Higher differentiability results in the scale of Besov Spaces to a class of double-phase obstacle problems

Abstract

We study the higher fractional differentiability properties of the gradient of the solutions to variational obstacle problems of the form $\min \left\{ {\int_\Omega {F\left( {x,w,Dw} \right){\rm{d}}x\,:\,w\, \in \,{\kappa _\psi }\left( \Omega \right)} } \right\},$ with F double phase functional of the form $F\left( {x,\,w,\,z} \right) = b\left( {x,w} \right)\left( {{{\left| z \right|}^p} + a\left( x \right){{\left| z \right|}^q}} \right),$ where Ω is a bounded open subset of ℝn, ψ ∈ W1,p(Ω) is a fixed function called obstacle and = {w ∈ W1,P(Ω) : w ≥ ψ a.e. in Ω} is the class of admissible functions. Assuming that the gradient of the obstacle belongs to a suitable Besov space, we are able to prove that the gradient of the solution preserves some fractional differentiability property.