\documentclass[twoside]{article} \usepackage{amssymb,amsmath} \pagestyle{myheadings} \markboth{\hfil Regularity for nonlinear elliptic systems\hfil EJDE--2002/20} {EJDE--2002/20\hfil Josef Dan\v{e}\v{c}ek \& Eugen Viszus \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2002}(2002), No. 20, pp. 1--13. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % $\mathcal{L}^{2,\varPhi}$ regularity for nonlinear elliptic systems of second order % \thanks{ {\em Mathematics Subject Classifications:} 49N60, 35J60. \hfil\break\indent {\em Key words:} Nonlinear equations, regularity, Morrey-Campanato spaces. \hfil\break\indent \copyright 2002 Southwest Texas State University. \hfil\break\indent Submitted May 31, 2001. Published February 19, 2002.\hfil\break\indent J. Dan\v{e}\v{c}ek was partially supported by the research project MSM no. 261100006 } } \date{} % \author{Josef Dan\v{e}\v{c}ek \& Eugen Viszus} \maketitle \begin{abstract} This paper is concerned with the regularity of the gradient of the weak solutions to nonlinear elliptic systems with linear main parts. It demonstrates the connection between the regularity of the (generally discontinuous) coefficients of the linear parts of systems and the regularity of the gradient of the weak solutions of systems. More precisely: If above-mentioned coefficients belong to the class $L^\infty(\Omega)\cap\mathcal{L}^{2,\varPsi}(\Omega)$ (generalized Campanato spaces), then the gradient of the weak solutions belong to $\mathcal{L}_{loc}^{2,\varPhi}(\Omega,\mathbb{R}^{nN})$, where the relation between the functions $\varPsi$ and $\varPhi$ is formulated in Theorems \ref{thm1} and \ref{thm2} below. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \numberwithin{equation}{section} \section{Introduction} In this paper, we consider the problem of the regularity of the first derivatives of weak solutions to the nonlinear elliptic system \begin{equation} -D_\alpha a_i^\alpha(x,u,Du)=a_i(x,u,Du),\quad i=1,\dots,N, \label{1.1} \end{equation} where $a_i^\alpha$, $a_i$ are Caratheodorian mappings from $(x,u,z)\in\Omega\times\mathbb{R}^N\times\mathbb{R}^{nN}$ into $\mathbb{R}$, $N>1$, $\Omega\subset\mathbb{R}^n$, $n\ge 3$ is a bounded open set. A function $u\in W_{{\rm loc}}^{1,2}(\Omega,\mathbb{R}^N)$ is called a weak solution of (\ref{1.1}) in $\Omega$ if $$\int_{\Omega}a_i^\alpha(x,u,Du)D_\alpha \varphi^i(x)\, dx =\int_{\Omega}a_i(x,u,Du)\varphi^i(x)\, dx, \quad\forall\,\varphi\in C_0^\infty(\Omega,\mathbb{R}^N). $$ We use the summation convention over repeated indices. As it is known, in case of a general system (\ref{1.1}), only partial regularity can be expected for $n>2$ (see e.g. \cite{Ca,Gia,Ne}). Under the assumptions below we will prove $\mathcal{L}^{2,\varPhi}$-regularity of gradient of weak solutions for the system (\ref{1.1}) whose coefficients $a_i^\alpha$ have the form \begin{equation} a_i^\alpha(x,u,Du) =A_{ij}^{\alpha\beta}(x) D_\beta u^j+g_i^\alpha(x,u,Du), \label{1.2} \end{equation} where $i$, $j=1,\dots,N$, $\alpha$, $\beta=1,\dots,n$, $A_{ij}^{\alpha\beta}$ is a matrix of functions, and the following condition of strong ellipticity \begin{equation} A_{ij}^{\alpha\beta}(x)\xi_\alpha^i\xi_\beta^j \ge\nu\vert\xi\vert^2,\quad \text{a.e.