\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2013 (2013), No. 121, pp. 1--17.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2013 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2013/121\hfil Regularity on the interior] {Regularity on the interior for the gradient of weak solutions to nonlinear second-order elliptic systems} \author[J. Dan\v{e}\v{c}ek, E. Viszus \hfil EJDE-2013/121\hfilneg] {Josef Dan\v{e}\v{c}ek, Eugen Viszus} % in alphabetical order \address{Josef Dan\v{e}\v{c}ek \newline Institute of Mathematics and Biomathematics, Faculty of Science, University of South Bohemia, Brani\v{s}ovsk\'{a} 31, 3705 \v{C}esk\'{e} Bud\v{e}jovice, Czech Republic} \email{josef.danecek@prf.jcu.cz} \address{Eugen Viszus \newline Department of Mathematical Analysis and Numerical Mathematics, Faculty of Mathematics, Physics and Informatics Comenius University, Mlynsk\'{a} dolina, 84248 Bratislava, Slovak Republic} \email{eugen.viszus@fmph.uniba.sk} \thanks{Submitted April 8, 2013. Published May 16, 2013.} \subjclass[2000]{35J47} \keywords{Nonlinear elliptic equations; weak solutions; regularity; \hfill\break\indent Campanato spaces} \begin{abstract} We consider weak solutions to the Dirichlet problem for nonlinear elliptic systems. Under suitable conditions on the coefficients of the systems we obtain everywhere H\"older regularity on the interior for the gradients of weak solutions. Our sufficient condition for the regularity works even though an excess of the gradient of solution is not very small. More precise partial regularity on the interior can be deduced from our main result. The main result is illustrated through examples at the end of this article. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} In this paper we give conditions guaranteeing that a weak solution to the Dirichlet problem for a nonlinear elliptic system \begin{equation}\label{R} \begin{gathered} -D_\alpha\big(A_i^{\alpha}(Du)\big)=0\quad \text{in } \Omega,\; i=1,\ldots,N,\\ u=g\quad \text{on } \partial\Omega \end{gathered} \end{equation} belongs to $C_{\mathrm{loc}}^{1,\gamma}(\Omega,\mathbb{R}^N)$ space. Here and in the following, summation over repeated indices is understood. By a weak solution to the Dirichlet problem \eqref{R}, we mean a function $u$ in $W^{1,2}(\Omega,\mathbb{R}^N)$ such that \begin{equation*} \int_\Omega A_i^{\alpha}(Du)D_{\alpha}\varphi^{i}\, dx=0, \quad\forall\,\varphi\in W_0^{1,2}(\Omega,\mathbb{R}^N) \end{equation*} and $u-g\in W_0^{1,2}(\Omega,\mathbb{R}^N)$. Here $\Omega\subset\mathbb{R}^n$ is a bounded open set, $n\ge 3$, the function $g$ belongs to the space $W^{1,2}(\Omega,\mathbb{R}^N)$, the coefficients $(A_i^{\alpha})_{i=1,\ldots,N,\alpha=1,\ldots,n}$ are differentiable, have the linear controlled growth and satisfy the strong uniform ellipticity condition. More precisely, denoting by \begin{equation*} A_{ij}^{\alpha\beta}(p)=\frac{\partial A_i^{\alpha}} {\partial p_{j}^{\beta}}(p) \end{equation*} and assuming that $A_i^{\alpha}(0)=0$ we require \begin{itemize} \item [(i)] there exists a constant $M>0$ such that for every $p\in\mathbb{R}^{nN}$ \begin{equation*} |A_i^{\alpha}(p)|\le M(1+|p|), \end{equation*} \item [(ii)] $ |A_{ij}^{\alpha\beta}(p)|\le M$, \item [(iii)] the strong ellipticity condition holds; i.e., there exists a constant $\nu>0$ such that for every $p$, $\xi\in\mathbb{R}^{nN}$, \begin{equation*} A_{ij}^{\alpha\beta}(p)\xi_{\alpha}^{i}\xi_{\beta}^{j}\ge\nu|\xi|^{2}, \end{equation*} \item [(iv)] there exists a real function $\omega$ defined and continuous on $[0,\infty)$, which is bounded, nondecreasing, increasing on a neighbourhood of zero, $\omega(0)=0$ and such that for all $p$, $q\in\mathbb{R}^{nN}$ \begin{equation*} |A_{ij}^{\alpha\beta}(p)-A_{ij}^{\alpha\beta}(q)| \le\omega(|p-q|). \end{equation*} We set $\omega_{\infty}=\lim\nolimits_{t\to\infty}\omega(t)\le 2M$. \end{itemize} Here it is worth to point out (see \cite[pg.\ 169]{Gia83}) that for uniformly continuous coefficients $A_{ij}^{\alpha\beta}$ there exists the real function $\omega$ satisfying the assumption (iv) and, viceversa, (iv) implies the uniform continuity of the coefficients and the absolute continuity of $\omega$ on $[0,\infty)$. It is clear that if $\omega(t)=0$ for $t\in[0,\infty)$, then the system \eqref{R} is reduced to the system with constant coefficients and in this case the regularity of weak solutions is well understood (see, e.g. \cite{Gia83} and references therein). The system \eqref{R} has been extensively studied (see, e.g. \cite{Ca80,Gia83,Kos95,Ne83}). It is well known that the Dirichlet problem has a unique solution $u\in W^{1,2}(\Omega,\mathbb{R}^N)$. Moreover, for boundary function $g\in W^{1,2}(\Omega,\mathbb{R}^N)$ it holds \begin{gather}\label{DI} \int_{\Omega} | Du|^{2}\, dx \le C_D\int_{\Omega} |Dg|^{2}\, dx, \\ \label{DIR} \int_\Omega\,|Du-(Du)_{\Omega}|^{2}\, dx \le C_D\int_\Omega\,|Dg-(Dg)_{\Omega}|^{2}\, dx \end{gather} where $(Dg)_{\Omega}=\frac{1}{m(\Omega)}\int_\Omega\,Dg\, dx$, $m(\Omega)=m_{n}(\Omega)$ is the $n$ - dimensional Lebesgue measure of $\Omega$ and $C_D=n^{2}N^{2}(M/\nu)^{2}$. The estimates \eqref{DI} and \eqref{DIR} can be proved by a standard technique (see \cite{Gia93}, Remark on pg.113). For reader's convenience the proofs of \eqref{DI} and \eqref{DIR} are given in Appendix to this paper. The first regularity results for $n=2$ and for nonlinear systems were established by Morrey (see \cite{Morr66}) and they state that the gradient of unique solution to \eqref{R} is locally H\"{o}lder continuous. If $n\geq 3$, it is known that the gradient $Du$ may be discontinuous and unbounded (see \cite{HaLeonaNe96,Leona99,Ne83}). For $n\geq 3$ and for the nonlinear systems many partial regularity results were obtained, i.e., it was proved that the gradient of any weak solution to \eqref{R} (or more general system) is locally H\"{o}lder continuous up to a singular set of the Hausdorff dimension $n-2$ (see, e.g. \cite{Ca80,Gia83,Ne83}). In the last two decades some new methods for proving the partial regularity of weak solutions to the nonlinear systems, based on a generalization of the technique of harmonic approximation, have appeared (see, e.g. \cite{Ham98,DuGro00} and