Regularity results for solutions to a class of obstacle problems

Abstract

In this paper we prove some regularity properties of solutions to variational inequalities of the form ∫Ω〈A(x,u,Du),D(φ−u)〉dx≥∫ΩB(x,u,Du)(φ−u)dx,∀φ∈Kψ(Ω).Here Ω is a bounded open set of Rn, n≥2, the function ψ:Ω→[−∞,+∞), called obstacle, belongs to the Sobolev class W1,p(Ω) and Kψ(Ω)={w∈W1,p(Ω):w≥ψq.o. inΩ} is the class of the admissible functions. First we establish a local Calderòn–Zygmund type estimate proving that the gradient of the solutions is as integrable as the gradient of the obstacle in the scale of Lebesgue spaces Lpq, for every q∈(1,∞), provided the partial map (x,u)↦A(x,u,ξ) is Hölder continuous and B(x,u,ξ) satisfies a suitable growth condition. Next, this estimate allows us to prove that a higher differentiability in the scale of Besov spaces of the gradient of the obstacle transfers to the gradient of the solutions.