Higher differentiability results for solutions to a class of non-autonomous obstacle problems with sub-quadratic growth conditions

Abstract

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{\{}\int_{\Omega}f(x,Dv(x)):v\in\mathcal{K}_{\psi}(\Omega)\biggr{\}}, where the function f satisfies p-growth conditions with respect to the gradient variable, for 1 < p < 2 {1<p<2} , and 𝒦 ψ ⁢ ( Ω ) {\mathcal{K}_{\psi}(\Omega)} is the class of admissible functions v ∈ u 0 + W 0 1 , p ⁢ ( Ω ) {v\in u_{0}+W^{1,p}_{0}(\Omega)} such that v ≥ ψ {v\geq\psi} a.e. in Ω, where u 0 ∈ W 1 , p ⁢ ( Ω ) {u_{0}\in W^{1,p}(\Omega)} 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\mapsto D_{\xi}f(x,\xi)} 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,\xi)\approx a(x)\lvert\xi\rvert^{p}} with 1 < p < 2 {1<p<2} , and where the map a belongs to a Sobolev or Besov–Lipschitz space.