Regularity results of solutions of quasilinear systems having singularities in the coefficients

Abstract

We consider non-degenerate elliptic systems of the type

where $u:\Omega \to {\mathbb{R}}^N$, $g\in L^2(\Omega ,{\mathbb{R}}^N)$ and $x\to A(x,\xi )$ has derivatives in the Marcinkiewicz class $L^{n,\infty }(\Omega )$ with sufficiently small distance to $L^{\infty }(\Omega )$. We prove that every weak solution $u\in W_{\mathrm{loc}}^{1,p}(\Omega ,{\mathbb{R}}^N)$ of the system is such that the nonlinear expression of its gradient $V_{\mu }(Du):=({\mu }^2+|Du{|}^2{)}^{\frac{p-2}{4}}Du$ is weakly differentiable with $D(V_{\mu }(Du))\in L_{\mathrm{loc}}^2(\Omega )$. Then, we deduce higher differentiability properties for $u$ itself and some higher integrability results for its gradient.