Elliptic Equations and Meyers Estimates

This work is connected with estimates of solutions to the Zaremba problem for elliptic equation in bounded Lipschitz domain $D\in \mathbb{R}^n$, where $n>1$, of the form
\begin{equation}\label{op} \mathcal{L}u:=\text{div} (|\nabla u|^{p-2}a(x)\nabla u) \end{equation}
with uniformly elliptic measurable and symmetric matrix $a(x)=\{a_{ij}(x)\}$, i.e. $a_{ij}=a_{ji}$ and
\begin{equation}\label{1} \alpha^{-1}|\xi|^2\le \sum\limits_{i,j=1}^na_{ij}(x)\xi_i\xi_j\le\alpha |\xi|^2~\text{for almost all}~x\in D~\text{and all}~\xi\in \mathbb{R}^n. \end{equation}
We assume that $F\subset\partial D$ is closed and $G=\partial D\setminus F$. Consider the Zaremba problem
\begin{equation}\label{2} \left\{\begin {array}{l} \mathcal{L}u=l\quad \text{in}\quad D,\\ u=0\quad \text{on}\quad F,\\ \frac{\partial u}{\partial \nu}=0\quad \text{on}\quad G, \end{array}\right. \end{equation}
where $\frac{\partial u}{\partial \nu}$ is the outer conormal derivative of $u$, and $l$ is a linear functional on $W^1_p(D, F)$, the completion of the set of infinitely differentiable in the closure of $D$ functions vanishing in the vicinity of $F$, by the norm
$$ \parallel u\parallel_{W^{1}_p(D, F)}=\biggl (~\int\limits_{D} u^p\,dx+\int\limits_{D}|\nabla u|^p\,dx\biggr )^{1/p}. $$
By the solution of the problem \eqref{2} we mean the function $u \in W^1_p (D, F)$ for which the integral identity
\begin{equation}\label{3} \int\limits_{D}|\nabla u|^{p-2}a\nabla u\cdot\nabla\varphi\,dx=\int\limits_{D} f\cdot\nabla\varphi\,dx \end{equation}
holds for all test-functions $\varphi\in W^1_p(D, F)$, the components of the vector-function $f=(f_1,\ldots,f_n)$ belong to $L_{p'}(D)$, $p'=\frac{p}{p-1}$. For the compact $K\subset \mathbb{R}^n$ we define the capacity $C_q(K)$, $1<q<n$, by the formula
\begin{equation}\label{hu} C_q(K)=\inf~ \biggl \{~ \int\limits_{\mathbb{R}^n}|\nabla\varphi|^q\,dx:~\varphi\in C^\infty_0 (\mathbb{R}^n),~\varphi\ge 1~\text{on}~K\biggr \}, \end{equation}
if $p\in (1, n/(n-1)]$, then $q=(p+1)/2$, but if $r\in (n/(n-1), n]$, where $n > 2$, then $q=np/(n+p)$.
$\bullet$ Case of linear equation ($p=2$).
Suppose $B^{x_0}_r$ is an open ball of the radius $r$ centered in $x_0$, and $mes_{n-1}(E)$ is $(n-1)$-measure of the set $E$. Assume also that $q=2n/(n+2)$ as $n>2$ and $q=3/2$ as $n=2$. We suppose one of the following conditions is fulfilled: for an arbitrary point $x_0\in F$ as $r\le r_0$ the inequality
\begin{equation}\label{g1} C_q( F\cap \overline B^{x_0}_r)\ge c_0 r^{n-q} \end{equation}
holds true or the inequality
\begin{equation}\label{g2} mes_{n -1}( F\cap \overline B^{x_0}_r)\ge c_0 r^{n-1} \end{equation}
holds, the positive constant $c_0$ does not depend on $x_0$ and $r$. Condition \eqref{g2} is universal (even for nonlinear equations).
Theorem. If $f\in L_{2+\delta_0}(D)$, where $\delta_0>0$, then there exist positive constants $\delta(n,\delta_0)<\delta_0$ and $C$, such that for a solution to the problem \eqref{2} the estimate
\begin{equation}\label{t} \int\limits_{D}|\nabla u|^{2+\delta}dx\leq C\int\limits_{D}|f|^{2+\delta}\ dx, \end{equation}
holds, where $C$ depends only on $\delta_0$, the dimension $n$, constant $c_0$ from \eqref{g1} and \eqref{g2}, and also the constant $r_0$.
$\bullet$ Case of $p$-elliptic equation ($p>1$).
A. If $1< p \le n$, then the following condition is assumed to hold: for an arbitrary point $x_0\in F$ for $r\le r_0$, the condition \eqref{g1} is true.
B. If $p> n$, then the set $F$ is assumed to be nonempty: $F\neq \emptyset$.
Theorem. If $f\in L_{p'+\delta_0}(\Omega)$, where $\delta_0>0$, then there exist positive constants $\delta(n, p,\delta_0)<\delta_0$ and $C$, such that for a solution to the problem \eqref{2} the estimate
\begin{equation}\label{tm} \int\limits_{\Omega}|\nabla u|^{p+\delta}dx\leq C\int\limits_{\Omega}|f|^{p'(1+\delta/p)}\ dx, \end{equation}
holds, where $C$ depends only on $p$, $\delta_0$, the dimension $n$, constant $c_0$ from \eqref{g1} and \eqref{g2}, and also the constant $r_0$.