references therein). These methods extend the previous partial regularity results in such a way that they allow to establish the optimal H\"{o}lder exponent for the gradients of weak solutions on their regular sets. In this place, it is worth to mention the papers \cite{SvY00,SvY02} where the authors through examples showed that (for $n=3$) the gradient of the unique minimizer of the convex and differentiable functional $F$ (in this case \eqref{R} is the Euler-Lagrange equation of $F$) can be discontinuous or unbounded. Thus full regularity cannot be achieved even in this special case. On the other hand, Campanato in \cite{Ca87} proved that the weak solution of the system \eqref{R} belongs to $W^{2,2+\epsilon}_{loc}(\Omega,\mathbb{R}^N)$ which implies that $Du\in C_{\mathrm{loc}}^{0,\gamma}(\Omega,\mathbb{R}^{nN})$ for $n=2$ and $u\in C_{\mathrm{loc}}^{0,\gamma}(\Omega,\mathbb{R}^N)$ for $2\leq n\leq 4$, $\gamma\in (0,1)$. Kristensen and Melcher have recently proved (using a method which avoids employing the Gehring's lemma) in \cite{KrMe08} that an analogous result is true under the strong monotonicity and the Lipschitz continuity of the coefficients. Moreover, they have stated the value of the last mentioned $\epsilon$ as $\epsilon=\delta\alpha/\beta$ where $\delta>1/50$ is a universal constant, $0<\alpha\leq\beta$ are the constant of the monotonicity and the Lipschitz continuity constant respectively. The aim of this paper is to extend the last mentioned results and the results of the paper \cite{DJS07}, giving some conditions sufficient for the everywhere interior regularity of the solutions to the systems \eqref{R} for $n\geq 3$. In the paper \cite{DJS07}, the first author with John and Star\'{a} gave conditions, expressed in terms of the continuity modulus of the first derivatives of the coefficients of \eqref{R}, that guarantee the local H\"{o}lder continuity of the gradients of solutions to \eqref{R} in $\Omega$. More precisely, they proved that there exists $\nu_0>0$ such that for every ellipticity constant $\nu\geq\nu_0$ with the ratio $M/\nu\leq P$, where $P>1$ is a given constant, the gradients of weak solutions to \eqref{R} are locally H\"{o}lder continuous in $\Omega$ (see \cite{Da02} as well). The point of the current paper is to give conditions guaranteeing the same quality of the solutions to \eqref{R} when the ratio $\omega_{\infty}/\nu$ is admitted to be arbitrary and no lower bound for the constant of ellipticity $\nu$ is needed (we remind that if the constant $M$ is given, then $\omega_{\infty}\leq 2M$). The main results are stated in two theorems. The first of them refers that if $\omega_{\infty}/\nu$ is small enough, the solutions to \eqref{R} are regular. This result is not very surprising but, moreover, an upper bound $C_{cr}$ (although probably not optimal) of $\omega_{\infty}/\nu$ is designed there (see \eqref{T} below). If $\omega_{\infty}/\nu>C_{cr}$, then a sufficient condition for regularity of solutions to the system \eqref{R} is given in Theorem \ref{Th2}. A basic advantage of condition \eqref{T1} below is that it