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

Abstract

Abstract We here establish the higher fractional differentiability for solutions to a class of obstacle problems with non-standard growth conditions. We deal with the case in which the solutions to the obstacle problems satisfy a variational inequality of the form ∫ Ω ⟨ A ⁢ ( x , D ⁢ u ) , D ⁢ ( φ - u ) ⟩ ⁢ d x ≥ 0 for all ⁢ φ ∈ K ψ ⁢ ( Ω ) , \int_{\Omega}\langle\mathcal{A}(x,Du),D(\varphi-u)\rangle\,dx\geq 0\quad\text{for all}\ \varphi\in\mathcal{K}_{\psi}(\Omega), where Ω is a bounded open subset of R n \mathbb{R}^{n} , ψ ∈ W 1 , p ⁢ ( Ω ) \psi\in W^{1,p}(\Omega) is a fixed function called obstacle and K ψ ⁢ ( Ω ) = { w ∈ W 1 , p ⁢ ( Ω ) : w ≥ ψ ⁢ a.e. in ⁢ Ω } \mathcal{K}_{\psi}(\Omega)=\{w\in W^{1,p}(\Omega):w\geq\psi\ \text{a.e. in}\ \Omega\} is the class of admissible functions. Assuming that the gradient of the obstacle belongs to some suitable Besov space, we are able to prove that some fractional differentiability property transfers to the gradient of the solution.