{PreREG}
In this section we prove a regularity result for more regular problems with respect to \eqref{functional} that will be needed in the approximation procedure. More precisely, for $\varepsilon \geq 0$ we consider
\begin{equation}\label{Eqconepsilon}
 f_\varepsilon(x,u, \xi)= \sum_{i=1}^{n} \, \dfrac{b_i(x)}{p_i} \lvert \xi_i \rvert^{p_i} + \varepsilon(1+|\xi|^\frac{p_n}{2})^2 -\overline{\omega}(x)u
\end{equation}
with
$b_i(x):\Omega \to [0,+\infty), \, i=1, \dots, n$, non-negative, Lipschitz continuous coefficients,
i.e., the following conditions hold
\begin{equation}\label{atilde}
L_i= \sup_{x,y \in \Omega, x \neq y} \frac{|b_i(x)-b_i(y)|}{|x-y|}< \infty , \quad i=1,..,n.
\end{equation}
Setting $\tilde{f}(x, \xi)= \sum_{i=1}^{n} \, b_i(x) \lvert \xi_i \rvert^{p_i},$ one can easily check that
\begin{equation}\label{A'4}
 |D_\xi \tilde{f}(x,\xi)-D_\xi \tilde{f}(y, \xi)| \leq K |x-y|\, \, \sum_{i=1}^{n} |\xi_i|^{p_i-1} \tag{A3'}
\end{equation}
\noindent for a.e.\ $x,y \in \Omega $ and every $\xi \in \mathbb{R}^{ n}$, where $K=K(L_i, p_i)$
.\\
We recall a Lipschitz regularity Theorem for the minimizers of the functional
\begin{equation}\label{functional2}
 \mathcal{F}_\varepsilon (x, Dv) =\int_\Omega f_\varepsilon(x,v, Dv) \, dx,
\end{equation}
that can be deduced by \cite[Theorem $8.9$]{giusti} in the case $p=p_n$, where $f_\varepsilon$ was defined at \eqref{Eqconepsilon}.
\begin{thm}\label{thm2Mascolo}
 Let $\tilde{f} $ satisfy \eqref{A2}, \eqref{A3} and \eqref{A'4} and $\overline{\omega} \in L^\infty(\Omega)$.
 Let $u_\varepsilon \in W^{1,p_n}_{\mathrm{loc}}(\Omega)$ be a local minimizer of \eqref{functional2}.
Then, $u_\varepsilon \in W^{1,\infty}_{\mathrm{loc}}(\Omega)$
\end{thm}

\begin{align}
 \sum_{i=1}^n \int_{B_{\rho}} |\tau_{j,h} V_{p_i} ((u_\varepsilon)_{x_i})|^2 dx & \leq \,|h|^2 \Biggl\{ \frac{c_\beta}{(t-\rho)^2} +
\frac{c(K)}{(t-\rho)}
+ c_{\beta}
 + \frac{c \varepsilon}{(t-\rho)^2} \notag\\
& \qquad + \sum_{i=1}^n \left( \int_{B_R}|\omega_{x_i}|^{\frac{p_i+2}{p_i+1}} \, dx\right)^\frac{p_i+1}{p_i+2} \Biggr\}, \notag
 \end{align}
 and so
\begin{align}
 &\sum_{i=1}^n \int_{B_{\rho}} |\tau_{j,h} V_{p_i} ((u_\varepsilon)_{x_i})|^2 dx \leq c |h|^{2} \,\left[ 1+ \sum_{i=1}^{n} \Vert \omega_{x_i} \Vert_{L^{\frac{p_i+2}{p_i+1}}(B_{R})} \right], \label{StimaLemma}
\end{align}
for a positive constant $c = c(n,p_i,\rho, R, \lambda, K, \lvert \lvert Du \rvert \rvert_{\infty})$.
Then we have that
 $$V_{p_i}((u_\varepsilon)_{x_i}) \in W^{1,2}_{\mathrm{loc}}(\Omega), \qquad \forall i=1, \dots, n,$$
 which implies in particular that $$|(u_\varepsilon)_{x_i x_j}|^2|(u_\varepsilon)_{x_i}|^{p_i-2} \in L^1_{\mathrm{loc}}(\Omega), \qquad \forall i=1, \dots, n \text{ and } \forall j=1, \dots, n. $$
\end{proof}

\section{A priori estimates}\label{Apriorisec}
The main aim of this section is to establish the following a priori estimate which is the main step in the proof of our main result.
\begin{thm}\label{AppThm}
Let $\omega \in W^{1,\frac{\mathbf{p}+2}{\mathbf{p}+1}}_{\mathrm{loc}}(\Omega)$ and let $u \in W^{1, \mathbf{p}}_{\mathrm{loc}}(\Omega) \cap L^\infty_{\mathrm{loc}}(\Omega)$ be a local minimizer of \eqref{functional} under assumptions \eqref{A2}--\eqref{A4}, with exponents $p_i\geq 2, \forall i=1,\dots,n $, such that $p_n < p_1+2$,
and with a function $g \in L^{r}_{\mathrm{loc}}{(\Omega)}$, where $r$ satisfies the condition
\eqref{Ipotesir}.
If we assume that
\begin{equation}\label{Apriori}
V_{p_i}(u_{x_i}) \in {W^{1,2}_{\mathrm{loc}}(\Omega)}, \qquad\forall i=1,\dots,n,
\end{equation}
then the following estimates
\begin{align}\label{uxistima}
 \sum_{i=1}^{n} \left( \int_{B_{R/4}} |u_{x_i}|^{p_i +2 } dx \right) & \leq c \left(1+ \sum_{i=1}^{n} \left(\Vert u_{x_i} \Vert_{L^{p_i}(B_R)} + \Vert \omega_{x_i} \Vert_{L^\frac{p_i+2}{p_i+1}(B_{R})} \right) + \Vert g \Vert_{L^r (B_{R})} \right)^\gamma
\end{align}
and
\begin{align}
 \sum_{i=1}^{n} \left( \int_{B_{R/4}} |u_{x_i x_j}|^2|u_{x_i}|^{p_i-2}dx \right) & \leq c \left( 1+\sum_{i=1}^{n} \left(\Vert u_{x_i} \Vert_{L^{p_i}(B_R)} + \Vert \omega_{x_i} \Vert_{L^\frac{p_i+2}{p_i+1}(B_{R})} \right)+ \Vert g \Vert_{L^r (B_{R})} \right)^\gamma
 \label{StimaTeo1}
\end{align}
hold for every pair of concentric balls $B_{R/4} \subset B_{R} \Subset \Omega$, where $c = c(n,p_i,\lambda, \Lambda,R, {\Vert u \Vert_{\infty}})$ and $\gamma= \gamma (n,p_i, {r})$ are positive constants.
\end{thm}
\begin{proof}
We start by observing that, thanks to assumption \eqref{Ipotesip}, Theorem \ref{ulimitatoCupini} and Lemma \ref{LemmaTroisi}, we have $u \in L^\infty_{\mathrm{loc}}(\Omega)$. Therefore, by virtue of the local boundedness of $u$ and of assumption \eqref{Apriori}, an application of Proposition \ref{LemmaBrasco} gives that
\begin{equation}
 u_{x_k} \in L^{p_k+2}_{\mathrm{loc}}(\Omega) \qquad \text{for every } k \in \mathbb{N}. \label{highint}
\end{equation}
Fix a ball $B_R \Subset \Omega$ and consider radii $\frac{R}{4}< \rho< s<t<t'<{\sigma}<R$, a cut-off function $\eta \in C_0^{\infty}(B_t)$, with $\eta=1$ on $B_{s}$, $0 \leq \eta \leq 1$, $|D \eta | \leq \frac{c}{t-s}$ and $|h|\leq \frac{t' - t}{2}$.
We test the Euler-Lagrange equation of \eqref{functional} with the function $$\varphi = \tau_{j,-h}(\eta^2 \tau_{j,h}u),$$
thus getting
\begin{align}\label{SommaIntera}
 0= \int_{\Omega} &\langle f_{\xi} (x+e_jh, Du(x+e_jh))- f_{\xi} (x+e_jh, Du(x)) , \eta^2 \tau_{j,h} Du \rangle \, dx \notag \\
 &+ \int_{\Omega} \langle f_{\xi} (x+e_jh, Du(x+e_jh))- f_{\xi} (x+e_jh, Du(x)) , 2 \eta D \eta \tau_{j,h} u \rangle \, dx \notag \\
 &+ \int_{\Omega} \langle f_{\xi} (x+e_jh, Du(x))- f_{\xi} (x, Du(x)) , 2 \eta D \eta \tau_{j,h} u \rangle \, dx \notag \\
 &+ \int_{\Omega} \langle f_{\xi} (x+e_jh, Du(x))- f_{\xi} (x, Du(x)) , \eta^2 \tau_{j,h} Du \rangle \, dx \notag \\
 &- \int_{\Omega} \tau_{j,h}\omega(x)\eta^2 \tau_{j,h} u(x) \, dx \notag\\
 & =: I_1 + I_2 + I_3 + I_4+I_5. \notag
\end{align}
From the previous equality, we deduce that
\begin{equation}\label{SommaInt}
 I_1 \leq |I_2| + |I_3| + |I_4|+|I_5| .
\end{equation}
By virtue of assumption \eqref{A2}, we infer
\begin{align}\label{I1}
 I_1 &= \int_{\Omega} \langle f_{\xi} (x+e_j h, Du(x+ e_jh))- f_{\xi} (x+ e_j h, Du(x)) , \eta^2 \tau_{j,h} Du \rangle \, dx \notag \\
 & \geq l \int_{\Omega} \eta^2 \sum_{i=1}^{n}\, \left( \, |u_{x_i}(x+e_jh)|^2+|u_{x_i}(x) |^2 \, \right)^\frac{p_i -2}{2}| \tau_{j,h} u_{x_i} |^2 dx =: l \textbf{\textbf{RHS}}.
\end{align}

\begin{rmk}
We observe that condition \eqref{Ipotesip} in Theorem \ref{thmBfinito} ensures that $u \in L^\infty_{\mathrm{loc}}(\Omega)$. Indeed, by Lemma \ref{LemmaTroisi}, the assumption $\omega \in  W_{\mathrm{loc}}^{1,\frac{\mathbf{p}+2}{\mathbf{p}+1}}(\Omega)$ implies $\omega \in L^t_{\mathrm{loc}}(\Omega)$, where $t$ was defined at \eqref{D1}. Thus, assuming that the exponents $t$ and $p_i$ satisfy the relation $p_n < \frac{\overline{p}^*}{t'}$, we obtain that local minimizers of \eqref{functional} are locally bounded in $\Omega$ (see Theorem \ref{ulimitatoCupini} below).
If $\omega \in L^\infty_{\mathrm{loc}}(\Omega)$, then this relation becomes $p_n  < \overline{p}^*$, which is the usual one when dealing with bounded minimizers (see \cite{CMMjota}).
If $\overline{p} \ge n$, this condition disappears, since according to \eqref{definizionePi} we can choose $\overline{p}^*$ large enough so that $p_1+2 < \frac{\overline{p}^*}{t'}$. \\
Moreover, since $p_i \ge 2$, for every $i=1,...,n$, it holds $\overline{q} < \overline{p}$. Therefore,
 if $\overline{p}<n$,  the exponent $t$ appearing in \eqref{Ipotesip} is equal to $\frac{n \overline{q}}{n-\overline{q}}$.\\
 Eventually, we note that the exponents $\overline{p}^*$ and $t$ depend on $p_i$, hence the condition $p_n < \frac{\overline{p}^*}{t'}$ is necessary, since is not satisfied for any choice of exponents $p_i$.
\end{rmk}

Now, we briefly describe the strategy of the proof, which, as usual, will consist of an approximation argument, a uniform a priori estimate, and a passage to the limit. It is well known that the a priori estimate needs to be established for sufficiently regular minimizers. Therefore, the first step is to establish higher differentiability for minimizers of functionals with Lipschitz continuous coefficients, since this result is not available in the literature. As usual the higher differentiability is achieved with the use of the well known difference quotient methods that is frequently used in this context.\\
More specifically, for a given $\varepsilon \geq 0$, we investigate the function
\[
f_\varepsilon(x,u, \xi) = \sum_{i=1}^{n} \dfrac{b_i(x)}{p_i} |\xi_i|^{p_i} + \varepsilon \left( 1 + |\xi|^{\frac{p_n}{2}} \right)^2-\omega(x)u
\]
where the coefficients $b_i(x): \Omega \to \mathbb{R}$ are non-negative and satisfy the following condition
\[
L_i = \sup_{x, y \in \Omega, x \neq y} \frac{|b_i(x) - b_i(y)|}{|x - y|} < \infty.
\]
The first goal is to analyze the regularity properties of minimizers of the following functional
\[
\mathcal{F}_\varepsilon(u, \Omega) = \int_\Omega f_\varepsilon(x,u, Du) \, dx.
\]
After, we aim to establish an uniform a priori estimate, which plays a crucial role in the proof of our main result. Finally, Theorem \ref{thmBfinito} follows by proving that the a priori estimate is preserved in passing to the limit.\\

The plan of the paper is briefly described. After summarizing some known results and fixing a few notations, we prove a lemma for approximating functionals to be used in the a priori estimates in Section \ref{PreREG}. Section \ref{Apriorisec} is devoted to the proof of some a priori estimates for solutions to a family of approximating problems. Next, in Section \ref{Appro}, we pass to the limit in the approximating problems.