admits (for sufficiently big ellipticity constant $\nu$) an arbitrary growth of the continuity modulus $\omega=\omega(t)$ when $t$ is near by zero. Here it is needful to note that Theorem \ref{Th2} works likewise when $\nu$ is small but, in this case, the modulus of continuity $\omega$ has to grow slowly enough. Many proofs of regularity results for systems like the system \eqref{R} are based on a certain excess-decay estimate for the excess function $U_r(x)$ (in our case this function is defined by \eqref{Exc} below). The key assumption of the excess-decay estimate is that $U_r(x)$ has to be sufficiently small on a ball $B_r(x)\Subset\Omega$. On the other hand, our condition \eqref{T1} does not suppose smallness of the excess function $U_r(x)$ (see Remark \ref{Re2} below). We would like to note that more delicate estimates and careful designing of some parameters in proofs allow us to state these results in a much simpler form than in \cite{DJS07}. Various conditions, guaranteeing the regularity of weak solutions, were studied by Giaquinta and Ne\v{c}as in \cite{GiaNe79,GiaNe81} (the Liouville's condition for regularity formulated through $L^{\infty}$-spaces), Dan\v{e}\v{c}ek in \cite{Da84} (the Liouville's condition for regularity formulated through $BMO$-spaces), Chipot and Evans in \cite{ChiEv86} and Koshelev in \cite{Kos95}. Koshelev's condition, interpreted according to the assumptions (ii) and (iii), is the following : If it is supposed that $nNM|\xi|^{2}\geq A_{ij}^{\alpha\beta}(p)\xi_{\alpha}^{i}\xi_{\beta}^{j} \ge\nu|\xi|^{2}$ for every $p$, $\xi\in\mathbb{R}^{nN}$, $A_{ij}^{\alpha\beta}=A_{ji}^{\beta\alpha}$ and \begin{equation*} \frac{M}{\nu}<\frac{1}{nN} \frac{\sqrt{1+\frac{(n-2)^2}{n-1}}+1} {\sqrt{1+\frac{(n-2)^2}{n-1}}-1}, \end{equation*} then any solution to \eqref{R} has the locally H\"{o}lder continuous gradient in $\Omega$. It is proved in \cite{Kos95} that the above condition is sharp. The same result is proved, by an another method which is based on an estimate of the gradient of solution in a suitable weighted Morrey space, in \cite{Leona99}. Further results concerning the local (and global as well) H\H older regularity of the solutions and the dispersion of the eigenvalues of the coefficients matrix of elliptic systems can be found in \cite{LeKoSt05,Leona04}. On the other hand, the last mentioned condition does not cover the linear systems with constant coefficients and the large dispersion of the eigenvalues of $A_{ij}^{\alpha\beta}$, while every linear system with constant coefficients satisfies the conditions \eqref{T} and \eqref{T1} as well. Chipot and Evans in \cite{ChiEv86} consider the variational problem and assume that $A_{ij}^{\alpha\beta}(p)$ tend to a constant matrix for $p$ tending to infinity. Thus the modulus of continuity of $A_{ij}^{\alpha\beta}(p)$ approaches zero for sufficiently large $p$ while our assumption requires that its changes are small enough. Herein we would like to note that, as far as we know, the above mentioned condition from the paper \cite{ChiEv86} was for the first time employed in \cite{Da84}. The