Weighted gradient estimates for elliptic problems with Neumann boundary conditions in Lipschitz and (semi-)convex domains

摘要

Let n >= 2 and Omega be a bounded Lipschitz domain in R-n. In this article, the authors investigate global (weighted) norm estimates for the gradient of solutions to Neumann boundary value problems of second order elliptic equations of divergence form with real-valued, bounded, measurable coefficients in Omega. More precisely, for any given p is an element of (2, infinity), two necessary and sufficient conditions for W-1,W-P estimates of solutions to Neumann boundary value problems, respectively, in terms of a weak reverse Holder inequality with exponent p or weighted W-1,W-q estimates of solutions with q is an element of [2, p] and some Muckenhoupt weights, are obtained. As applications, for any given p is an element of (1, infinity) and omega is an element of A(p) (R-n) (the class of Muckenhoupt weights), the authors establish weighted W-omega(1,p) estimates for solutions to Neumann boundary value problems of second order elliptic equations of divergence form with small BMO coefficients on bounded (semi-)convex domains. As further applications, the global gradient estimates are obtained, respectively, in (weighted) Lorentz spaces, (Lorentz-)Morrey spaces, (weighted) Orlicz spaces, and variable Lebesgue spaces. (C) 2019 Elsevier Inc. All rights reserved.