} \ x\in\Omega,\ \forall\,\xi\in\mathbb{R}^{nN}; \nu>0 \label{1.3} \end{equation} holds, and $g_i^\alpha$ are functions with sublinear growth in $z$. In what follows, we formulate the conditions on the smoothness and the growth of the functions $A_{ij}^{\alpha\beta}$, $g_i^\alpha$ and $a_i$ precisely. It is well known (see \cite{Ca}) that, in the case of linear elliptic systems with continuous coefficients $A_{ij}^{\alpha\beta}$, the gradient of weak solutions has the $L^{2,\lambda}$-regularity and, if the coefficients $A_{ij}^{\alpha\beta}$ belong to some H\"{o}lder class, then the gradient of weak solutions belongs to the BMO-class (functions with bounded mean oscillations, see Definition 2.1). These results were generalized in \cite{Da} where the first author has proved the $L^{2,\lambda}$-regularity of the gradient of weak solutions to (\ref{1.1})-(\ref{1.3}) in the situation where the coefficients $A_{ij}^{\alpha\beta}$ are continuous and the BMO-regularity of gradient in the case where coefficients $A_{ij}^{\alpha\beta}$ are H\"{o}lder continuous. In the case of linear elliptic systems when the coefficients $A_{ij}^{\alpha\beta}$ are ``small multipliers of $BMO(\Omega)$", a class neither containing nor contained in $C(\overline{\Omega})$, Acquistapace in \cite{Ac} proved global (under Dirichlet boundary condition) and local BMO-regularity for the gradient of solutions. In \cite{Ac} the local BMO-regularity does not follow in a standard way from the global one, because there are no regularity results in the Morrey spaces $L^{2,\lambda}$, $0<\lambda2$), we obtain $\mathcal{L}^{2,\varPhi}$-regularity of the gradient. \section{Notation and definitions} We consider the bounded open set $\Omega\subset\mathbb{R}^n$ with points $x=(x_1,\dots x_n)$, $n\ge 3$, $u\:\Omega\to\mathbb{R}^N$, $N\ge 1$, $u(x)=(u^1(x),\dots,u^N(x))$ is a vector-valued function, $Du=(D_1u,\dots,D_nu)$, $D_\alpha=\partial/\partial x_\alpha$. The meaning of $\Omega_0\subset\subset\Omega$ is that the closure of $\Omega_0$ is contained in $\Omega$, i.e. $\overline\Omega_0\subset\Omega$. For the sake of simplicity we denote by $\vert\cdot\vert$ the norm in $\mathbb{R}^n$ as well as in $\mathbb{R}^N$ and $\mathbb{R}^{nN}$. If $x\in\mathbb{R}^n$ and $r$ is a positive real number, we write $B_r(x)=\lbrace{y\in\mathbb{R}^n : |y-x|0$ such that, for every $x\in\overline\Omega$ and all $00}\frac{1}{r^\lambda} \int_{\Omega (x,r)}\vert u(y)\vert^q\, dy<\infty. $$ Let $\lambda\in [0,n+q]$, $q\in [1,\infty)$. The Campanato space $\mathcal{L}^{q,\lambda}(\Omega,\mathbb{R}^N)$ is the subspace of such functions $u\in L^q(\Omega,\mathbb{R}^N)$ for which $$[u]_{\mathcal{L}^{q,\lambda}(\Omega,\mathbb{R}^N)}^q =\sup_{r>0,x\in\Omega} \frac{1}{r^\lambda}\int_{\Omega(x,r)} \vert u(y)-u_{x,r}\vert^q\, dy<\infty. $$ Let $Q_0\subset\mathbb{R}^n$ is a cube whose edges are parallel with the coordinate axes. The $BMO(Q_0,\mathbb{R}^N)$ space (bounded mean oscillation space) is the subspace of such functions $u\in L^1(Q_0,\mathbb{R}^N)$ for which $$\langle u\rangle_{Q_0} =\sup_{Q\subset Q_0}\frac{1}{\vert Q\vert}\int_{Q} \vert u(y)-u_Q\vert\, dy<\infty, $$ where $u_Q=\int_{Q}\kern-14pt-\ u(y)\, dy$ and $Q\subset Q_0$ is the cube homotetic with $Q_0$. \paragraph{Remark} $u\in L_{{\rm loc}}^{q,\lambda}(\Omega,\mathbb{R}^N)$ if and only if $u\in L^{q,\lambda}(\Omega_0,\mathbb{R}^N)$ for each $\Omega_0\subset\subset\Omega$. \begin{proposition} \label{prop1} For a domain $\Omega\subset\mathbb{R}^n$ of the class $\mathcal{C}^{0,1}$ we have the following \begin{enumerate} \item[(a)] With the norms $\Vert u\Vert_{L^{q,\lambda}}$ and $\|u\|_{\mathcal{L}^{q,\lambda}}=\|u\|_{L^q}+[u]_{L^{q,\lambda}}$, $\|u\|_{BMO}=\|u\|_{L^1}+\langle u\rangle_{Q}$, $L^{q,\lambda}(\Omega,R^N)$, $\mathcal{L}^{q,\lambda}(\Omega,R^N)$ and $BMO(Q_0,\mathbb{R}^N)$ are Banach spaces. \item[(b)] $L^{q,\lambda}(\Omega,\mathbb{R}^N)$ is isomorphic to the $\mathcal{L}^{q,\lambda}(\Omega,\mathbb{R}^N)$, $1\le q<\infty$, $0\le\lambda0$ is almost decreasing, i.e. there exists $k_\psi\ge 1$ and $d_0\le d$ such that \begin{equation}k_\psi\frac{\psi(r)}{r^\xi}\geq\frac{\psi(R)}{R^\xi}, \quad\forall\ 00$. \section{Main results} Suppose that for all $(x,u,z)\in\Omega\times\mathbb{R}^N\times\mathbb{R}^{nN}$ the following conditions hold: \begin{align} \vert a_i(x,u,z)\vert &\le f_i(x)+L\vert z\vert^{\gamma_0},\label{3.1}\\ \vert g_i^\alpha(x,u,z)\vert &\le f_i^\alpha(x) +L\vert z\vert^\gamma,\label{3.2}\\ g_i^\alpha(x,u,z)z_\alpha^i &\ge\nu_1\vert z\vert^{1+\gamma}-f^2(x) \label{3.3} \end{align} for almost all $x\in\Omega$ and all $u\in\mathbb{R}^{N}$, $z\in\mathbb{R}^{nN}$. Here $L$, $\nu_1$ are positive constants, $1\le\gamma_0<(n+2)/n$, $0\le\gamma<1$, $f$, $f_i^\alpha\in L^{\sigma,\lambda}(\Omega)$, $\sigma>2$, $0<\lambda\le n$, $f_i\in L^{\sigma q_0,\lambda q_0}(\Omega)$, $q_0=n/(n+2)$. We set $A=(A_{ij}^{\alpha\beta})$, $g=(g_i^\alpha)$, $a=(a_i)$, $\widetilde f=(f_i)$, $\widetilde{\widetilde{f}}=(f_i^\alpha)$. The next theorem is slightly generalizing the main result from \cite{DV1}. \begin{theorem} \label{thm1} Let $u\in W_{{\rm loc}}^{1,2}(\Omega,\mathbb{R}^N)$ be a weak solution to the system (\ref{1.1}) and the conditions (\ref{1.2}), (\ref{1.3}), (\ref{3.1}), (\ref{3.2}) and (\ref{3.3}) be satisfied. Suppose further that $A_{ij}^{\alpha\beta}\in L^\infty(\Omega) \cap\mathcal{L}^{2,\varPsi}(\Omega)$, $i$, $j=1,\dots,N$, $\alpha$, $\beta=1,\dots,n$ and $\varPsi$ is a function satisfying the condition (\ref{2.1}) with $\zeta=n$. Then $$ Du\in \begin{cases} L_{{\rm loc}}^{2,\lambda}(\Omega,R^{nN}) &\text{if }\lambda 2$, $q_0=n/(n+2)$. It is not difficult to see that from assumptions (\ref{3.4})--(\ref{3.6}) follow (\ref{3.1})--(\ref{3.3}) with $\lambda=n$. We can now formulate the main result of this paper. \begin{theorem}\label{thm2} Let $u\in