Higher differentiability of solutions for a class of obstacle problems with variable exponents

Abstract

In this paper we prove a higher differentiability result for the solutions to a class of obstacle problems in the formmin⁡{∫ΩF(x,Dw)dx:w∈Kψ(Ω)} where ψ∈W1,p(x)(Ω) is a fixed function called obstacle and Kψ(Ω)={w∈W01,p(x)(Ω)+u0:w≥ψa.e. in Ω} is the class of the admissible functions, for a suitable boundary value u0. We deal with a convex integrand F which satisfies the p(x)-growth conditions|ξ|p(x)≤F(x,ξ)≤C(1+|ξ|p(x)),p(x)>1.