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

¹ Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli “Federico II”, Via Cintia, 80126 Napoli, Italy
² University of Modena and Reggio Emilia, Dipartimento di Scienze Fisiche, Informatiche e Matematiche, via Campi 213/b, 41125 Modena, Italy

^* Corresponding author: erica.ipocoana@unipr.it

Received: 3 March 2022
Accepted: 4 July 2022

Abstract

We study the higher fractional differentiability properties of the gradient of the solutions to variational obstacle problems of the form

with F double phase functional of the form

where Ω is a bounded open subset of ℝⁿ, ψ ∈ W^1,p(Ω) is a fixed function called obstacle and = {w ∈ W^1,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.