Higher differentiability for solutions of nonhomogeneous elliptic obstacle problems

Abstract

This paper focuses on weak solutions to the following variational inequality∫Ω〈a(x,Du),D(φ−u)〉dx≥∫Ω〈|F|p−2F,D(φ−u)〉dx for all φ∈Aψ(Ω)={w∈W1,p(Ω)|w≥ψa.e. inΩ}, where the vector field a satisfies some growth and ellipticity conditions. We derive that the higher differentiability property of the weak solution u is related to the regularity of the assigned obstacle ψ and the nonhomogeneous term F under a suitable integer or fractional differentiability assumption on a(x,ξ) with respect to x.