To prove Theorem~\ref{thmBfinito} the strategy is described as follows: it “will consist of an approximation argument, a uniform a priori estimate, and a passage to the limit.” Since “the a priori estimate needs to be established for sufficiently regular minimizers,” the first step is “to establish higher differentiability for minimizers of functionals with Lipschitz continuous coefficients,” a result the authors note “is not available in the literature.” This higher differentiability is obtained “with the use of the well known difference quotient methods.”

More specifically, for “a given $\varepsilon \ge 0$,” they introduce the approximating integrand
\[
f_\varepsilon(x,u,\xi)=\sum_{i=1}^n \frac{b_i(x)}{p_i}|\xi_i|^{p_i}+\varepsilon\bigl(1+|\xi|^{\frac{p_n}{2}}\bigr)^2-\omega(x)u,
\]
with nonnegative coefficients $b_i$ that are Lipschitz, i.e. $\sup_{x\ne y}\frac{|b_i(x)-b_i(y)|}{|x-y|}<\infty$, and the functional
\[
\mathcal F_\varepsilon(u,\Omega)=\int_\Omega f_\varepsilon(x,u,Du)\,dx.
\]
The “first goal is to analyze the regularity properties of minimizers” of $\mathcal F_\varepsilon$. After that, they “aim to establish an uniform a priori estimate,” and finally Theorem~\ref{thmBfinito} is obtained by “proving that the a priori estimate is preserved in passing to the limit.”

To prove the main theorem, the strategy is described as follows: it “will consist of an approximation argument, a uniform a priori estimate, and a passage to the limit.” Since “the a priori estimate needs to be established for sufficiently regular minimizers,” the first step is “to establish higher differentiability for minimizers of functionals with Lipschitz continuous coefficients,” a result the authors note “is not available in the literature.” This higher differentiability is obtained “with the use of the well known difference quotient methods.”

More specifically, for “a given $\varepsilon \ge 0$,” they introduce the approximating integrand
\[
f_\varepsilon(x,u,\xi)=\sum_{i=1}^n \frac{b_i(x)}{p_i}|\xi_i|^{p_i}+\varepsilon\bigl(1+|\xi|^{\frac{p_n}{2}}\bigr)^2-\omega(x)u,
\]
with nonnegative coefficients $b_i$ that are Lipschitz, i.e. $\sup_{x\ne y}\frac{|b_i(x)-b_i(y)|}{|x-y|}<\infty$, and the functional
\[
\mathcal F_\varepsilon(u,\Omega)=\int_\Omega f_\varepsilon(x,u,Du)\,dx.
\]
The “first goal is to analyze the regularity properties of minimizers” of $\mathcal F_\varepsilon$. After that, they “aim to establish an uniform a priori estimate,” and finally the main theorem is obtained by “proving that the a priori estimate is preserved in passing to the limit.”,

\label{thmBfinito}
Let $\omega \in  W_{\mathrm{loc}}^{1,\frac{\mathbf{p}+2}{\mathbf{p}+1}}(\Omega) $ and let $u \in {W_{\mathrm{loc}}^{1,\mathbf{p}}}(\Omega)$ be a local minimizer of
\begin{equation}\label{functional}
 \mathcal{F}(u,\Omega):= \, \int_\Omega \left[f(x, Du)-\omega(x)u(x) \right] dx,
\end{equation}
under assumptions
\begin{equation}\label{A2}
    \langle D_\xi f(x, \xi)-D_\xi f(x, \eta),\xi-\eta\rangle  \geq  l \sum_{i=1}^{n}(|\xi_i|^2+|\eta_i |^2)^\frac{p_i -2}{2}|\xi_i - \eta_i |^2 \tag{A1}
\end{equation}
and
\begin{equation}\label{A4}
    |D_\xi f(x,\xi)-D_\xi f(y, \xi)| \leq |x-y|\,(g(x)+g(y)) \, \sum_{i=1}^{n} |\xi_i|^{p_i-1} \tag{A3}
\end{equation}
with exponents $p_i\geq 2, \forall i=1,\dots,n $, such that
\begin{equation}\label{Ipotesip}
p_n <
\begin{cases}
    \min \Big\{ \dfrac{\overline{p}^*}{t'} ,p_1 +2 \Big\} \, &\text{if } \overline{p} < n,  \\
     p_1 +2 \quad &\text{otherwise},
    \end{cases}
\end{equation}
and with a function $g \in {L^{r}_{\mathrm{loc}}(\Omega)}$, where $r$ satisfies the condition
\begin{equation}\label{Ipotesir}
   r> p_n + 2 .
\end{equation}
Then,
\begin{equation*}
{V_{p_i}(u_{x_i}) \in W^{1,2}_{\mathrm{loc}}(\Omega)} \qquad\forall i=1,\dots,n,
\end{equation*}
and the following estimates
\begin{align*}
    &\sum_{i=1}^{n} \left(  \int_{B_{R/4}} |u_{x_i}|^{p_i +2 } dx \right) \notag\\
    &\leq c  \left( 1+\sum_{i=1}^{n} \left(\Vert u_{x_i} \Vert_{L^{p_i}(B_R)} + {\Vert \omega_{x_i} \Vert_{L^\frac{p_i+2}{p_i+1}(B_{R})}} \right) +\Vert \omega\Vert_{L^t(B_R)}+ \Vert g \Vert_{L^r (B_{R})} \right)^\sigma
\end{align*}
and
\begin{align*}
        &\sum_{i=1}^{n}  \left( \int_{B_{R/4}} |u_{x_i x_j}|^2|u_{x_i}|^{p_i-2}dx \right) \notag\\
         &\leq c  \left( 1+\sum_{i=1}^{n} \left(\Vert u_{x_i} \Vert_{L^{p_i}(B_R)} + {\Vert \omega_{x_i} \Vert_{L^\frac{p_i+2}{p_i+1}(B_{R})}} \right) +\Vert \omega\Vert_{L^t(B_R)}+ \Vert g \Vert_{L^r (B_{R})} \right)^\sigma
\end{align*}
hold for every pair of concentric balls $B_{R/4} \subset B_{R} \Subset \Omega$, where $c = c(n,p_i,\lambda, \Lambda,R, \Vert u \Vert_{\infty})$ and $\sigma= \sigma (n,p_i, r)$ are positive constants.,

Question:
Let \(\Omega\subset \mathbb{R}^n\), let \(\mathbf p=(p_1,\dots,p_n)\) with \(p_i\ge 2\) for all \(i\), and define the anisotropic Sobolev space
\[
W^{1,\mathbf p}_{\mathrm{loc}}(\Omega)=\{u\in W^{1,1}_{\mathrm{loc}}(\Omega): u_{x_i}\in L^{p_i}_{\mathrm{loc}}(\Omega)\ \text{for }i=1,\dots,n\}.
\]
Let
\[
\frac1{\overline p}=\frac1n\sum_{i=1}^n \frac1{p_i},
\qquad
\overline p^*=\frac{n\overline p}{n-\overline p}\ \text{if }\overline p<n,
\]
and let \(t'\) denote the H\"older conjugate of \(t\). Assume \(\omega\in L^t_{\mathrm{loc}}(\Omega)\) and \(\omega\in W^{1,\frac{\mathbf p+2}{\mathbf p+1}}_{\mathrm{loc}}(\Omega)\), meaning \(\omega_{x_i}\in L^{\frac{p_i+2}{p_i+1}}_{\mathrm{loc}}(\Omega)\) for each \(i\). Let \(g\in L^r_{\mathrm{loc}}(\Omega)\) with \(r>p_n+2\), and let \(u\in W^{1,\mathbf p}_{\mathrm{loc}}(\Omega)\) be a local minimizer of
\[
\mathcal F(u;\Omega)=\int_\Omega \bigl[f(x,Du)-\omega(x)u(x)\bigr]\,dx.
\]
Assume furthermore that for all \(x,y\in\Omega\) and \(\xi,\eta\in\mathbb R^n\),
\[
\langle D_\xi f(x,\xi)-D_\xi f(x,\eta),\xi-\eta\rangle
\ge l\sum_{i=1}^n (|\xi_i|^2+|\eta_i|^2)^{\frac{p_i-2}{2}}|\xi_i-\eta_i|^2,
\]
and
\[
|D_\xi f(x,\xi)-D_\xi f(y,\xi)|
\le |x-y|\,(g(x)+g(y))\sum_{i=1}^n |\xi_i|^{p_i-1}.
\]
Assume also that
\[
p_n<
\begin{cases}
\min\left\{\dfrac{\overline p^*}{t'},\,p_1+2\right\}, & \overline p<n,\\[4pt]
p_1+2, & \overline p\ge n.
\end{cases}
\]
For \(s\in\mathbb R\), write \(V_{p_i}(s)=|s|^{\frac{p_i-2}{2}}s\). Under these assumptions, which quantitative estimate holds?

For every \(i=1,\dots,n\),
\[
V_{p_i}(u_{x_i})\in W^{1,2}_{\mathrm{loc}}(\Omega).
\]
Moreover, for every pair of concentric balls \(B_{R/4}\subset B_R\Subset \Omega\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_i}|^{p_i+2}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^r(B_R)}\right)^\sigma,
\]
and, for each fixed \(j=1,\dots,n\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_i x_j}|^2\,|u_{x_i}|^{p_i-2}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^r(B_R)}\right)^\sigma,
\]
where \(c>0\) depends on \(n\), the exponents \(p_i\), the structural constants in the hypotheses, \(R\), and \(\|u\|_\infty\), while \(\sigma>0\) depends only on \(n\), the exponents \(p_i\), and \(r\).