W_{{\rm loc}}^{1,2}(\Omega,\mathbb{R}^N)$ be a weak solution to the system (\ref{1.1}) and suppose that the conditions (\ref{1.2}), (\ref{1.3}), (\ref{3.4}), (\ref{3.5}) and (\ref{3.6}) hold. Let further $A_{ij}^{\alpha\beta}\in L^\infty(\Omega) \cap\mathcal{L}^{2,\varPsi}(\Omega)$, for each $i$, $j=1,\dots ,N$, $\alpha$, $\beta=1,\dots ,n$ and $\varPsi$ be a function satisfying the conditions (\ref{2.1}) and (\ref{2.2}) with $\zeta=n$ and $0<\xi\le 2$. Then $Du\in\mathcal{L}_{{\rm loc}}^{2,\varPhi}(\Omega,\mathbb{R}^{nN})$ with $\varPhi(R)=R^{n/2}$ in the case when the function $\psi$ has a form of some power function and $\varPhi(R)=R^{\lambda/2}\psi^{(r-2)/2r}(R)$ for some $r>2$ and arbitrary $\lambda\sup_{r\in [Kd,d]} \frac{\phi(r)}{r^\beta\psi(r)} $$ and there exists a sequence $\left\{r_n\right\}_{n=n_0}^\infty$ such that $1/n2$ such that $u\in W_{{\rm loc}}^{1,r}(\Omega,\mathbb{R}^N)$. Moreover there exists a constant $c=c(\nu,\nu_1,L,\Vert A\Vert_{L^\infty})$ and $\widetilde{R}>0$ such that, for all balls $B_R(x)\subset\Omega$, $R<\widetilde{R}$, the following inequality is satisfied \begin{align*} \big(\int_{B_{R/2}(x)}\kern-37pt-\vert Du\vert^r \, dy\big)^{1/r} \le& c\Big\{\big(\int_{B_R(x)}\kern-29pt-\vert Du\vert^2 \, dy\big)^{1/2}\\ & +\big(\int_{B_R(x)}\kern-29pt- (\vert f\vert^r+\vert\widetilde{\widetilde{f}}\vert^r) \, dy\big)^{1/r}+R\big(\int_{B_R(x)}\kern-29pt- \vert\widetilde{f}\vert^{rq_0} dy\big)^{1/rq_0}\Big\}. \end{align*} \end{proposition} \section{Proof of Theorems} \paragraph{Proof of Theorem \ref{thm1}.} Let $B_{R/2}(x_0)\subset B_R(x_0)\subset\Omega$ be an arbitrary ball and let $w\in W_0^{1,2}(B_{R/2}(x_0),\mathbb{R}^N)$ be a solution of the following system \begin{equation}\begin{aligned} \int_{B_{R/2}(x_0)}& (A^{\alpha\beta}_{ij})_{x_0,R/2}D_\beta w^j D_\alpha\varphi^i \, dx\\ =&\int_{B_{R/2}(x_0)} \big((A^{\alpha\beta}_{ij})_{x_0,R/2}-A^{\alpha\beta}_{ij}(x)\big) D_\beta u^j D_\alpha \varphi^i\, dx\\ & -\int_{B_{R/2}(x_0)} \, g_i^\alpha(x,u,Du)D_\alpha\varphi^i\, dx +\int_{B_{R/2}(x_0)}a_i(x,u,Du)\varphi^i\, dx \end{aligned}\label{5.1} \end{equation} for all $\varphi\in W_0^{1,2}(B_{R/2}(x_0),\mathbb{R}^N)$. It is known that, under the assumption of this theorem, such solution exists and it is unique for all $R2$, H\"{o}lder inequality ($r'=r/(r-2)$) and from the fact that, for a BMO-function, all $L^r$ norms, $1\le r<\infty$ are equivalent (see Proposition \ref{prop1}(d)) we obtain \begin{equation} I\le c\Big(\int_{B_{R/2}(x_0)} \vert A(x)-A_{x_0,R/2}\vert^{2r'}\, dx\Big)^{1/r'} \Big(\int_{B_{R/2}(x_0)}\vert Du\vert^{r} \, dx\Big)^{2/r} \label{5.2} \end{equation} From the assumptions of this theorem and taking into account the properties of matrix $A=(A^{\alpha\beta}_{ij})$ we can estimate the first term on the right hand side of (\ref{5.2}) \begin{eqnarray} &&\int_{B_{R/2}(x_0)} \vert A(x)-A_{x_0,R/2}\vert^{2r'}\, dx \le c\Big(\int_{B_{R/2}(x_0)} \vert A(x)-A_{x_0,R/2}\vert^2\, dx\Big)^{1/2}\times \nonumber\\ &&\times\Big(\int_{B_{R/2}(x_0)} \vert