Interior integral estimates on weak solutions of certain degenerate elliptic systems (Q809271)

The paper concerns the differentiability of weak solutions of the system \[ (1)\quad (\partial /\partial x_{\alpha})A^{\alpha}_ i(\nabla u)=0\text{ in } \Omega \quad (i=1,...,N) \] (\(\Omega \subset {\mathbb{R}}^ n\) open, \(u=\{u^ 1,...,u^ N\})\), where \(A^{\alpha}_ i\) are continuous functions on \({\mathbb{R}}^{nN}\) which satisfy the usual growth condition and \[ (A^{\alpha}_ i(\xi)-A^{\alpha}_ i(\eta))(\xi^ i_{\alpha}-\eta^ i_{\alpha})\geq c_ 1(1+| \xi |^ 2+| \eta |^ 2)^{(p-2)/2}| \xi -\eta |^ 2 \] for all \(\xi,\eta \in {\mathbb{R}}^{nN} (c_ 1=const>0\), \(1<p<2).\) The main results are as follows: (1) Let \(u\in W^{1,p}(\Omega;{\mathbb{R}}^ N)\) be any weak solution of (1). Then \[ \partial^ 2u/\partial x_{\alpha}\partial x_{\alpha}\in L^ p_{loc}(\Omega;{\mathbb{R}}^ N),\quad (1+| \nabla u|^ 2)^{(p-2)/4}| \nabla^ 2u| \in L^ 2_{loc}(\Omega)\quad (\alpha,\beta =1,...,n). \] (2) In addition, suppose that \(A^{\alpha}_ i\) satisfies a ``weighted'' Lipschitz condition. Then there exists a \(t>2\) such that \[ (1+| \nabla u|^ 2)^{(p- 2)/4}| \nabla^ 2u| \in L^ t_{loc}(\Omega). \] In particular, \(\nabla^ 2u\in L^{t_ 1}_{loc}(\Omega;{\mathbb{R}}^{4N})\) if \(n=2\) (\(\forall 2\leq t_ 1<t)\). The higher integrability result is based on the argument by Gehring-Giaquinta-Modica via a weighted Sobolev- Poincaré inequality.