Regularity results for a class of non-differentiable obstacle problems

Abstract

In this paper we prove the higher differentiability in the scale of Besov spaces of the solutions to a class of obstacle problems of the type min∫ΩF(x,z,Dz):z∈Kψ(Ω). Here Ω is an open bounded set of Rn, n≥2, ψ is a fixed function called obstacle and Kψ(Ω) is set of admissible functions z∈W1,p(Ω) such that z≥ψ a.e. in Ω. We assume that the gradient of the obstacle belongs to a suitable Besov space. The main novelty here is that we are not assuming any differentiability on the partial maps x↦F(x,z,Dz) and z↦F(x,z,Dz), but only their Hölder continuity.