Regularity results of solutions of quasilinear systems having singularities in the coefficients (Q6149219)

Summary: We consider non-degenerate elliptic systems of the type \[ - \operatorname{div} A(x, Du) = g(x) \quad \text{in } \Omega \subset \mathbb{R}^n, \] 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 \(u \in W^{1,p}_{\text{loc}} (\Omega, \mathbb{R}^N)\) 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 \(D(V_\mu (Du)) \in L^{2}_{\text{loc}} (\Omega)\). Then, we deduce higher differentiability properties for itself and some higher integrability results for its gradient.