Regularity results for a class of obstacle problems with <i>p</i>, <i>q</i>−growth conditions
Michele Caselli, Michela Eleuteri, Antonia Passarelli di Napoli
Abstract
In this paper we prove the the local Lipschitz continuity for solutions to a class of obstacle problems of the type min{ ∫ Ω F ( x, Dz ) : z ∈ 𝛫 ψ (Ω)}. Here 𝛫 ψ (Ω) is the set of admissible functions z ∈ u 0 + W 1, p (Ω) for a given u 0 ∈ W 1, p (Ω) such that z ≥ ψ a.e. in Ω, ψ being the obstacle and Ω being an open bounded set of ℝ n , 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.