methods of proving main results follow the standard procedures used in the direct proofs of the partial regularity. The novelty is an employment of special complementary Young functions which allows us (through a modification of the Natanson's Lemma - see Lemma \ref{L6} below) to get some key estimates. As a consequence of our proof of the main result (Theorem \ref{Th2} below) we obtain the partial regularity result concerning the more precise identification of the singular set of the weak solution to \eqref{R}. As it is known (see \cite{Gia83,Ne83,Ham98,DuGro00}), the singular set of the weak solution to \eqref{R} is characterized as follows \begin{equation*} \Omega_{\rm sing}=\big\{x\in\Omega : \liminf_{r\to 0} -\hskip-10pt \int_{B_{r}(x)} |Du(y)-(Du)_{x,r}|^{2}\, dy>0\big\}. \end{equation*} Our description of the singular set $\Omega\setminus\Omega_{\mathcal{R}}$, from Theorem \ref{Th3} below, indicates clearly that $\Omega\setminus\Omega_{\mathcal{R}}\subsetneq \Omega_{\rm sing}$ and the constant which describes $\Omega\setminus\Omega_{\mathcal{R}}$ is computable. Four examples, illustrating above mentioned results, are given at the end of the paper. The first one presents a system which our results can be applied to. The second and the third of them show typical samples of modulus of continuity that our main result deals with. The fourth one indicates that the regularity of gradient of boundary data, which is considerably weaker than the Campanato's one, does not admit the singularities of the weak solutions to \eqref{R} in a subdomain. \section{Main results} By $\Omega_0\Subset\Omega$ we will understand any bounded subdomain $\Omega_0$ which is compactly embedded into $\Omega$ (i.e. $\Omega_0\subset\overline{\Omega}_0\subset\Omega$) and the boundary $\partial\Omega_0$ is smooth. For $x\in\Omega$, $r>0$ such that $B_{r}(x)=\left\{y\in\mathbb{R}^n: |y-x|0$ be given. Let $u$ be a weak solution to the Dirichlet problem \eqref{R} where $g\in W^{1,2}(\Omega)$ and the hypotheses (i), (ii), (iii), (iv) be satisfied with $M$, $\nu$ and the function $\omega$ for which \begin{equation}\label{T} \frac{\omega_{\infty}}{\nu} \le\frac{1}{\sqrt{8n^{2}N^{2}(2^{n+5}L)^{\frac{\vartheta} {n+2-\vartheta}}}}:=C_{cr} \end{equation} where the constant $L$ is given in Lemma \ref{L5} below. Then \begin{equation}\label{TTT} \|Du\|_{\mathcal{L}^{2,\vartheta}(\Omega_0,\mathbb{R}^{nN})} \le cd_{\vartheta}^{-\vartheta}\|Dg\|_{L^{2}(\Omega,\mathbb{R}^{nN})} \end{equation} for some $0 C_{cr}$. \begin{theorem}\label{Th2} Let $\Omega_0\Subset\Omega$ with $\operatorname{dist}(\Omega_0,\partial\Omega)\ge 2d>0$ and $n\le\varthetan$ and $Du\in BMO(\Omega_0,\mathbb{R}^{nN})$ for $\vartheta=n$. Here $t_0>0$, $\omega(t_0)=\sqrt{\varepsilon}$, $\varepsilon>0$ is specified in \eqref{EPS} where the constant $\epsilon_0=\frac{1}{4(2^{n+5}L)^{\vartheta/(n+2-\vartheta)}}$ ($L$ is the constant from Lemma \ref{L5}) and $\mu\ge 2$. \end{theorem} \begin{remark}\label{Re2} {\rm As it is visible from the condition \eqref{T1}, an appropriate choice of the constant $c_0$ guarantees the regularity even if