Weighted regularity estimates in Orlicz spaces for fully nonlinear elliptic equations. (October 2017) - Byun, Sun-Sig; Lee, Mikyoung; Ok, Jihoon
Abstract: In this paper we develop a global W 2, p estimate for the viscosity solution of the Dirichlet problem of fully nonlinear elliptic equations F ( D 2 u, D u, u, x ) = f ( x ) in Ω, u = 0 on ∂ Ω to a more general function space. Given an N -function Φ and a Muckenhoupt weight w, we prove that if f belongs to the associated weighted Orlicz space L w Φ ( Ω ), then D 2 u ∈ L w Φ ( Ω ) and u satisfies a global W w 2, Φ estimate, under a minimal regularity requirement on F in the variable x and a basic geometric assumption on ∂ Ω . The correct condition on the couple, Φ and w, is also addressed. This result generalizes the W 2, p estimate (Caffarelli, 1989, Escauriaza, 1993, Winter, 2009) of Calderón and Zygmund as well as an analogous one (Byun et al., 2016) in the weighted L p setting.