Choices:
B.
For every \(i=1,\dots,n\),
\[
V_{p_i}(u_{x_i})\in W^{1,2}_{\mathrm{loc}}(\Omega).
\]
Moreover, for every pair of concentric balls \(B_{R/4}\subset B_R\Subset \Omega\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_i}|^{p_i+2}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^{p_n+2}(B_R)}\right)^\sigma,
\]
and, for each fixed \(j=1,\dots,n\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_i x_j}|^2\,|u_{x_i}|^{p_i-2}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^{p_n+2}(B_R)}\right)^\sigma,
\]
where \(c>0\) depends on \(n\), the exponents \(p_i\), the structural constants in the hypotheses, \(R\), and \(\|u\|_\infty\), while \(\sigma>0\) depends only on \(n\), the exponents \(p_i\), and \(r\).
C.
For every pair of concentric balls \(B_{R/4}\subset B_R\Subset \Omega\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_i}|^{p_i+2}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^r(B_R)}\right)^\sigma,
\]
where \(c>0\) depends on \(n\), the exponents \(p_i\), the structural constants in the hypotheses, \(R\), and \(\|u\|_\infty\), while \(\sigma>0\) depends only on \(n\), the exponents \(p_i\), and \(r\).
D.
For every \(i=1,\dots,n\),
\[
V_{p_i}(u_{x_i})\in W^{1,2}_{\mathrm{loc}}(\Omega).
\]
Moreover, there exist constants \(c>0\) and \(\sigma>0\), depending only on \(n\), the exponents \(p_i\), the structural constants in the hypotheses, and \(r\), such that for every pair of concentric balls \(B_{R/4}\subset B_R\Subset \Omega\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_i}|^{p_i+2}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^r(B_R)}\right)^\sigma,
\]
and, for each fixed \(j=1,\dots,n\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_i x_j}|^2\,|u_{x_i}|^{p_i-2}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^r(B_R)}\right)^\sigma.
\]
E.
For every \(i=1,\dots,n\),
\[
V_{p_i}(u_{x_i})\in W^{1,2}_{\mathrm{loc}}(\Omega).
\]
Moreover, for every pair of concentric balls \(B_{R/4}\subset B_R\Subset \Omega\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_i}|^{p_i+2}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^r(B_R)}\right)^\sigma,
\]
and, for every \(j=1,\dots,n\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_i x_j}|^2\,|u_{x_i}|^{p_i}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^r(B_R)}\right)^\sigma,
\]
where \(c>0\) depends on \(n\), the exponents \(p_i\), the structural constants in the hypotheses, \(R\), and \(\|u\|_\infty\), while \(\sigma>0\) depends only on \(n\), the exponents \(p_i\), and \(r\).

dropped the higher-differentiability conclusion and second estimate

dependence of constants on R and \|u\|_\infty

weighted second-derivative structure coming from V_{p_i}


Question:
Let \(\Omega\subset \mathbb{R}^n\), let \(\mathbf p=(p_1,\dots,p_n)\), and let
\[
W^{1,\mathbf p}_{\mathrm{loc}}(\Omega)=\{u\in W^{1,1}_{\mathrm{loc}}(\Omega):u_{x_i}\in L^{p_i}_{\mathrm{loc}}(\Omega)\ \text{for }i=1,\dots,n\}.
\]
Define the harmonic average \(\bar p\) by
\[
\frac1{\bar p}=\frac1n\sum_{i=1}^n \frac1{p_i},
\]
and its Sobolev conjugate by
\[
\bar p^*=
\begin{cases}
\dfrac{n\bar p}{n-\bar p}, & \bar p<n,\\[2mm]
\text{any exponent in }[1,\infty), & \bar p\ge n.
\end{cases}
\]
For an exponent \(s\), write \(s'\) for its H\"older conjugate. Assume
\(\omega\in W^{1,\frac{\mathbf p+2}{\mathbf p+1}}_{\mathrm{loc}}(\Omega)\), i.e.
\(\omega_{x_i}\in L^{\frac{p_i+2}{p_i+1}}_{\mathrm{loc}}(\Omega)\) for every \(i\), and let
\(u\in W^{1,\mathbf p}_{\mathrm{loc}}(\Omega)\) be a local minimizer of
\[
\mathcal F(u;\Omega)=\int_\Omega \bigl[f(x,Du)-\omega(x)u(x)\bigr]\,dx.
\]
Suppose that, for all \(x,y\in\Omega\) and all \(\xi,\eta\in\mathbb R^n\),
\[
\langle D_{\xi}f(x,\xi)-D_{\xi}f(x,\eta),\xi-\eta\rangle
\ge l\sum_{i=1}^n (|\xi_i|^2+|\eta_i|^2)^{\frac{p_i-2}{2}}|\xi_i-\eta_i|^2,
\]
and
\[
|D_{\xi}f(x,\xi)-D_{\xi}f(y,\xi)|
\le |x-y|\,(g(x)+g(y))\sum_{i=1}^n |\xi_i|^{p_i-1},
\]
where \(p_i\ge 2\) for all \(i=1,\dots,n\), \(g\in L^r_{\mathrm{loc}}(\Omega)\), \(r>p_n+2\), and
\[
p_n<
\begin{cases}
\min\!\left\{\dfrac{\bar p^*}{t'},\,p_1+2\right\}, & \bar p<n,\\[2mm]
p_1+2, & \bar p\ge n.
\end{cases}
\]
If \(V_{p_i}(s):=|s|^{\frac{p_i-2}{2}}s\), which statement about \(u\) holds under these hypotheses?

For every \(i=1,\dots,n\),
\[
V_{p_i}(u_{x_i})\in W^{1,2}_{\mathrm{loc}}(\Omega).
\]
Moreover, for every pair of concentric balls \(B_{R/4}\subset B_R\Subset\Omega\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_i}|^{p_i+2}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^r(B_R)}\right)^{\sigma},
\]
and, for each \(j=1,\dots,n\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_ix_j}|^2|u_{x_i}|^{p_i-2}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^r(B_R)}\right)^{\sigma},
\]
where \(c=c(n,p_i,\lambda,\Lambda,R,\|u\|_{\infty})>0\) and \(\sigma=\sigma(n,p_i,r)>0\).

Choices:
B.
For every \(i=1,\dots,n\),
\[
V_{p_i}(u_{x_i})\in W^{1,2}_{\mathrm{loc}}(\Omega).
\]
Moreover, for every pair of concentric balls \(B_{R/4}\subset B_R\Subset\Omega\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_i}|^{p_i+2}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^{p_n+2}(B_R)}\right)^{\sigma},
\]
and, for each \(j=1,\dots,n\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_ix_j}|^2|u_{x_i}|^{p_i-2}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^{p_n+2}(B_R)}\right)^{\sigma},
\]
where \(c=c(n,p_i,\lambda,\Lambda,R,\|u\|_{\infty})>0\) and \(\sigma=\sigma(n,p_i,r)>0\).
C.
For every \(i=1,\dots,n\),
\[
V_{p_i}(u_{x_i})\in W^{1,2}_{\mathrm{loc}}(\Omega).
\]
D.
There exists a constant \(c=c(n,p_i,\lambda,\Lambda,R)>0\) and an exponent \(\sigma=\sigma(n,p_i)>0\), independent of \(u\), \(g\), \(r\), and the choice of concentric balls \(B_{R/4}\subset B_R\Subset\Omega\), such that for every pair of concentric balls \(B_{R/4}\subset B_R\Subset\Omega\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_i}|^{p_i+2}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^r(B_R)}\right)^{\sigma},
\]
and, for each \(j=1,\dots,n\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_ix_j}|^2|u_{x_i}|^{p_i-2}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^r(B_R)}\right)^{\sigma},
\]
and consequently \(V_{p_i}(u_{x_i})\in W^{1,2}_{\mathrm{loc}}(\Omega)\) for all \(i=1,\dots,n\).
E.
For every \(i=1,\dots,n\),
\[
u_{x_i}\in L^{p_i+2}_{\mathrm{loc}}(\Omega)
\quad\text{and}\quad
u_{x_ix_j}\in L^2_{\mathrm{loc}}(\Omega)\ \text{for each }j=1,\dots,n.
\]
Moreover, for every pair of concentric balls \(B_{R/4}\subset B_R\Subset\Omega\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_i}|^{p_i+2}\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^r(B_R)}\right)^{\sigma},
\]
and, for each \(j=1,\dots,n\),
\[
\sum_{i=1}^n \int_{B_{R/4}} |u_{x_ix_j}|^2\,dx
\le c\left(1+\sum_{i=1}^n\Bigl(\|u_{x_i}\|_{L^{p_i}(B_R)}+\|\omega_{x_i}\|_{L^{\frac{p_i+2}{p_i+1}}(B_R)}\Bigr)+\|\omega\|_{L^t(B_R)}+\|g\|_{L^r(B_R)}\right)^{\sigma},
\]
where \(c=c(n,p_i,\lambda,\Lambda,R,\|u\|_{\infty})>0\) and \(\sigma=\sigma(n,p_i,r)>0\).

strict integrability threshold r>p_n+2 replaced by endpoint r=p_n+2

dropped the quantitative local estimates on \(|u_{x_i}|^{p_i+2}\) and weighted second derivatives

dependence of constants/exponents made uniform by removing dependence on \(\|u\|_\infty\) and \(r\)

weighted second-derivative control confused with unweighted \(L^2\) Hessian regularity


\begin{theorem}
\label{thm:opt-iid-homo}
Assume {\rm(A1), (A2), (A3)}, and $d\ge 3$. Let $\bar a,\bar\psi$ be as in Theorem~\ref{thm:opt-quant-homo}. Suppose that $f\in C^{4,\alpha}(\bar\B_1)$ and $g\in C^{6,\alpha}(\partial\B_1)$ for some $\alpha\in (0,1)$. Then, there exists a $C^6(\bar\B_1)$-extension of $g$ such that the solutions  $u, \bar u$ of \eqref{eq:elliptic-dirich}, \eqref{eq:effective-ellip} satisfy
\[
\max_{x\in B_R}|u(x)-\bar u(\tfrac{x}{R})|
\lesssim
\left\{
  \begin{array}{rl}
  	R^{-3/2}\norm{\bar u}_{C^6(\bar\B_1)}\tx &\text{ when } d=3,\\
  	R^{-2}(\log R)\norm{\bar u}_{C^6(\bar\B_1)}\tx&\text{ when }d\ge 4.
  \end{array}
\right.
\]
An example of such an extension is $g=\bar u$.
\end{theorem}

\begin{remark}\label{rmk:optimality}
This proposition identifies $O(R^{-1})$ as the optimal homogenization rate unless $\bar z \equiv 0$, and so no improvement beyond $O(R^{-1})$ should be expected in general.
         We further expect that Proposition~\ref{lem:two-scale-exp} remains valid for environments with finite range of dependence, for the same reasons discussed in the comments following Theorem~\ref{thm:opt-quant-homo}.
\end{remark}

\begin{align}
&\lambda^k_j(x)=\lambda^k_j(x;\omega) := a_j(x)\left[v^k(x+e_j)-v^k(x-e_j)\right],\;\;
\bar{\lambda}^k_j(x) :=E_{\mb P}\left[\rho(x)\lambda^k_j(x)\right],\label{eq:def-lambda}\\
&\eta_j(x)=\eta_j(x;\omega)\; :=a_j(x)\left[\xi(x+e_j)-\xi(x-e_j)\right],
\quad\;\;\,
\bar{\eta}_j(x) :=E_{\mb P}\left[\rho(x)\eta_j(x)\right],\label{eq:def-eta}
\end{align}

\label{sec:intro}
In this paper, we consider the homogenization convergence rates of purely second-order non-divergence form difference operators in an independent and identically distributed (i.i.d.) random environment on $\Z^d$ for dimensions $d\ge 3$.

Large-scale behavior of random walks in i.i.d.\ environments has been intensively studied; see \cite{BoSz-02, OZ-04, MB-11, DreRam-14, Kumagai-14} and the references therein. Since the dynamics of random walks in random environments (RWRE) can be described by difference equations, stochastic homogenization of difference equations in random environments is crucial for understanding the large-scale behavior of RWRE.
For environments in which spectral-gap or Efron--Stein type inequalities are available, the strong concentration properties of the medium can be exploited to obtain quantitative homogenization results; see, for example, \cite{GNO-15, AL-17, GT-23}. In this paper, we show that, for non-divergence form difference operators in the i.i.d.\ setting, the reflection symmetry of the law of the environment can in fact be used to extract additional cancellations. Somewhat unexpectedly, this yields homogenization rates for the Dirichlet problem that are strictly better than the standard optimal rate $O(R^{-1})$ obtained in \cite{GT-23}. To achieve this, we require a more delicate analysis of the fluctuations of the homogenization error, relying in particular on a deeper understanding of higher-order correctors.

