Weighted regularity estimates in Orlicz spaces for fully nonlinear elliptic equations

Abstract

In this paper we develop a global W-2,W-p estimate for the viscosity solution of the Dirichlet problem of fully nonlinear elliptic equations F(D(2)u, Du, u, x) = f(x) in Omega, u = 0 on partial derivative Omega to a more general function space. Given an N-function Phi and a Muckenhoupt weight w, we prove that if f belongs to the associated weighted Orlicz space L-w(Phi) (Omega), then D(2)u is an element of L-w(Phi) (Omega) and u satisfies a global W-w(2,Phi) estimate, under a minimal regularity requirement on F in the variable x and a basic geometric assumption on partial derivative Omega. The correct condition on the couple, Phi and w, is also addressed. This result generalizes the W-2,W-p estimate (Caffarelli, 1989, Escauriaza, 1993, Winter, 2009) of Calderon and Zygmund as well as an analogous one (Byun et al., 2016) in the weighted L-p setting. (C) 2017 Elsevier Ltd. All rights reserved.