Regularity results for a class of obstacle problems with p, q−growth conditions

¹ ETH Zürich, Department of Mathematics, Rämistrasse 101, 8092 Zürich, Switzerland.
² Dipartimento di Scienze Fisiche, Informatiche e Matematiche, Università degli Studi di Modena e Reggio Emilia, via Campi 213/b, 41125 Modena, Italy.
³ Dipartimento di Matematica e Applicazioni “R. Caccioppoli”, Università degli Studi di Napoli, “Federico II” Via Cintia, 80126 Napoli, Italy.

^* Corresponding author: antpassa@unina.it

Received: 14 April 2020
Accepted: 8 February 2021

Abstract

In this paper we prove the the local Lipschitz continuity for solutions to a class of obstacle problems of the type

Here 𝛫_ψ(Ω) is the set of admissible functions z ∈ u₀ + W^1,p(Ω) for a given u₀ ∈ W^1,p(Ω) such that z ≥ ψ a.e. in Ω, ψ being the obstacle and Ω being an open bounded set of ℝⁿ, n ≥ 2. The main novelty here is that we are assuming that the integrand F(x, Dz) satisfies (p, q)-growth conditions and as a function of the x-variable belongs to a suitable Sobolev class. We remark that the Lipschitz continuity result is obtained under a sharp closeness condition between the growth and the ellipticity exponents. Moreover, we impose less restrictive assumptions on the obstacle with respect to the previous regularity results. Furthermore, assuming the obstacle ψ is locally bounded, we prove the local boundedness of the solutions to a quite large class of variational inequalities whose principal part satisfies non standard growth conditions.