A Hölder continuity result for a class of obstacle problems under non standard growth conditions

✍ Scribed by Michela Eleuteri; Jens Habermann

• Publisher: John Wiley and Sons
• Year: 2011
• Tongue: English
• Weight: 272 KB
• Volume: 284
• Category: Article
• ISSN: 0025-584X
• DOI: 10.1002/mana.201190024

✦ Synopsis

Abstract

We prove C^0, α^ regularity for local minimizers u of functionals with p(x)‐growth of the type

in the class \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$K :=\lbrace w \in W^{1,p(\cdot )}(\Omega ;{\mathbb R}): w \ge \psi \rbrace$\end{document}, where the exponent function p : Ω → (1, ∞) is assumed to be continuous with a modulus of continuity satisfying

and 1 < γ~1~ ⩽ p(x) ≤ γ~2~ < +∞. Moreover, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\psi \in W^{1,1}_{\textnormal {loc}}(\Omega )$\end{document} is a given obstacle function, whose gradient __D__ψ belongs to a Morrey space \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$L_{\textnormal {loc}}^{q,\lambda }(\Omega )$\end{document} with n − γ~1~ < λ < n and q > γ~2~. We do not assume any quantitative continuity of the integrand function f.