\subsection{Settings}
Let $\mb S_{d\times d}$ denote the set of real $d\times d$ positive definite diagonal matrices. A map
\[
\omega:\Z^d\to\mb S_{d\times d}
\] is called an {\it environment}, and the set of all such environments is denoted by $\Omega$.
Let $\mb P$ be a probability measure on $\Omega$ such that
\[
\left\{\omega(x)=\mathrm{diag}[\omega_1(x),\ldots, \omega_d(x)], \;x\in\Z^d\right\}
\]
are i.i.d. under $\mb P$. We denote the expectation with respect to $\mb P$ by $\mb E$ (or $E_{\mb P}$).

\begin{definition}\label{def:differences}
Let $\{e_1,\ldots,e_d\}$ be the canonical basis for $\R^d$, and let
\[
U:=\left\{e\in\Z^d:|e|=1\right\}=\left\{\pm e_1,\ldots,\pm e_d\right\}
\]
be the set of unit vectors in $\Z^d$. We define the difference operators $\nabla=(\nabla_e)_{e\in U}$  and
$\nabla^2= \mathrm{diag}[\nabla_1^2,\ldots,\nabla_d^2]$ by
\begin{equation}\label{eq:def-nabla}
\nabla_e u(x) := u(x+e)-u(x), \qquad
\nabla_i^2u(x) := u(x+e_i)+u(x-e_i)-2u(x)
\end{equation}
for $e\in U$ and $i\in\{1,\ldots,d\}$.
Clearly, $\nabla$ and $\nabla^2$ are linear operators.
\end{definition}

For $r>0$ and $y\in\R^d$, we write
\[
\B_r(y) := \left\{x\in\R^d: |x-y|<r\right\}, \qquad
B_r(y):=\B_r(y)\cap\Z^d
\]
to denote the continuous and discrete ball with center $y$ and radius $r$, respectively.
When $y=0$, we write $\B_r := \B_r(0)$ and $B_r := B_r(0)$ for simplicity.
For any $B\subset\Z^d$, its {\it discrete boundary} $\partial B$ and {\it discrete closure} $\bar B$ are defined as
\[
\partial B:=\left\{z\in\Z^d\setminus B: |z-x|=1 \text{ for some }x\in B\right\},\qquad \bar B := B\cup\partial B.
\]
By abuse of notation, we also use $\partial A$ and $\bar A$ to denote the usual continuous boundary and closure of $A\subset\R^d$, respectively, if there is no confusion.
The interior of $B_R$ is denoted by
\[
\inB{R} := \{x\in B_R\,:\,  |x-y| \ge 1 \text{ for all }y\in\partial\B_R\}.
\]
Note that for any $R\ge 1$, there holds
\[
\bar{\B}_{R-1}\cap\Z^d\subset\inB
R\subset B_R\subset\binB{R}\subset\bar\B_R\cap\Z^d\subset\bar B_R.
\]

 For $x\in\Z^d$, a {\it spatial shift} is defined by
 \[
 \theta_x:\Omega\to\Omega,\qquad \theta_x\omega := \omega(x+\cdot).
 \]
In a random environment $\omega\in\Omega$ in dimensions $d\ge 2$,  we consider the discrete {\it non-divergence form}   elliptic Dirichlet problem
\begin{equation}\label{eq:elliptic-dirich}
\left\{
\begin{array}{lr}
\tfrac 12\tr\big(\omega(x)\nabla^2u(x)\big)=\frac{1}{R^2}f\left(\tfrac{x}{R}\right)\zeta(\theta_x\omega) & x\in \inB R,\\[5 pt]
u(x)=g\left(\tfrac{x}{R}\right) & x\in \partial \inB R,
\end{array}
\right.
\end{equation}
where $f,g\in\R^{\bar\B_1}$ are sufficiently regular, and $\zeta\in \R^\Omega$ satisfies suitable integrability conditions.
As $R\to\infty$, it is expected that $\vert u - \bar u(\frac{\cdot}{R})\rvert$ converges to $0$, where $\bar u$ solves
\begin{equation}\label{eq:effective-ellip}
\left\{
\begin{array}{lr}
\tfrac 12\tr(\bar a\, D^2\bar{u}(x))
=f(x)\,\bar\psi &x\in\B_1,\\
\bar u(x)=g(x) &x\in\partial \B_1.
\end{array}
\right.
\end{equation}
Here, $D^2 \bar{u}$ denotes the Hessian matrix of $\bar{u}$, and $\bar a=\bar a(\mb P)\in\mb S_{d\times d}$ and $\bar\psi=\bar\psi(\mb P,\zeta)\in\R$ are {\it deterministic} and do not depend on the realization of the random environment; see Theorem~\ref{thm:QCLT} and Theorem~\ref{thm:opt-quant-homo} for formulas of $\bar{a}$ and $\bar{\psi}$.

\medskip

We remark that in the continuous PDE setting, one only needs a function $g\in C^{2,\alpha}(\partial\B_1)$ on the boundary $\partial\B_1$ to consider Dirichlet problems of types \eqref{eq:elliptic-dirich} and \eqref{eq:effective-ellip}. However, in the discrete setting,
the discrete and continuous boundaries do not agree, and therefore the function $g$ in $\eqref{eq:elliptic-dirich}$ should be extended to $\frac{1}{R}\partial\inB R\not\subset\partial\B_1$.  This discrepancy typically induces an error of order $\tfrac{1}{R}$ which automatically becomes the ceiling of the optimal homogenization error in the discrete setting,  cf.  \cite{GT-23}.
Indeed, this $O(R^{-1})$ rate is the optimal rate obtained in \cite{GT-23} for $d\geq 3$.
Whereas, when such a discretization issue is not present, e.g., in the continuous periodic setting \cite{GTY-19, ST-21, Spr-24},  one needs to go deeper into higher order correctors to justify the optimality of the rate $O(R^{-1})$ for the homogenization of the Dirichlet problem. We refer to \cite{CSS-20, FGP-24, SSZ-25} for recent developments in numerical homogenization of PDEs in non-divergence form.

\medskip

A goal of this article is to isolate and examine the homogenization rate that does {\bf not} originate from boundary discretization.
To this end, we assume that the boundary data $g$ is properly extended to be a function on $\frac{1}{R}\partial\inB R$ in the Dirichlet problem \eqref{eq:elliptic-dirich}.

\medskip

The non-divergence form difference equation \eqref{eq:elliptic-dirich} is used to describe random walks in a random environment (RWRE) in $\Z^d$. To be specific, for $x,y\in \Z^d$, we set
\begin{equation}\label{def:omegaxei}
\omega(x,x\pm e_i):=\frac{\omega_i(x)}{2\,\tr\omega(x)} \quad  \text{ for } i=1,\ldots, d,
\end{equation}
 and
 \[
 \omega(x,y) := 0 \quad \text{ if $|x-y|\neq 1$},
 \]
 that is, we normalize $\omega$ to get a transition probability.
 We note that the configuration of $\{\omega(x,y):x,y\in\Z^d\}$ is also called a {\it balanced environment} in the literature \cite{L-82,GZ-12,BD-14,DGR-15, BCDG-18,DG-19}.
\begin{definition}\label{def:rwre-discrete}
For each fixed $\omega\in\Omega$, the random walk  $(X_n)_{n\ge 0}$ in the environment $\omega$ with $X_0=x$ is a Markov chain  in $\Z^d$ with transition probability $P_\omega^x$ specified by
\begin{equation}\label{eq:def-RW}
P_\omega^x\left(X_{n+1}=z|X_n=y\right) := \omega(y,z).
\end{equation}
The {\it continuous-time} RWRE $(Y_t)_{t\ge 0}$ in $\omega$ is defined to be the Markov process on $\Z^d$ with jump rates $\omega(x,y)$ from $x$ to $y$. With abuse of notations, we still denote by $P_\omega^x$  the quenched law of $(Y_t)$ if there is no ambiguity from the context.
\end{definition}
The expectation with respect to $P_\omega^x$ is denoted by $E_\omega^x$. When the starting point of the random walk is $0$, we sometimes omit the superscript and write $P_\omega^0$ and $E_\omega^0$ as $P_\omega$ and $E_\omega$, respectively. Note that for random walks $(X_n)$ in an environment $\omega$, we have that
\begin{equation}\label{def:omegabar}
\bar\omega^i := \theta_{X_i}\omega\in\Omega, \qquad i\ge 0,
\end{equation}
is also a Markov chain, called the {\it environment viewed from the particle} process. By abuse of notation, we enlarge our probability space so that $P_\omega$ still denotes the joint law of the random walks and $(\bar\omega^i)_{i\ge 0}$.

For $\omega\in\Omega$ and $i\in \{1,\ldots,d\}$, we set $a_i(x)=a_i^\omega(x) := \frac{\omega_i(x)}{\tr\omega(x)} = 2\omega(x,x+e_i)$ and
\begin{equation}
\label{eq:def-a}
a(x)=a^\omega(x):=\diag[a_1(x),\ldots, a_d(x)]=\frac{\omega(x)}{\tr\omega(x)}, \quad x\in\Z^d.
\end{equation}
Finally, for $\omega\in \Omega$, we introduce the difference operator $L_{\omega}$ given by
\begin{align*}
  L_\omega u(x) := \sum_{y\in \Z^d}\omega(x,y)[u(y)-u(x)]=\tfrac{1}{2}\tr(a(x)\nabla^2 u(x)).
\end{align*}
That is, $L_\omega$ is the generator of the process $(Y_t)_{t\ge 0}$.

\subsection{Main assumptions and some notations}
Throughout the paper, the following assumptions are always in force:
\begin{enumerate}[(\text{A}1)]
\item $\left\{\omega(x),\, x\in \Z^d \right\}$ are i.i.d. under the probability measure $\mb P$.
\item\label{item:ellip-cond} $\frac{\omega}{\tr\omega}\ge 2\kappa{\rm I}$ for $\mb P$-almost every $\omega$ and some constant $\kappa\in(0,\tfrac{1}{2d}]$.
\item $\psi(\omega):=\tfrac{\zeta(\omega)}{\tr\omega(0)}$ is a $L^\infty(\mb P)$-bounded measurable function of $\omega(0)$.
\end{enumerate}
Throughout this paper, we use $c, C$ to denote positive constants which may change from line to line but only depend on the dimension $d$ and the ellipticity constant $\kappa$ unless stated otherwise. We write $A
\lesssim B$ if $A\le CB$, and $A\asymp B$ if $A\lesssim B$ and $A\gtrsim B$.
We also use the notations $A\lesssim_j B$ and $A\asymp_j B$ to indicate that the multiplicative constant depends on the variable $j$ other than $(d,\kappa)$.
We will use the convention of summation over repeated indices.
For example, $a_kb_k$ represents $\sum_{k=1}^d a_kb_k$.
This convention is not used in definitions \eqref{eq:def-lambda} and \eqref{eq:def-eta}.
For simplicity, we use $\tx=\tx(\omega)$ to denote a generic {\it stretched-exponentially integrable} random variable (i.e., $\mb E[\exp(\tx^c)]<C$) whose value may differ from line to line.
We write $\tx_x(\omega):=\tx(\theta_x\omega)$.
We note that $\tx$ may depend on other parameters as well, although the constants $c, C$ in the stretched-exponential moment bound only depend on $(d,\kappa)$.

\subsection{Earlier results in the literature}

We first recall the following quenched central limit theorem (QCLT) proved in \cite{L-82}, which is a discrete version of \cite{PV-82}.
\begin{thmx}\label{thm:QCLT}
Assume \rm{(A2)} and that law $\mb P$ of the environment is ergodic under spatial shifts $\{\theta_x:x\in\Z^d\}$. Then, the following assertions hold.
\begin{enumerate}[(i)]
\item\label{item:ergodic} There exists a probability measure $\mb Q\approx\mb P$ such that $(\evp{i})_{i\ge 0}$ is an ergodic (with respect to time shifts) sequence under law $\mb Q\times P_\omega$.
\item For $\mb P$-almost every $\omega$, the rescaled path $X_{n^2t}/n$ converges weakly (under law $P_\omega$) to a Brownian motion with covariance matrix
\begin{equation}
\label{eq:def-abar}
\bar a=\diag[\bar a_1,\ldots,\bar a_d]:=E_\Q[a]=E_\Q[\tfrac{\omega(0)}{\tr\omega(0)}]>0.
\end{equation}
\end{enumerate}
\end{thmx}
We denote the Radon--Nikodym derivative of $\Q$ with respect to $\mb P$ by
\begin{equation}\label{eq:def-rho}
\rho(\omega)=\dd\mb Q/\dd\mb P.
\end{equation}
If the environment is i.i.d., it was proved in \cite[Theorem 1.5]{GT-22} that
\begin{equation}
    \label{eq:rho-bounds}
    E[\exp(\rho^{-c})]+E[\exp(\rho^{c})]<\infty
\end{equation}
for some constant $c=c(\kappa,d)>0$.
For any $x\in\Z^d$, we write
\[
\rho_\omega(x):=\rho(\theta_x\omega).
\]
For any $\alpha\in(0,1]$ and any function $f$ on a set $A$, we define
\[
\osc_A f:=\sup_{x,y\in A}|f(x)-f(y)|,
\qquad
[f]_{\alpha;A} :=\sup_{x,y\in A,\; x\neq y}\frac{|f(x)-f(y)|}{|x-y|^\alpha},
\]
and, if $A$ is a finite set, for $p\in(0,\infty)$, we define
\[
\norm{f}_{p;A}:=\left(\frac{1}{\#A}\sum_{x\in A}|f(x)|^p\right)^{1/p}, \quad
\norm{f}_{\infty;A} :=\max_{x\in A}|f(x)|.
\]

For any integer $j\ge 0$, let $\fct_j$ denote the set of $j$-th order polynomials, with $\fct_0=\R$. In fact, in our paper, we will only use the cases $j=0,1$.

