Higher differentiability of solutions for a class of obstacle problems with non standard growth conditions

Abstract

We are going to prove a higher differentiability result for the solutions to a class of variational obstacle problems of the formmin⁡{∫ΩF(x,Dω)dx:ω∈Kψ(Ω)} where the function F satisfies non standard growth conditions of the type ν|z|p≤F(x,z)≤L(1+|z|q) with 1<p<q and 0<ν<L; in particular we will consider the subquadratic case: 1<p<q<2.Moreover Ω⊂Rn,n>2 is a bounded open domain, the function ψ:Ω→[−∞,+∞) is called obstacle and it belongs to the Sobolev class W1,p(Ω), p>1 and Kψ(Ω) is the set of admissible functions, i.e.Kψ(Ω)={ω∈u0+W01,p(Ω):ω≥ψa.e.inΩ} where u0 is a fixed boundary value. The main result consists in showing thatDψ∈Wloc1,r(Ω)⇒(1+|Du|2)p−24Du∈Wloc1,2(Ω) under the conditions:qp<1+1n−1r,1<p<q<2<n<r.