A(x)-A_{x_0,R/2}\vert^{2(2r'-1)}\, dx\Big)^{1/2} \nonumber\\ &&\le\, c(n,[A]_{2,\Psi,\Omega})\Vert A\Vert_{L^\infty (\Omega,\mathbb{R}^{n^2N^2})}^{2r'-1}\, R^n\, \psi(R). \label{5.3} \end{eqnarray} To estimate the last integral in (\ref{5.2}) we use Proposition \ref{prop3} obtaining \begin{equation}\begin{aligned} \Big(\int_{B_{R/2}(x_0)} &\vert Du\vert^r\, dx\Big)^{2/r} \le c\Big\{\frac{1}{R^{n(1-2/r)}} \int_{B_R(x)}\vert Du\vert^2\, dy\\ & +\Big(\int_{B_R(x)}(\vert f\vert^r +\vert\widetilde{\widetilde{f}}\vert^r) \, dy\Big)^{2/r}+R^{2(1-2/r)}\Big(\int_{B_R(x)} \vert\widetilde{f}\vert^{rq_0}\, dy\Big)^{2/rq_0}\Big\}\\ &\le c\,\Big(\frac{1}{R^{n(1-2/r)}} \int_{B_R(x)}\vert Du\vert^2\, dy +R^{2\lambda/r}+R^{2(r-2+\lambda)/r)}\Big), \label{5.4} \end{aligned} \end{equation} where $c=c(r,\Vert f\Vert_{L^{r,\lambda}(\Omega)}, \Vert\widetilde{\widetilde{f}}\Vert_{L^{r,\lambda}(\Omega)}, \Vert\widetilde{f}\Vert_{L^{rq_0,\lambda q_0}(\Omega)})$. From (\ref{5.2}), (\ref{5.3}) and (\ref{5.4}) we obtain \begin{align*} I &\le c\,\Big(\psi^{1/r'}(R)\int_{B_R(x_0)} \vert Du\vert^2 dx+(R^{2\lambda/r} +R^{2(r-2+\lambda)/r)}) R^{n/r'}\psi^{1/r'}(R)\Big)\\ &\le c\,\Big(\psi^{1/r'}(R)\int_{B_R(x_0)} \vert Du\vert^2 dx +\, R^{n-2(n-\lambda)/r}\psi^{1/r'}(R)\Big), \end{align*} where $c=c(n,r,[A]_{2,\Psi,\Omega}, \Vert A\Vert_{L^\infty(\Omega,\mathbb{R}^{n^2N^2})}, \Vert f\Vert_{L^{r,\lambda}(\Omega)}, \Vert\widetilde{\widetilde{f}}\Vert_{L^{r,\lambda}(\Omega)}, \Vert\widetilde{f}\Vert_{L^{rq_0,\lambda q_0}(\Omega)})$. We can estimate II and III by means of Lemma \ref{lm3} (with $\tau=0$) and we have \begin{equation} \nu^2\int_{B_{R/2} (x_0)}\vert Dw\vert^2\, dx \le c\Big\{(\varepsilon+\psi^{1/r'}(R)) \int_{B_R(x_0)}\vert Du\vert^2 \, dx+R^{\mu}\Big\}, \label{5.5} \end{equation} where $\mu=\min\{n,n-2(n-\lambda)/r,n+2-n\gamma_0\}$. The function $v=u-w\in W^{1,2}(B_{R/2}(x_0),\mathbb{R}^N)$ is the solution of the system \begin{equation} \int_{B_{R/2}(x_0)} (A^{\alpha\beta}_{ij})_{x_0,R/2} D_\beta v^j D_\alpha\varphi^i\, dx=0, \qquad\forall\varphi\in W_0^{1,2}(B_{R/2}(x_0),\mathbb{R}^N). \label{5.6} \end{equation} From Lemma \ref{lm1} we have, for $0<\sigma\le R/2$, $$\int_{B_\sigma(x_0)}\vert Dv\vert^2\, dx \le c\, \Big(\frac{\sigma}{R}\Big)^n \int_{B_{R/2}(x_0)}\vert Dv\vert^2\, dx. $$ By means of (\ref{5.5}) and the last estimate we obtain, for all $0<\sigma\le R$ and $\varepsilon\in(0,1)$, the following estimate $$\int_{B_\sigma(x_0)}\vert Du\vert^2\, dx \le c_1\left[\Big(\frac{\sigma}{R}\Big)^n +\varepsilon+\psi^{1/r'}(R)\right] \int_{B_R(x_0)}\vert Du\vert^2 \, dx+c_2\, R^{\mu}, $$ where the constants $c_1$ and $c_2$ only depend on the above-mentioned parameters. Now, in a way analogous to that from \cite{DV2}, we obtain the result. \paragraph{Proof of Theorem \ref{thm2}.} By Theorem \ref{thm1}, $Du\in L_{{\rm loc}}^{2,\lambda}(\Omega,\mathbb{R}^{nN})$ for arbitrary $\lambda0$ exists such that $Bk_\psi K^{-\lambda-\xi} =c_1\varepsilon k_\psi K^{-\lambda-\xi}<1/2$ and then, for all $0<\sigma\le R