The following large-scale regularity result can be found in \cite{GT-23}, which is a discrete version of the $C^{1,1}$ regularity estimate in \cite[Theorem 3.1]{AL-17}.
This was first done in the periodic setting in \cite{AL-87,AL-89}.

\begin{thmx}
\label{thm:c11}
Assume {\rm(A1), (A2)},  and that $\psi$ is a {\bf local} function. Let $R\ge 1$.
There exists $\alpha=\alpha(d,\kappa)\in(0,\tfrac13)$ such that,
for any $u$ with $L_\omega u(x)=\psi(\theta_x\omega)+f(x)$ on $B_R$, $j\in\{1,2\}$, $\tx\le r<R$, we have that
\begin{equation}
\label{eq:oscillation-estimates}
\frac{1}{r^{j}}\inf_{p\in\fct_{j-1}}\osc_{B_r}\,(u-p)
\lesssim
\frac{1}{R^{j}}
\inf_{p\in\fct_{j-1}}\osc_{B_R}\,(u-p)+A_{j},
\end{equation}
where the terms $A_1=A_1(R)$ and $A_2=A_2(R,r)$ obey the bounds
\begin{align*}
&A_1\le R^{1-\alpha}\norm{\psi}_\infty+R\norm{f}_\infty
\quad\text{ and }\quad
A_1\le R^{1-\alpha}\norm{\psi+f(0)}_\infty+R^{1+\sigma}[f]_{\sigma;B_R},
\\
&A_2\le r^{-\alpha}\norm{\psi}_\infty+\log(\tfrac{R}{r})\norm{f}_\infty
\quad\text{ and }\quad
A_2\le r^{-\alpha}\norm{\psi+f(0)}_\infty+R^\sigma[f]_{\sigma;B_R}
\end{align*}
for any $\sigma\in(0,1]$. In particular, for any $R>1$ and $j\in\{1,2\}$, there holds
\begin{equation}\label{eq:c2}
|\nabla^ju(0)|\lesssim \left(\frac{\tx}{R}\right)^j
\left(
\norm{u-u(0)}_{1;B_R}+R^2\norm{\psi+f(0)}_\infty+R^{2+\sigma}[f]_{\sigma;B_{R}}
\right).
\end{equation}
\end{thmx}

Introducing the quantities
\begin{equation}\label{eq:def-mu}
\mu(R):=
\left\{
\begin{array}{lr}
R & d=2,\\
R^{1/2} &d=3,\\
(1\vee \log R)^{1/2} &d=4,\\
1 &d\ge 5,
\end{array}
\right.
\quad
\delta(R):=\left\{
\begin{array}{lr}
(\log (R\vee 2))^{3/2} &d=2,\\
1 &d\ge 3,
\end{array}
\right.
\end{equation}
the following theorems on properties of the (global) correctors and quantitative homogenization of the Dirichlet problem were established in \cite{GT-23}.

\begin{thmx}
\label{thm:global_krt}
Let $\psi$ be an $L^\infty(\mb P)$-bounded function of $\omega(0)$ with $\norm{\psi}_\infty=1$.
For each $d\ge 2$ and $\mb P$-a.e. $\omega$, there exists a function $\krt=\krt_\omega:\Z^d\to\R$ that solves
\[
L_\omega\krt(x)=\psi(\theta_x\omega)-\bar\psi \quad\text{ for }x\in\Z^d
\]
with the following properties:
\begin{enumerate}[(i)]
\item\label{item:gkrt-2} When $d\ge 5$,\quad $\mb E[\exp(c|\krt(x)/\mu(|x|)|^p)]<C_p$ for any $p\in(0,\tfrac{2d}{3d+2}),x\in\Z^d$;

When $d=3,4$,\; $\mb E[\exp(c|\krt(x)/\mu(|x|)|^p)]<C_p$ for any $p\in(0,\tfrac{2d}{3d+4}),x\in\Z^d$;

When $d=2$,\quad
$\mb E\left[
\exp\big(
c\abs{\tfrac{\krt(x)}{|x|\delta(|x|)}}^p
\big)
\right]\le C_p$ for any $p\in(0,\tfrac13), x\in\Z^d$;

\item\label{item:gkrt-3} $\mb E[\exp(c|\nabla\krt(x)/\delta(|x|)|^q)]\le C_q$ for any $q\in(0,\tfrac{d}{2d+2})$ and $x\in\Z^d$;
\item\label{item:gkrt-2nd-der}
 $\mb E[\exp(c|\nabla^2\krt(x)|^r)]\le C_r$ for any $r\in(0,\tfrac{d}{2})$ and $x\in\Z^d$;
\item\label{item:gkrt-4}(Stationarity properties)
\begin{itemize}
\item When $d\ge 5$,  the field $\{\krt_\omega(x):x\in\Z^d\}$ is stationary (under $\mb P$);
\item When $d\ge 3$, we have $\nabla\krt_\omega(x)=\nabla\krt_{\theta_x\omega}(0)$ for all $x\in\Z^d$, and the field $\{\nabla\krt_\omega(x):x\in\Z^d\}$  is stationary;
\item When $d=2$, we have $\nabla^2\krt_\omega(x)=\nabla^2\krt_{\theta_x\omega}(0)$ for all $x\in\Z^d$, and the field $\{\nabla^2\krt_\omega(x):x\in\Z^d\}$  is stationary.
\end{itemize}
\end{enumerate}
Moreover,  such a corrector $\krt$ is unique up to an additive constant when $d\ge 3$,  and is unique up to an affine transformation when $d=2$.
\end{thmx}

\begin{thmx}
\label{thm:opt-quant-homo}
Assume {\rm(A1)}, {\rm(A2)}, and {\rm(A3)}. Let $\bar a :=E_{\Q}[\frac{\omega(0)}{\tr\omega(0)}]>0$ and $\bar\psi :=E_{\Q}[\frac{\zeta}{\tr\omega(0)}]$. Suppose $f,g$ are both in $C^4(\R^d)$. Then, for any $\error\in(0,1)$ and $R\ge 2$, there exists a random variable $\ms Y=\ms Y(R,\error, \omega)>1$ with $\mb E[\exp(\ms Y^{{d}/{(2d+2)}-\error})]<C$ such that the solution $u$ of \eqref{eq:elliptic-dirich} satisfies
\[
\max_{x\in B_R}|u(x)-\bar{u}(\tfrac{x}{R})|
\lesssim
\left\{
\begin{array}{rl}
(\log R)^{2+\error}R^{-1}\norm{\bar u}_{C^4(\bar\B_1)}\ms Y &\text{ when }d=2,\\
R^{-1}\norm{\bar u}_{C^4(\bar\B_1)}\ms Y &\text{ when }d\ge 3,
\end{array}
\right.
\]
where $\bar u$ is the solution of the Dirichlet problem \eqref{eq:effective-ellip}.
\end{thmx}

We point out that Theorem \ref{thm:opt-quant-homo} also remains valid for certain finite range dependent environments that are not i.i.d.\  (for example, the ``checkerboard" environment considered in \cite{AL-17}), provided that an Efron--Stein-type concentration inequality holds. Indeed, to obtain the heat-kernel bounds with exponential integrability \cite{GT-22} and the optimal convergence rates \cite{GT-23}, the only requirements are an Efron--Stein-type concentration inequality and a finite range of dependence.
We refer the reader to \cite{CaSu10, AS-14, AL-17, BCDG-18, GPT-19, GT-22, AFL-22, JZ-23,GT-23} for recent developments of quantitative stochastic homogenization of non-divergence form elliptic PDE in both discrete and continuous settings.

\subsection{Main results}

For $x\in\Z^d$, a {\it spatial shift} is defined by
 \[
 \theta_x:\Omega\to\Omega,\qquad \theta_x\omega := \omega(x+\cdot).
 \]
In a random environment $\omega\in\Omega$ in dimensions $d\ge 2$, we consider the discrete {\it non-divergence form} elliptic Dirichlet problem
\begin{equation}\label{eq:elliptic-dirich}
\left\{
\begin{array}{lr}
\tfrac 12\tr\big(\omega(x)\nabla^2u(x)\big)=\frac{1}{R^2}f\left(\tfrac{x}{R}\right)\zeta(\theta_x\omega) & x\in \inB R,\\[5 pt]
u(x)=g\left(\tfrac{x}{R}\right) & x\in \partial \inB R,
\end{array}
\right.
\end{equation}
where $f,g\in\R^{\bar\B_1}$ are sufficiently regular, and $\zeta\in \R^\Omega$ satisfies suitable integrability conditions.
As $R\to\infty$, it is expected that $\vert u - \bar u(\frac{\cdot}{R})\rvert$ converges to $0$, where $\bar u$ solves
\begin{equation}\label{eq:effective-ellip}
\left\{
\begin{array}{lr}
\tfrac 12\tr(\bar a\, D^2\bar{u}(x))
=f(x)\,\bar\psi &x\in\B_1,\\
\bar u(x)=g(x) &x\in\partial \B_1.
\end{array}
\right.
\end{equation}
Here, $D^2 \bar{u}$ denotes the Hessian matrix of $\bar{u}$, and $\bar a=\bar a(\mb P)\in\mb S_{d\times d}$ and $\bar\psi=\bar\psi(\mb P,\zeta)\in\R$ are {\it deterministic} and do not depend on the realization of the random environment; see Theorem~\ref{thm:QCLT} and Theorem~\ref{thm:opt-quant-homo} for formulas of $\bar{a}$ and $\bar{\psi}$.

We first recall the following quenched central limit theorem (QCLT) proved in \cite{L-82}, which is a discrete version of \cite{PV-82}.
\begin{thmx}\label{thm:QCLT}
Assume \rm{(A2)} and that law $\mb P$ of the environment is ergodic under spatial shifts $\{\theta_x:x\in\Z^d\}$. Then, the following assertions hold.
\begin{enumerate}[(i)]
\item\label{item:ergodic} There exists a probability measure $\mb Q\approx\mb P$ such that $(\evp{i})_{i\ge 0}$ is an ergodic (with respect to time shifts) sequence under law $\mb Q\times P_\omega$.
\item For $\mb P$-almost every $\omega$, the rescaled path $X_{n^2t}/n$ converges weakly (under law $P_\omega$) to a Brownian motion with covariance matrix
\begin{equation}
\label{eq:def-abar}
\bar a=\diag[\bar a_1,\ldots,\bar a_d]:=E_\Q[a]=E_\Q[\tfrac{\omega(0)}{\tr\omega(0)}]>0.
\end{equation}
\end{enumerate}
\end{thmx}
We denote the Radon--Nikodym derivative of $\Q$ with respect to $\mb P$ by
\begin{equation}\label{eq:def-rho}
\rho(\omega)=\dd\mb Q/\dd\mb P.
\end{equation}
If the environment is i.i.d., it was proved in \cite[Theorem 1.5]{GT-22} that
\begin{equation}
 \label{eq:rho-bounds}
 E[\exp(\rho^{-c})]+E[\exp(\rho^{c})]<\infty
\end{equation}
for some constant $c=c(\kappa,d)>0$.
For any $x\in\Z^d$, we write
\[
\rho_\omega(x):=\rho(\theta_x\omega).
\]
For any $\alpha\in(0,1]$ and any function $f$ on a set $A$, we define
\[
\osc_A f:=\sup_{x,y\in A}|f(x)-f(y)|,
\qquad
[f]_{\alpha;A} :=\sup_{x,y\in A,\; x\neq y}\frac{|f(x)-f(y)|}{|x-y|^\alpha},
\]
and, if $A$ is a finite set, for $p\in(0,\infty)$, we define
\[
\norm{f}_{p;A}:=\left(\frac{1}{\#A}\sum_{x\in A}|f(x)|^p\right)^{1/p}, \quad
\norm{f}_{\infty;A} :=\max_{x\in A}|f(x)|.
\]