the excess $U_{2d}$ is not assumed to be very small in $\Omega_0$. Moreover, the term $(U_{2d}(x))^{1/2}$ in \eqref{T1} can be replaced with $\|Du\|_{L^{2}(\Omega,\mathbb{R}^{nN})}/(2d)^{n/2}$ or, in the case of the Dirichlet problem \eqref{R}, with $C_D^{1/2}\|Dg\|_{L^{2}(\Omega,\mathbb{R}^{nN})}/(2d)^{n/2}$ where $C_D$ is from \eqref{DI}. See Example \ref{Exam2} and \ref{Exam3} for additional information. } \end{remark} \begin{remark}\label{Re3} {\rm It can be seen (according to the assumption (iii)) that $\mathcal{M}$ is finite. On the parameter $\mu$ we only quote that its main goal is to damp the exponential growth. A structure of the Young functions in \eqref{YF} and the estimates \eqref{I1} - \eqref{II2} below indicate a role of $\mu$. It is visible from these estimates that it is possible to find a value of the parameter $\mu$ which is optimal in some measure. } \end{remark} The next theorem is a straightforward consequence of Theorem \ref{Th2}. It presents the well-known partial regularity result but unlike the other partial regularity results this theorem describes the so-called singular set a little bit more precisely. \begin{theorem}\label{Th3} Let $n<\vartheta0,x\in\Omega} \frac{1}{r^\lambda}\int_{\Omega_{r}(x)} | u(y)-u_{x,r}|^q\, dy<\infty $$ where $u_{x,r}=-\hskip-9pt \int_{\Omega_{r}(x)} u(y)\, dy$ and $\Omega_{r}(x)=\Omega\cap B_{r}(x)$. The norm in the space $\mathcal{L}^{q,\lambda}(\Omega,\mathbb{R}^N)$ is defined by $\|u\|_{\mathcal{L}^{q,\lambda}(\Omega,\mathbb{R}^N)} =\|u\|_{L^q(\Omega,\mathbb{R}^N)} +[u]_{\mathcal{L}^{q,\lambda}(\Omega,\mathbb{R}^N)}$. \end{definition} \begin{proposition}[\cite{Ca80,Gia83,KJF77}]\label{prop1} For a bounded domain $\Omega\subset\mathbb{R}^n$ with a Lipschitz boundary, for $q\in[1,\infty)$ and $0<\lambda<\mu<\infty$ the relation $\mathcal{L}^{q,\mu}(\Omega,\mathbb{R}^N) \subset\mathcal{L}^{q,\lambda}(\Omega,\mathbb{R}^N)$ holds and $\mathcal{L}^{q,\lambda}(\Omega,\mathbb{R}^N)$ is isomorphic to the $C^{0,(\lambda-n)/q}(\overline\Omega,\mathbb{R}^N)$, for $n<\lambda\le n+q$. \end{proposition} Now, let $\Phi$, $\Psi$ be a pair of the complementary Young functions \begin{equation}\label{YF} \Phi(u)=u\ln_{+}^{\mu}(au),\quad \Psi(u)\le\overline{\Psi}(u) =\frac{1}{a}u\mathrm{e}^{(\frac{u}{2\sqrt{\mu}})^{2/(2\mu-1)}} \quad\text{for } u\ge 0 \end{equation} where $a>0$ and $\mu\ge 2$ are constants, \begin{equation} \ln_{+}(au)= \begin{cases} 0 &\text{for } 0\le u< 1/a,\\ \ln(au) &\text{for } u\ge 1/a. \end{cases} \end{equation} Then the Young inequality for $\Phi$, $\Psi$ reads \begin{equation}\label{YIN} uv\le\Phi(u)+\Psi(v),\quad u,v\ge 0. \end{equation} \begin{lemma}[{\cite[pg.37]{Zi89}}]\label{Zi} Let $\phi: [0,\infty)\to[0,\infty)$ be a nondecreasing function which is absolutely continuous on every closed interval of finite length, $\phi(0)=0$. If $w\ge 0$ is measurable and $E(t)=\{y\in\mathbb{R}^n: w(y)>t\}$ then \[ \int_{\mathbb{R}^n}\phi\circ w\, dy =\int_0^{\infty}\, m\big(E(t)\big)\phi'(t)\, dt. \] \end{lemma} The next Lemma will be employed in the proof of Theorem \ref{Th2}. \begin{lemma}[{\cite[pg.388]{Da02}}] \label{LDA3} Let $v\in L^{2}_{\mathrm{loc}}(\Omega,\mathbb{R}^N)$, $N\ge 1$, $B_{r}(x)\Subset\Omega$, $b>0$ and $s\in(1,+\infty)$. Then \begin{equation*} \int_{B_{r}(x)}\,\ln_{+}^{s}(b|v|^{2})\, dy \le s\big(\frac{s-1}{e}\big)^{s-1}b \int_{B_{r}(x)}\,|v|^{2}\, dy. \end{equation*} \end{lemma} The following Lemma is a small modification of \cite[Lemma 1.IV]{Ca80}. \begin{lemma}\label{L3} Let $A$, $R_0\le R_1$ be positive numbers, $n\le\varthetaa$ and $\lim_{k\to\infty}b_{k}=\infty$ put \begin{equation*} f_{k}(t)= \begin{cases} f(t) &\text{for } a\leq t\le b_{k}\\ 0 &\text{for } b_{k}0$, $\mu\ge 2$, $c_1$, $c_2\in\mathbb{R}$ we have \begin{align*} &\int_{B_{R}(x)}|Du(y)-(Du)_{B_{R}(x)}|^{2} \ln_{+}^{\mu}(b|Du(y)-c_1|^{2})\, dy\\ &\le C_{P}^{2}C_{Cacc}\Big(C_{q\mu}b -\hskip-10pt \int_{B_{R}(x)} |Du(y)-c_1|^{2}\, dy\Big)^{1-1/q} \int_{B_{2R}(x)}|Du(y)-c_2|^{2}\, dy \end{align*} where $10$ and let $00$, $B_{2R}(x_0)\subset\Omega$. Following the first part of the proof of Theorem \ref{Th1} step by step, we obtain the estimate \eqref{PTh1}. To estimate the last integral in \eqref{PTh1} we use the Young inequality \eqref{YIN} (here complementary functions are defined through \eqref{YF}) and for any $0<\varepsilon<\omega_{\infty}^{2}$ we obtain \begin{equation}\label{P2} \begin{aligned} &\int_{B_{R}}\omega^{2}(|Du-(Du)_{R}|) |Du-(Du)_{R}|^{2}\, dx \\ &\le\varepsilon\int_{B_{R}}|Du-(Du)_{R}|^{2} \ln_{+}^{\mu}\big(a\varepsilon|Du-(Du)_{R}|^{2}\big)\, dx +\int_{B_{R}}\overline{\Psi} (\frac{\omega_{R}^{2}}{\varepsilon})\, dx \\ &=\varepsilon I_1+I_2 \end{aligned} \end{equation} where $\omega_{R}^{2}(x)=\omega^{2}(|Du(x)-(Du)_{R}|)$. The term $I_1$ can be estimated by means of Proposition \ref{prop2} and we obtain \begin{equation} \label{I1} I_1\le C_{P}^{2}C_{Cacc}C_{q\mu}^{1-1/q} (2^na\varepsilon U_{2R})^{1-1/q}\phi(2R) =K(a\varepsilon U_{2R})^{1-1/q}\phi(2R) \end{equation} where $10 \end{equation*} and $m_{R}(t)=m(\{y\in B_{R}(x_0) :|Du-(Du)_{R}|>t\})$. Using the estimate $m_{R}(t)\le\kappa_{n}R^n$, $\kappa_{n}$ is the Lebesgue measure of the unit ball, we have (we use Lemma \ref{L6}) \begin{align} \widetilde{I}_2&\le\int_0^{t_0} \,\frac{d}{dt}\widetilde{\Psi} (\frac{\omega^{2}(t)}{\varepsilon})m_{R}(t)\, dt +\int_{t_0}^{\infty} \,\frac{d}{dt}\widetilde{\Psi} (\frac{\omega^{2}(t)}{\varepsilon})m_{R}(t)\, dt \nonumber \\ &\le\kappa_{n}R^n\int_0^{t_0} \,\frac{d}{dt}\widetilde{\Psi} (\frac{\omega^{2}(t)}{\varepsilon})\, dt +\sup_{t_00$ the average $U_{R}=0$ then it is clear that $x_0$ is the regular point. So in the next we can suppose $U_{R}$ is positive for all $R>0$. Inserting \eqref{P2}--\eqref{II2} into \eqref{PTh1} yields \begin{equation} \label{I3} \begin{aligned} \phi(\sigma) &\le 4L(\frac{\sigma}{R})^{n+2}\phi(R) +2n^{2}N^{2}(1+2L(\frac{\sigma}{R})^{n+2}) \\ &\times[\frac{\varepsilon\, K} {\nu^{2}}(2^na\varepsilon U_{2R})^{1-1/q} +\frac{1}{a\nu^{2}}(\frac{\widetilde{\Psi} \big(\frac{\omega^{2}(t_0)}{\varepsilon}\big)}{U_{2R}} +\frac{\mathcal{M}}{\sqrt{U_{2R}}})]\phi(2R). \end{aligned} \end{equation} In \eqref{I3} we can choose \begin{equation*} a=\frac{16\mathrm{e}n^{2}N^{2}}{\epsilon_0\nu^{2}c_0\, U_{2R}} \quad\text{for } U_{2R}>0 \end{equation*} where $00$, we obtain \begin{equation} \label{I5} \begin{aligned} \phi(\sigma) &\le 4L(\frac{\sigma}{R})^{n+2}\phi(R) +\frac{1}{2}\Big(1+2L(\frac{\sigma}{R})^{n+2}\Big)\\ &\quad\times\big[KK_1\epsilon_0^{\alpha+(\alpha-1)(1-1/q)} \nu^{(\beta-2)(2-1/q)} +\frac{\epsilon_0}{4\mathrm{e}^{2}} \big(\mathrm{e}+\mathcal{M}\,\sqrt{U_{2R}}\big)\big]\phi(2R) \\ &=4L\big(\frac{\sigma}{R}\big)^{n+2}\phi(R) +\frac{1}{2}\Big(1+2L(\frac{\sigma}{R})^{n+2}\Big)\\ &\quad\times \big[KK_1(\epsilon_0^{\alpha-1}\nu^{\beta-2})^{2-1/q} +\frac{c_0}{4}+\frac{\mathcal{M}}{4\mathrm{e}}c_0\,\sqrt{U_{2R}}\big] \epsilon_0\phi(2R) \end{aligned} \end{equation} where $K_1=4n^{2}N^{2}(2^{n+4} \mathrm{e}n^{2}N^{2}/c_0)^{1-1/q}$. The constants $\alpha$ and $\beta$ can be always chosen in such a way that \begin{equation*} KK_1(\epsilon_0^{\alpha-1}\nu^{\beta-2})^{2-1/q} \le\frac{1}{4} \end{equation*} and finally we have \begin{equation}\label{I7} \phi(\sigma)\le 4L(\frac{\sigma}{R})^{n+2}\phi(R) +\frac{1}{2}\big(1+2L(\frac{\sigma}{R})^{n+2}\big) \big(\frac{1}{2} +\frac{1}{10}\mathcal{M}c_0\,\sqrt{U_{2R}}\big)\epsilon_0\phi(2R). \end{equation} We can put \begin{equation*} B_1=\frac{1}{2}\epsilon_0, \quad B_2=\frac{1}{10}\mathcal{M}\epsilon_0 \end{equation*} and if we take into account assumption \eqref{T1} of Theorem \ref{Th2} we can use Lemma \ref{L3}. \end{proof} \begin{proof}[Proof of Theorem \ref{Th3}] Let $x_0\in\Omega_{\mathcal{R}}$ and $R_1>0$ be chosen in such a way that $B_{2R_1}(x_0)\subset\Omega$ and let $04. \end{cases} \end{equation*} If $m$ is chosen in a suitable way (with respect to $\lambda$) then our results can guarantee the interior regularity of the gradient of weak solution to the Dirichlet problem \eqref{R}. } \end{example} \begin{example}\label{Exam2} {\rm To illustrate some parameters from the proof of Theorem \ref{Th2} we can consider the following modulus of continuity \begin{equation*} \omega(t)= \begin{cases} \omega_0(t)=\frac{(1+s)^{s}\sqrt{\varepsilon}} {(1+\ln\frac{t_0\mathrm{e}^{s}}{t})^{s}} \quad &\text{for } 00,\\ \omega_1(t)=\sqrt{\varepsilon}\, kt^{\gamma}, &\text{for } t_00\\ \omega_{\infty}\ &\text{for } t>t_1 \end{cases} \end{equation*} where $\varepsilon>0$ is from \eqref{EPS}, $\omega_0(t_0)=\omega_1(t_0)=\sqrt{\varepsilon}<\omega_{\infty}$. For $\mathcal{M}$ from \eqref{Nat} (see \eqref{I2} and \eqref{I3} as well) where $\omega$ is the above function we obtain the estimate \begin{align*} \mathcal{M} &=\sup_{t_00, \\ \omega_1(t)=\sqrt{\varepsilon\ln(1+\theta(t))}, &\text{for}\quad t_0t_1 \end{cases} \end{equation} where $\varepsilon>0$ is from \eqref{EPS}, $\omega_0(t_0)=\omega_1(t_0)=\sqrt{\varepsilon}<\omega_{\infty}$, $\theta(t)$ is a suitable increasing function such that $\lim_{t\to t_0^{+}}\theta(t)=\mathrm{e}-1$. For $\mathcal{M}$ defined by \eqref{Nat}, where $\omega$ is the above function, we obtain \begin{align*} \mathcal{M} &=\sup_{t_00$ is a constant), we can see that $\mathcal{M}\le 1$ for $t_00$ holds. Then, choosing $\Omega_0=B_{r}(0)$, $0