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

ESAIM: COCV 27 (2021) 19 ESAIM: Control, Optimisation and Calculus of Variations https://doi.org/10.1051/cocv/2021017 www.esaim-cocv.org REGULARITY RESULTS FOR A CLASS OF OBSTACLE PROBLEMS WITH p, q−GROWTH CONDITIONS M. Caselli1, M. Eleuteri2 and A. Passarelli di Napoli3,* Abstract. In this paper we prove the the local Lipschitz continuity for solutions to a class of obstacle problems of the type min {ˆ Ω F (x, Dz) : z ∈ Kψ (Ω) } . Here Kψ (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.