\begin{thmx}
\label{thm:c11}
Assume {\rm(A1), (A2)}, and that $\psi$ is a {\bf local} function. Let $R\ge 1$.
There exists $\alpha=\alpha(d,\kappa)\in(0,\tfrac13)$ such that,
for any $u$ with $L_\omega u(x)=\psi(\theta_x\omega)+f(x)$ on $B_R$, $j\in\{1,2\}$, $\tx\le r<R$, we have that
\begin{equation}
\label{eq:oscillation-estimates}
\frac{1}{r^{j}}\inf_{p\in\fct_{j-1}}\osc_{B_r}\,(u-p)
\lesssim
\frac{1}{R^{j}}
\inf_{p\in\fct_{j-1}}\osc_{B_R}\,(u-p)+A_{j},
\end{equation}
where the terms $A_1=A_1(R)$ and $A_2=A_2(R,r)$ obey the bounds
\begin{align*}
&A_1\le R^{1-\alpha}\norm{\psi}_\infty+R\norm{f}_\infty
\quad\text{ and }\quad
A_1\le R^{1-\alpha}\norm{\psi+f(0)}_\infty+R^{1+\sigma}[f]_{\sigma;B_R},
\\
&A_2\le r^{-\alpha}\norm{\psi}_\infty+\log(\tfrac{R}{r})\norm{f}_\infty
\quad\text{ and }\quad
A_2\le r^{-\alpha}\norm{\psi+f(0)}_\infty+R^\sigma[f]_{\sigma;B_R}
\end{align*}
for any $\sigma\in(0,1]$. In particular, for any $R>1$ and $j\in\{1,2\}$, there holds
\begin{equation}\label{eq:c2}
|\nabla^ju(0)|\lesssim \left(\frac{\tx}{R}\right)^j
\left(
\norm{u-u(0)}_{1;B_R}+R^2\norm{\psi+f(0)}_\infty+R^{2+\sigma}[f]_{\sigma;B_{R}}
\right).
\end{equation}
\end{thmx}

\begin{thmx}
\label{thm:opt-quant-homo}
Assume {\rm(A1)}, {\rm(A2)}, and {\rm(A3)}. Let $\bar a :=E_{\Q}[\frac{\omega(0)}{\tr\omega(0)}]>0$ and $\bar\psi :=E_{\Q}[\frac{\zeta}{\tr\omega(0)}]$. Suppose $f,g$ are both in $C^4(\R^d)$. Then, for any $\error\in(0,1)$ and $R\ge 2$, there exists a random variable $\ms Y=\ms Y(R,\error, \omega)>1$ with $\mb E[\exp(\ms Y^{{d}/{(2d+2)}-\error})]<C$ such that the solution $u$ of \eqref{eq:elliptic-dirich} satisfies
\[
\max_{x\in B_R}|u(x)-\bar{u}(\tfrac{x}{R})|
\lesssim
\left\{
\begin{array}{rl}
(\log R)^{2+\error}R^{-1}\norm{\bar u}_{C^4(\bar\B_1)}\ms Y &\text{ when }d=2,\\
R^{-1}\norm{\bar u}_{C^4(\bar\B_1)}\ms Y &\text{ when }d\ge 3,
\end{array}
\right.
\]
where $\bar u$ is the solution of the Dirichlet problem \eqref{eq:effective-ellip}.
\end{thmx}

We point out that Theorem \ref{thm:opt-quant-homo} also remains valid for certain finite range dependent environments that are not i.i.d.\ (for example, the ``checkerboard" environment considered in \cite{AL-17}), provided that an Efron--Stein-type concentration inequality holds. Indeed, to obtain the heat-kernel bounds with exponential integrability \cite{GT-22} and the optimal convergence rates \cite{GT-23}, the only requirements are an Efron--Stein-type concentration inequality and a finite range of dependence.
We refer the reader to \cite{CaSu10, AS-14, AL-17, BCDG-18, GPT-19, GT-22, AFL-22, JZ-23,GT-23} for recent developments of quantitative stochastic homogenization of non-divergence form elliptic PDE in both discrete and continuous settings.

\subsection{Main results}

\subsection{Main results}

\begin{proposition}
\label{lem:two-scale-exp}
Suppose that $d\geq 3$, $f\in C^{4,\alpha}(\bar\B_1)$, and $g\in C^{6,\alpha}(\partial\B_1)$. Let $u,\bar u$ be the solutions of \eqref{eq:elliptic-dirich}, \eqref{eq:effective-ellip}, respectively. Recall $a,\bar a$ from \eqref{eq:def-a}, \eqref{eq:def-abar}, and let $v^k,\xi,\lambda^k_j, \bar\lambda^k_j,\eta_j,\bar\eta_j$ be as above. We extend $g$ so that $g=\bar{u}$ on $\bar\B_1$. Let $\bar z:\bar\B_1\to\R$ be the solution of
\[
\left\{
\begin{array}{lr}
\frac{1}{2}\tr(\bar{a}D^2 \bar{z}(x))=\tfrac{1}{2}\bar{\lambda}^k_i\partial_{kki}\bar{u}(x)-\tfrac{1}{2}\bar\eta_j\partial_j f(x) & x\in\B_1,\\
\bar z(x)=0 &x\in\partial\B_1.
\end{array}
\right.
\]
For $\omega\in\Omega$, $R>1$, and $j,k\in\{1,\ldots,d\}$, let $p^k_j, s_j:\Z^d \rightarrow \R$ be solutions of
\begin{align}
L_\omega p^k_j(x)&=\lambda^k_j(x)-\bar\lambda^k_j \quad\text{ for }x\in B_R,\label{eq:def-p}\\
L_\omega s_j(x)&=\eta_j(x)-\bar\eta_j \quad\;\text{ for }x\in B_R\label{eq:def-sj},
\end{align}
respectively. Then, for $\omega\in\Omega$, $R>1$, we have
\begin{align}\label{eq:two-scale-further}
&\max_{x\in B_R}\left|u(x)-\bar{u}(\tfrac xR)-\tfrac 1R\bar z(\tfrac xR)\right|\nn\\
&\qquad\lesssim
\tfrac 1{R^2}\norm{\bar u}_{C^6(\bar{\B}_1)}\max_{i,j,k=1}^d
\bigg[1+\left\| |\nabla p^k_j|+|\nabla s_j|+(|\bar\lambda_j^i|+|\bar\eta_j|)|\nabla v^k|\right\|_{d;B_R}
\nn\\
&\qquad
\qquad\qquad\qquad\qquad\quad\;\;\,+\max_{\bar B_R}\left(|\xi|+|v^k|+\tfrac{1}{R}(|p_i^k|+|s_i|)\right)\bigg].
\end{align}
\end{proposition}

\begin{proof}
Let $z:\binB R\to\R$ be a solution of the Dirichlet problem
\[
\left\{
\begin{array}{lr}
L_\omega z(x)=\tfrac{1}{R^2}\left[ \tfrac{1}{2}\bar{\lambda}^k_i\partial_{kki}\bar{u}(\tfrac xR)-\tfrac{1}{2}\bar\eta_i\partial_i f(\tfrac xR)\right] & x\in\inB R,\\
z(x)=0 &x\in\partial\inB R.
\end{array}
\right.
\]
Consider the function
\begin{align}\label{eq:def-w}
&w(x)=u(x)-\bar u(\tfrac{x}{R})-\tfrac 1R z(x)\nn\\
&
+\tfrac{1}{R^2}\left[v^k(x)\partial_{kk}\bar u(\tfrac{x}{R})
-f(\tfrac{x}R)\xi(x)\right]- {\tfrac{1}{2}} \tfrac{1}{R^3}\left[\partial_{kki}\bar{u}(\tfrac{x}{R})p_i^k(x)-\partial_i f(\tfrac{x}{R})s_i(x)\right]
\end{align}
for $x\in\binB R\subset\bar\B_R\cap\Z^d$. Then, applying the formula
\begin{align*}
L_\omega(uv)(x)&=u(x)L_\omega v(x)+v(x) L_\omega u(x)+
\sum_{y:y\sim x}\omega(x,y)[u(y)-u(x)][v(y)-v(x)]\\
&=u(x)L_\omega v(x)+v(x) L_\omega u(x) \\ &\qquad+ \tfrac{1}{2}
a_i(x)\left(\nabla_{e_i}u(x)[v(x+e_i)-v(x-e_i)]
+\nabla_i^2u(x)\nabla_{-e_i}v(x)
\right)
\end{align*}
to the products within \eqref{eq:def-w}, we obtain in $\inB R$ that
\begin{align}
&L_\omega[v^k\,\partial_{kk}\bar u(\tfrac{\cdot}{R})]\nn\\
&=
L_\omega v^k \,\partial_{kk}\bar u(\tfrac{\cdot}{R})
+v^k \, L_\omega[\partial_{kk}\bar u(\tfrac{\cdot}{R})]+\tfrac{1}{2}\lambda^k_i\nabla_{e_i}[\partial_{kk}\bar u(\tfrac{\cdot}{R})]+\tfrac{1}{2}a_i \,\nabla_i^2[\partial_{kk}\bar u(\tfrac{\cdot}{R})]\,\nabla_{-e_i}v^k
\nn\\
&=\tfrac1{2}(a_k -\bar a_k)\partial_{kk}\bar u(\tfrac{\cdot}{R})+\tfrac{1}{2}\tfrac{1}{R}\lambda^k_i\partial_{kki}\bar u(\tfrac{\cdot}{R})+O(\tfrac 1{R^2})\norm{\bar u}_{C^5(\bar\B_1)}\max_{\binB R}\sum_{k = 1}^d|v^k|,\label{eq:term1}
\end{align}
where we used the fact that $|\lambda^k_i|\lesssim|\nabla v^k|\lesssim\max_{\binB R}|v^k|$ in $\inB R$. Similarly,
\begin{align}
L_\omega[f(\tfrac{\cdot}R)\xi ]
&=(\psi-\bar\psi)f(\tfrac{\cdot}R)+{\tfrac{1}{2}}\tfrac{1}{R}\eta_i\partial_i f(\tfrac{\cdot}{R})+O(\tfrac 1{R^2})\norm{\bar u}_{C^5(\bar\B_1)}\max_{\inB R}|\xi|,\label{eq:term2}\\
L_\omega[\partial_{kki}\bar{u}(\tfrac{\cdot}{R})p_i^k ]
&=(\lambda^k_i-\bar\lambda^k_i)\partial_{kki}\bar{u}(\tfrac{\cdot}{R})
+{\tfrac{1}{2}}\tfrac{1}{R}\partial_{kkij}\bar{u}(\tfrac{\cdot}{R})a_j[p^k_i(\cdot+e_j)-p^k_i(\cdot-e_j)]\nn\\
&+O(\tfrac 1{R^2})\norm{\bar u}_{C^6(\bar\B_1)}\max_{\binB R}\sum_{i,k = 1}^d|p^k_i|,\label{eq:term3}\\
L_\omega[\partial_i f(\tfrac{\cdot}{R})s_i ]
&=(\eta_i-\bar\eta_i)\partial_i f(\tfrac{\cdot}{R})
+{\tfrac{1}{2}}\tfrac{1}{R}\partial_{ij}f(\tfrac{\cdot}{R})a_j [s_i(\cdot+e_j)-s_i(\cdot-e_j)]\nn\\
&+O(\tfrac 1{R^2})\norm{\bar u}_{C^6(\bar\B_1)}\max_{\binB R}\sum_{i=1}^d|s_i|.\label{eq:term4}
\end{align}
Note that
$\abs{L_\omega[\bar u(\tfrac{\cdot}{R})]-\tfrac1{2R^2}\tr[a D^2\bar{u}(\tfrac{\cdot}{R})]}\lesssim\tfrac1{R^4}\norm{\bar u}_{C^4(\bar\B_1)}$, and
\begin{align*}
L_\omega u
=\tfrac1{R^2}f(\tfrac{\cdot}{R})(\psi -\bar\psi)+\tfrac1{R^2}f(\tfrac{\cdot}{R})\bar\psi =\tfrac1{R^2}\big[f(\tfrac{\cdot}{R})L_\omega\xi +
\tfrac1{2}\bar{a}_k\partial_{kk}\bar u(\tfrac{\cdot}{R})\big].
\end{align*}
Then, recalling the definition of $w$ in \eqref{eq:def-w} and writing
\[
A:=\max_{\binB R}\sum_{i,k=1}^d\left[|\xi|+|v^k|+\tfrac{1}{R}(|p_i^k|+|s_i|)\right],
\]
by \eqref{eq:term1}, \eqref{eq:term2},\eqref{eq:term3}, \eqref{eq:term4},
we have in $\inB R$ that
\begin{align*}
&\abs{L_\omega w } \\
&=
\tfrac1{R^2}\Abs{
f(\tfrac{\cdot}{R})L_\omega\xi +\tfrac1{2}\bar{a}_k\partial_{kk}\bar u(\tfrac{\cdot}{R})-R^2L_\omega[\bar{u}(\tfrac{\cdot}{R})]-{\tfrac{1}{2}}\tfrac{1}{R} \bar{\lambda}^k_i\partial_{kki}\bar{u}(\tfrac{\cdot}{R})+{\tfrac{1}{2}}\tfrac 1R\bar\eta_i\partial_i f(\tfrac{\cdot}{R})
\\
&\qquad+\tfrac1{2}(a_k-\bar a_k)\partial_{kk}\bar u(\tfrac{\cdot}{R})+{\tfrac{1}{2}}\tfrac{1}{R}\lambda^k_i\partial_{kki}\bar u(\tfrac{\cdot}{R})
\nn\\
&\qquad-(\psi-\bar\psi)f(\tfrac{\cdot}{R})-{\tfrac{1}{2}}\tfrac{1}{R}\eta_i\partial_i f(\tfrac{\cdot}{R})+O(\tfrac{1}{R^2})\norm{\bar u}_{C^6(\bar\B_1)}A\\
&\qquad-{\tfrac{1}{2}}\tfrac{1}{R}(\lambda^k_i-\bar\lambda^k_i)\partial_{kki}\bar{u}(\tfrac{\cdot}{R})
-{\tfrac{1}{4}}\tfrac{1}{R^2}\partial_{kkij}\bar{u}(\tfrac{\cdot}{R})a_j[p^k_i(\cdot+e_j)-p^k_i(\cdot-e_j)]\\
&\qquad+{\tfrac{1}{2}}\tfrac{1}{R}(\eta_i-\bar\eta_i)\partial_i f(\tfrac{\cdot}{R})
+{\tfrac{1}{4}}\tfrac{1}{R^2}\partial_{ij}f(\tfrac{\cdot}{R})a_j[s_i(\cdot+e_j)-s_i(\cdot-e_j)]
}
\nn\\
&\lesssim
\tfrac1{R^4}\norm{\bar u}_{C^6(\bar\B_1)}\sum_{i,k=1}^d\left[1+
|\nabla p_i^k|+|\nabla s_i|
+\max_{\binB R}\left(|\xi|+|v^k|+\tfrac{1}{R}(|p_i^k|+ |s_i|)\right)
\right].
\end{align*}
Thus, by the above inequality, the definition of $w$ from \eqref{eq:def-w}, and the ABP maximum principle, we obtain that
\begin{align}
&\max_{\binB R}|u -\bar u(\tfrac{\cdot}{R})-\tfrac 1R z |\label{eq:ineq1}\\
&\le \max_{\binB R}|w|+
\max_{\binB R}\Abs{\tfrac{1}{R^2}[v^k\partial_{kk}\bar u(\tfrac{\cdot}{R})
-f(\tfrac{\cdot}{R})\xi ]
-{\tfrac{1}{2}}\tfrac{1}{R^3}[\partial_{kki}\bar{u}(\tfrac{\cdot}{R})p_i^k -\partial_i f(\tfrac{\cdot}{R})s_i ]}\nn\\
&\lesssim
R^2\norm{L_\omega w}_{d;\inB R}
+\tfrac1{R^2}\norm{\bar u}_{C^3}\max_{\binB R}\left(|\xi|+|v^k|+\tfrac{1}{R}(|p_i^k|+ |s_i|)\right)\nn
\\&
\lesssim
\tfrac1{R^2}\norm{\bar u}_{C^6}\sum_{i,k=1}^d
\big[1+
\left\||\nabla p_i^k|+|\nabla s_i|\right\|_{d;B_R}+\max_{\binB R}\left(|\xi|+|v^k|+\tfrac{1}{R}(|p_i^k|+|s_i|)\right)
\big].\nn
\end{align}
Furthermore, by \cite[Lemma~30]{GT-23},
\[
\max_{\binB R}\abs{z-\bar z(\tfrac{\cdot}{R})}
\lesssim
\tfrac1{R^2}\max_{j,\ell=1}^d(|\bar\lambda_j^\ell|+|\bar\eta_j|)\norm{\bar u}_{C^5(\bar\B_1)}
\sum_{k=1}^d(1+R \norm{\nabla v^k}_{d;\inB R}+\osc_{\binB R}v^k).
\]
This inequality, together with \eqref{eq:ineq1}, yields the claimed bound \eqref{eq:two-scale-further}.
\end{proof}

