Regularity results for solutions to obstacle problems with Sobolev coefficients

Abstract

We establish the higher differentiability of solutions to a class of obstacle problems of the typemin⁡{∫Ωf(x,Dv(x))dx:v∈Kψ(Ω)}, where ψ is a fixed function called obstacle, Kψ(Ω)={v∈Wloc1,p(Ω,R):v≥ψ a.e. in Ω} and the convex integrand f satisfies p-growth conditions with respect to the gradient variable. We derive that the higher differentiability property of the weak solution v is related to the regularity of the assigned ψ, under a suitable Sobolev assumption on the partial map x↦Dξf(x,ξ). The main novelty is that such assumption is independent of the dimension n and this, in the case p≤n−2, allows us to manage coefficients in a Sobolev class below the critical one W1,n.