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

created by eleuteri on 15 Oct 2009
modified on 02 Dec 2013

[BibTeX]

Published Paper

Inserted: 15 oct 2009
Last Updated: 2 dec 2013

Journal: Math. Nachr.
Volume: 284
Number: 11-12
Pages: 1404-1434
Year: 2011

Abstract:

We prove $C^{0,\alpha}$ regularity for local minimizers $u$ of functionals with $p(x)$-growth of the type $$F(w,\Omega) := \int\Omega f(x,w(x),Dw(x))\, dx,$$ in the class $K := \{ w \in W^{1,p(\cdot)}(\Omega;\R): w \ge \psi\}$, where the exponent function $p:\Omega \to (1,\infty)$ is assumed to be continuous with a modulus of continuity satisfying $$\limsup{\rho \to 0} \omega(\rho)\log \left(1\rho}\right) < +\infty,$$ and $1 < \gamma_1 \le p(x) \le \gamma_2 < + \infty$. Moreover, $\psi \in W^{1,1}_{\loc}(\Omega)$ is a given obstacle function, whose gradient $D\psi$ belongs to a Morrey space $L_{\loc}^{q,\lambda}(\Omega)$ with $n - \gamma_1 < \lambda < n$ and $q > \gamma_2$. We do not assume any quantitative continuity of the integrand function $f$.