Gradient estimates of general nonlinear singular elliptic equations with measure data

Abstract

We develop a global Calderón-Zygmund estimate for the gradients of renormalized solutions to the general nonlinear singular elliptic equations −divA(x,u,Du)=μ on a Reifenberg flat domain with the homogeneous Dirichlet boundary condition, while μ is a finite signed Radon measure. The associated nonlinearity behaves as the elliptic p-Laplacian with respect to Du for the singular case p∈(1,2−1/n], whose discontinuity in the x-variable is measured in terms of small BMO, and the Lipschitz continuity is required with respect to the u-variable. We prove it in two folds: the perturbation technique and the weighted Vitali type covering are first employed to establish the weighted good-λ type inequality, then such inequality is used to prove the desired global gradient estimates in weighted Lorentz spaces and Lorentz-Morrey spaces. As a direct consequence, finally we obtain a global gradient regularity in weighted Orlicz spaces.