Volume 27, 2021
|Number of page(s)||26|
|Published online||22 March 2021|
Regularity results for a class of obstacle problems with p, q−growth conditions
ETH Zürich, Department of Mathematics,
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: firstname.lastname@example.org
Accepted: 8 February 2021
In this paper we prove the the local Lipschitz continuity for solutions to a class of obstacle problems of the type
Here 𝛫ψ(Ω) is the set of admissible functions z ∈ u0 + W1,p(Ω) for a given u0 ∈ W1,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.
Mathematics Subject Classification: 35J87 / 49J40 / 47J20
Key words: Variational inequalities / obstacle problems / local boundedness / local Lipschitz continuity
© EDP Sciences, SMAI 2021
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