2021/03/13 by Andrea Gentile
paper · doi:10.1515/forum-2020-0299
Abstract We establish some higher differentiability results of integer and fractional order for solutions to non-autonomous obstacle problems of the form min ∫ Ω f ( x , D v ( x ) ) : v ∈ 𝒦 ψ ( Ω ) , min\biggl\∫Ωf(x,Dv(x)):v\inKψ(Ω)\biggr\, where the function f satisfies p -growth conditions with respect to the gradient variable, for 1 p 2 1 , and 𝒦 ψ ( Ω ) Kψ(Ω) is the class of admissible functions v ∈ u 0 + W 0 1 , p ( Ω ) v∈ u0+W1,p0(Ω) such that v ≥ ψ v≥ψ a.e. in Ω, where u 0 ∈ W 1 , p ( Ω ) u0∈ W1,p(Ω) is a fixed boundary datum. Here we show that a Sobolev or Besov–Lipschitz regularity assumption on the gradient of the obstacle ψ transfers to the gradient of the solution, provided the partial map x ↦ D ξ f ( x , ξ ) x↦ Dξf(x,ξ) belongs to a suitable Sobolev or Besov space. The novelty here is that we deal with sub-quadratic growth conditions with respect to the gradient variable, i.e. f ( x , ξ ) ≈ a ( x ) | ξ | p f(x,ξ)≈ a(x)|ξ|p with 1 p 2 1 , and where the map a belongs to a Sobolev or Besov–Lipschitz space.