In the i.i.d. setting with well-prepared boundary conditions, Theorem \ref{thm:opt-iid-homo} yields improved convergence rates compared to the rate $O(R^{-1})$ in Theorem \ref{thm:opt-quant-homo} when $d\geq 3$.
Our approaches and contributions are as follows.
We employ the two-scale asymptotic expansion to obtain the estimate \eqref{eq:two-scale-further} in Proposition \ref{lem:two-scale-exp}.
Due to the i.i.d. structure, the environment enjoys a reflection symmetry; that is, the random fields $\omega(\cdot)$ and $\omega(-\,\cdot)$ are identically distributed.
(Of course, this symmetry fails for environments with only a finite range of dependence.)
This reflection symmetry implies that the third-order homogenized tensor vanishes, which in turn suggests that a homogenization error bound sharper than
 $O(R^{-1})$ may be attainable.

We remark that such improved rates should not be expected in environments with correlations even slightly weaker than in the point-wise i.i.d.\ case, since the distribution's reflective symmetry is no longer present.

In our i.i.d.\ setting, without the presence of the third-order tensor, the homogenization rate is completely determined by the $C^{0,1}$-bounds of higher-order correctors.
Roughly speaking, the (first-order) correctors “correct” local coefficients, while the higher-order correctors account for the derivatives of the lower-order correctors.
Hence, unlike the correctors themselves, the source terms in the equations for higher-order correctors are highly nonlocal, and thus the large-scale regularity estimates in Theorem~\ref{thm:c11}, which apply only to equations with {\it local} source terms, no longer hold.

Our improved convergence rates in Theorem \ref{thm:opt-iid-homo} are inspired by earlier results in the periodic setting \cite{GST-22}, where it was proved that if the environment has ``one degree of freedom" $a(x)$, i.e., the coefficient matrix is of the form $\omega(x)=C+a(x)M$ for some constant symmetric matrices $C$ and $M$, then the third-order homogenized tensor vanishes, leading to an improved rate of $O(R^{-2})$.
The periodic setting, however, is quite rigid.
We provided examples in \cite{GST-22} that show that if the environment has two degrees of freedom, then the third-order homogenized tensor does not vanish in general, and the optimal convergence rate in such cases is only $O(R^{-1})$.

In contrast, in our random setting, $\omega(x)=\mathrm{diag}[\omega_1(x),\ldots, \omega_d(x)]$ has $d$ degrees of freedom, which is fundamentally different from the periodic case considered in \cite{GST-22}.
Moreover, in the periodic setting, the $C^{0,1}$-boundedness of higher-order correctors is automatically guaranteed by periodicity.

One of the main contributions of our work is to establish $C^{0,1}$ bounds for higher-order correctors in the random setting. Our approach is based on comparing the true higher-order correctors with certain {\it localized} higher-order correctors that possess good regularity properties and {\it approximate} higher-order correctors that are stationary.

\begin{remark}
	When $d=2$, the corrector $\upsilon^k$ grows super-linearly (Theorem~\ref{thm:global_krt}). Consequently, via the expansion \eqref{eq:two-scale-further}, it produces an error term of order $R^{-2}|v^k|$ that dominates the desired $O(R^{-1})$ scale. Therefore, in two dimensions, we should not expect an improvement beyond the $O(R^{-1})$ rate in i.i.d.\ environments.

    All in all, this demonstrates that, in the absence of additional structural assumptions on the random environment, the boundary conditions, or the dimension, the best convergence rate that one can expect is $O(R^{-1})$, as established in \cite{GT-23}. We refer to Remark~\ref{rmk:optimality} for a discussion of the optimality of the $O(R^{-1})$ rate.
\end{remark}

To prove Theorem~\ref{thm:opt-iid-homo}, the authors describe the following strategy and key ingredients.

- They “employ the two-scale asymptotic expansion to obtain the estimate \eqref{eq:two-scale-further} in Proposition~\ref{lem:two-scale-exp}.”

- They use a special feature of the point-wise i.i.d. setting: “Due to the i.i.d. structure, the environment enjoys a reflection symmetry; that is, the random fields $\omega(\cdot)$ and $\omega(-\,\cdot)$ are identically distributed.” This “reflection symmetry implies that the third-order homogenized tensor vanishes,” which “suggests that a homogenization error bound sharper than $O(R^{-1})$ may be attainable.”

- With “the absence of the third-order tensor,” they explain that “the homogenization rate is completely determined by the $C^{0,1}$-bounds of higher-order correctors.” They motivate the difficulty: while first-order correctors “correct local coefficients,” “the higher-order correctors account for the derivatives of the lower-order correctors,” and “the source terms in the equations for higher-order correctors are highly nonlocal,” so “the large-scale regularity estimates in Theorem~\ref{thm:c11}, which apply only to equations with {\it local} source terms, no longer hold.”

- A main technical step is then to “establish $C^{0,1}$ bounds for higher-order correctors in the random setting,” by “comparing the true higher-order correctors with certain {\it localized} higher-order correctors that possess good regularity properties and {\it approximate} higher-order correctors that are stationary.”

- They also note limitations: the improved rates rely on i.i.d.-driven reflection symmetry (“such improved rates should not be expected in environments with correlations… since the distribution's reflective symmetry is no longer present”), and in $d=2$ the corrector “grows super-linearly,” so the expansion yields an error term that “dominates the desired $O(R^{-1})$ scale,” hence “we should not expect an improvement beyond the $O(R^{-1})$ rate” in two dimensions.

