Regularity Results for Bounded Solutions to Obstacle Problems with Non-standard Growth Conditions

Abstract

In this paper, we consider a class of obstacle problems of the type min∫Ωf(x,Dv)dx:v∈Kψ(Ω)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\min \\left\\{ \\int _{\\Omega }f(x, Dv)\\, {\\mathrm d}x\\,:\\, v\\in {\\mathcal {K}}_\\psi (\\Omega )\\right\\} \\end{aligned}$$\\end{document}where psi is the obstacle, {mathcal {K}}_psi (Omega )={vin u_0+W^{1, p}_{0}(Omega , {mathbb {R}}): vge psi text { a.e. in }Omega }, with u_0 in W^{1,p}(Omega ) a fixed boundary datum, the class of the admissible functions and the integrand f(x, Dv) satisfies non standard (p, q)-growth conditions. We prove higher differentiability results for bounded solutions of the obstacle problem under dimension-free conditions on the gap between the growth and the ellipticity exponents. Moreover, also the Sobolev assumption on the partial map xmapsto A(x, xi ) is independent of the dimension n and this, in some cases, allows us to manage coefficients in a Sobolev class below the critical one W^{1,n}.