\label{thm:opt-iid-homo}
Assume {\rm(A1), (A2), (A3)}, and $d\ge 3$. Let $\bar a,\bar\psi$ be as in the following theorem.

\begin{thmx}
\label{thm:opt-quant-homo}
Assume {\rm(A1)}, {\rm(A2)}, and {\rm(A3)}. Let $\bar a :=E_{\Q}[\frac{\omega(0)}{\tr\omega(0)}]>0$ and $\bar\psi :=E_{\Q}[\frac{\zeta}{\tr\omega(0)}]$. Suppose $f,g$ are both in $C^4(\R^d)$. Then, for any $\error\in(0,1)$ and $R\ge 2$, there exists a random variable $\ms Y=\ms Y(R,\error, \omega)>1$ with $\mb E[\exp(\ms Y^{{d}/{(2d+2)}-\error})]<C$ such that the solution $u$ of
\begin{equation}\label{eq:elliptic-dirich}
\left\{
\begin{array}{lr}
\tfrac 12\tr\big(\omega(x)\nabla^2u(x)\big)=\frac{1}{R^2}f\left(\tfrac{x}{R}\right)\zeta(\theta_x\omega) & x\in \inB R,\\[5 pt]
\nu(x)=g\left(\tfrac{x}{R}\right) & x\in \partial \inB R,
\end{array}
\right.
\end{equation}
satisfies
\[
\max_{x\in B_R}|u(x)-\bar{u}(\tfrac{x}{R})|
\lesssim
\left\{
\begin{array}{rl}
(\log R)^{2+\error}R^{-1}\norm{\bar u}_{C^4(\bar\B_1)}\ms Y &\text{ when }d=2,\\
R^{-1}\norm{\bar u}_{C^4(\bar\B_1)}\ms Y &\text{ when }d\ge 3,
\end{array}
\right.
\]
where $\bar u$ is the solution of the Dirichlet problem
\begin{equation}\label{eq:effective-ellip}
\left\{
\begin{array}{lr}
\tfrac 12\tr(\bar a\, D^2\bar{u}(x))
=f(x)\,\bar\psi &x\in\B_1,\\
\bar u(x)=g(x) &x\in\partial \B_1.
\end{array}
\right.
\end{equation}
\end{thmx}

Suppose that $f\in C^{4,\alpha}(\bar\B_1)$ and $g\in C^{6,\alpha}(\partial\B_1)$ for some $\alpha\in (0,1)$. Then, there exists a $C^6(\bar\B_1)$-extension of $g$ such that the solutions $u, \bar u$ of
\begin{equation}\label{eq:elliptic-dirich}
\left\{
\begin{array}{lr}
\tfrac 12\tr\big(\omega(x)\nabla^2u(x)\big)=\frac{1}{R^2}f\left(\tfrac{x}{R}\right)\zeta(\theta_x\omega) & x\in \inB R,\\[5 pt]
\nu(x)=g\left(\tfrac{x}{R}\right) & x\in \partial \inB R,
\end{array}
\right.
\end{equation}
and
\begin{equation}\label{eq:effective-ellip}
\left\{
\begin{array}{lr}
\tfrac 12\tr(\bar a\, D^2\bar{u}(x))
=f(x)\,\bar\psi &x\in\B_1,\\
\bar u(x)=g(x) &x\in\partial \B_1.
\end{array}
\right.
\end{equation}
satisfy
\[
\max_{x\in B_R}|u(x)-\bar u(\tfrac{x}{R})|
\lesssim
\left\{
  \begin{array}{rl}
  	R^{-3/2}\norm{\bar u}_{C^6(\bar\B_1)}\tx &\text{ when } d=3,\\
  	R^{-2}(\log R)\norm{\bar u}_{C^6(\bar\B_1)}\tx&\text{ when }d\ge 4.
  \end{array}
\right.
\]
An example of such an extension is $g=\bar u$.

Question:
Assume \((A1)\), \((A2)\), and \((A3)\), and let \(d\ge 3\). For a random environment \(\omega\), define the spatial shift by \(\theta_x\omega=\omega(x+\cdot)\). Let \(\bar a:=E_{\mathbb Q}[\omega(0)/\operatorname{tr}\omega(0)]>0\) and \(\bar\psi:=E_{\mathbb Q}[\zeta/\operatorname{tr}\omega(0)]\). Suppose \(f\in C^{4,\alpha}(\overline{\mathbb B}_1)\) and \(g\in C^{6,\alpha}(\partial\mathbb B_1)\) for some \(\alpha\in(0,1)\). Let \(u\) solve the discrete Dirichlet problem on the lattice ball \(B_R\subset\mathbb Z^d\),
\[
\tfrac12\operatorname{tr}(\omega(x)\nabla^2u(x))=\frac1{R^2}f\!\left(\tfrac{x}{R}\right)\zeta(\theta_x\omega)\quad \text{for }x\in B_R,\qquad
u(x)=g\!\left(\tfrac{x}{R}\right)\quad \text{for }x\in \partial B_R,
\]
and let \(\bar u\) solve on the unit ball \(\mathbb B_1\subset\mathbb R^d\),
\[
\tfrac12\operatorname{tr}(\bar a\,D^2\bar u(x))=f(x)\bar\psi\quad \text{for }x\in\mathbb B_1,\qquad
\bar u(x)=g(x)\quad \text{for }x\in\partial\mathbb B_1.
\]
Which quantitative estimate holds for the homogenization error \(\max_{x\in B_R}|u(x)-\bar u(x/R)|\)?

There exists a \(C^6(\overline{\mathbb B}_1)\)-extension of \(g\) such that the corresponding solutions satisfy
\[
\max_{x\in B_R}\left|u(x)-\bar u\!\left(\tfrac{x}{R}\right)\right|
\lesssim
\begin{cases}
R^{-3/2}\,\|\bar u\|_{C^6(\overline{\mathbb B}_1)}\,\tilde X, & d=3,\\[4pt]
R^{-2}(\log R)\,\|\bar u\|_{C^6(\overline{\mathbb B}_1)}\,\tilde X, & d\ge 4.
\end{cases}
\]

Choices:
B.
There exists a \(C^6(\overline{\mathbb B}_1)\)-extension of \(g\) such that the corresponding solutions satisfy
\[
\max_{x\in B_R}\left|u(x)-\bar u\!\left(\tfrac{x}{R}\right)\right|
\lesssim
\begin{cases}
R^{-2}\,\|\bar u\|_{C^6(\overline{\mathbb B}_1)}\,\tilde X, & d=3,\\[4pt]
R^{-2}\,\|\bar u\|_{C^6(\overline{\mathbb B}_1)}\,\tilde X, & d\ge 4.
\end{cases}
\]
C.
There exists a \(C^6(\overline{\mathbb B}_1)\)-extension of \(g\) such that the corresponding solutions satisfy
\[
\max_{x\in B_R}\left|u(x)-\bar u\!\left(\tfrac{x}{R}\right)\right|
\lesssim R^{-1}\,\|\bar u\|_{C^6(\overline{\mathbb B}_1)}\,\tilde X,
\qquad d\ge 3.
\]
D.
For every \(\varepsilon\in(0,1)\), there exists a random variable \(\tilde X=\tilde X(R,\varepsilon,\omega)>1\) with stretched-exponential moments such that the corresponding solutions satisfy
\[
\max_{x\in B_R}\left|u(x)-\bar u\!\left(\tfrac{x}{R}\right)\right|
\lesssim
\begin{cases}
R^{-3/2+\varepsilon}\,\|\bar u\|_{C^6(\overline{\mathbb B}_1)}\,\tilde X, & d=3,\\[4pt]
R^{-2+\varepsilon}\,\|\bar u\|_{C^6(\overline{\mathbb B}_1)}\,\tilde X, & d\ge 4.
\end{cases}
\]
E.
There exists a \(C^4(\overline{\mathbb B}_1)\)-extension of \(g\) such that the corresponding solutions satisfy
\[
\max_{x\in B_R}\left|u(x)-\bar u\!\left(\tfrac{x}{R}\right)\right|
\lesssim
\begin{cases}
R^{-3/2}(\log R)\,\|\bar u\|_{C^4(\overline{\mathbb B}_1)}\,\tilde X, & d=3,\\[4pt]
R^{-2}\,\|\bar u\|_{C^4(\overline{\mathbb B}_1)}\,\tilde X, & d\ge 4.
\end{cases}
\]

dimension-dependent rates and logarithmic factor

sharp improved rates dropped in favor of the coarser \(R^{-1}\) decay

required \(C^6\)-extension/norm and placement of logarithmic correction across dimensions


Question:
Assume the standing hypotheses \((A1),(A2),(A3)\) on the random environment, and let \(d\ge 3\). For \(x\in\mathbb Z^d\), define the spatial shift by \(\theta_x\omega:=\omega(x+\cdot)\). Let
\[
\bar a:=E_{\mathbb Q}\!\left[\frac{\omega(0)}{\operatorname{tr}\omega(0)}\right]>0,
\qquad
\bar\psi:=E_{\mathbb Q}\!\left[\frac{\zeta}{\operatorname{tr}\omega(0)}\right].
\]
Suppose \(f\in C^{4,\alpha}(\overline{B}_1)\) and \(g\in C^{6,\alpha}(\partial B_1)\) for some \(\alpha\in(0,1)\). Consider the discrete Dirichlet problem on the lattice ball \(B_R\subset \mathbb Z^d\),
\[
\frac12\operatorname{tr}(\omega(x)\nabla^2 u(x))=\frac1{R^2}f\!\left(\frac{x}{R}\right)\zeta(\theta_x\omega)
\quad (x\in B_R),
\]
with boundary values on \(\partial B_R\) obtained by evaluating a \(C^6(\overline B_1)\)-extension of \(g\), and the effective continuum problem
\[
\frac12\operatorname{tr}(\bar a\,D^2\bar u(x))=f(x)\bar\psi\quad (x\in B_1),
\qquad \bar u=g\quad (x\in\partial B_1).
\]
Which existence statement holds?

There exists a function \(\tilde g\in C^6(\overline B_1)\) with trace \(\tilde g|_{\partial B_1}=g\) such that, if \(u\) solves the discrete problem with boundary condition \(u(x)=\tilde g(x/R)\) on \(\partial B_R\) and \(\bar u\) solves the effective problem above, then
\[
\max_{x\in B_R}\left|u(x)-\bar u\!\left(\frac{x}{R}\right)\right|
\lesssim
\begin{cases}
R^{-3/2}\,\|\bar u\|_{C^6(\overline B_1)}\,\tx, & d=3,\\[4pt]
R^{-2}(\log R)\,\|\bar u\|_{C^6(\overline B_1)}\,\tx, & d\ge 4.
\end{cases}
\]
Moreover, one such extension may be taken to be \(\tilde g=\bar u\).

Choices:
B.
There exists a function \(\tilde g\in C^6(\overline B_1)\) with trace \(\tilde g|_{\partial B_1}=g\) such that, if \(u\) solves the discrete problem with boundary condition \(u(x)=\tilde g(x/R)\) on \(\partial B_R\) and \(\bar u\) solves the effective problem above, then
\[
\max_{x\in B_R}\left|u(x)-\bar u\!\left(\frac{x}{R}\right)\right|
\lesssim
R^{-2}\,\|\bar u\|_{C^6(\overline B_1)}\,\tx
\qquad\text{for all }d\ge 3.
\]
Moreover, one such extension may be taken to be \(\tilde g=\bar u\).
C.
There exists a function \(\tilde g\in C^6(\overline B_1)\) with trace \(\tilde g|_{\partial B_1}=g\) such that, if \(u\) solves the discrete problem with boundary condition \(u(x)=\tilde g(x/R)\) on \(\partial B_R\) and \(\bar u\) solves the effective problem above, then
\[
\max_{x\in B_R}\left|u(x)-\bar u\!\left(\frac{x}{R}\right)\right|
\lesssim
\begin{cases}
R^{-1}\,\|\bar u\|_{C^6(\overline B_1)}\,\tx, & d=3,\\[4pt]
R^{-1}\,\|\bar u\|_{C^6(\overline B_1)}\,\tx, & d\ge 4.
\end{cases}
\]
Moreover, one such extension may be taken to be \(\tilde g=\bar u\).
D.
For every choice of \(C^6(\overline B_1)\)-extension \(\tilde g\) of \(g\), if \(u\) solves the discrete problem with boundary condition \(u(x)=\tilde g(x/R)\) on \(\partial B_R\) and \(\bar u\) solves the effective problem above, then
\[
\max_{x\in B_R}\left|u(x)-\bar u\!\left(\frac{x}{R}\right)\right|
\lesssim
\begin{cases}
R^{-3/2}\,\|\bar u\|_{C^6(\overline B_1)}\,\tx, & d=3,\\[4pt]
R^{-2}(\log R)\,\|\bar u\|_{C^6(\overline B_1)}\,\tx, & d\ge 4.
\end{cases}
\]
In particular, the estimate is independent of the particular extension chosen.
E.
There exists a function \(\tilde g\in C^6(\overline B_1)\) with trace \(\tilde g|_{\partial B_1}=g\) such that, if \(u\) solves the discrete problem with boundary condition \(u(x)=\tilde g(x/R)\) on \(\partial B_R\) and \(\bar u\) solves the effective problem above, then
\[
\max_{x\in B_R}\left|u(x)-\bar u\!\left(\frac{x}{R}\right)\right|
\lesssim
\begin{cases}
R^{-3/2}(\log R)\,\|\bar u\|_{C^6(\overline B_1)}\,\tx, & d=3,\\[4pt]
R^{-2}\,\|\bar u\|_{C^6(\overline B_1)}\,\tx, & d\ge 4.
\end{cases}
\]
Moreover, one such extension may be taken to be \(\tilde g=\bar u\).

sharp rates R^{-3/2} and R^{-2}\log R replaced by coarser R^{-1} bound

existential choice of extension replaced by uniform statement over all extensions

placement of logarithmic loss across dimensions