# Calderón-Zygmund type estimates for a class of obstacle problems with p(x) growth

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: J. Math. Anal. Appl.
Volume: 372
Number: (1)
Pages: 140-161
Year: 2010

Abstract:

For minimizers $u \in W^{1,p(x)}(\Omega)$ of quasiconvex integral functionals of the type $${\mathcal F}u := \int\Omega f(x,Du(x))\, dx$$ with $p(x)$ growth in the class ${\mathcal K} := \{ u \in W^{1,p(x)}(\Omega) : u \ge \psi\}$, where $\psi \in W^{1,p(x)}(\Omega)$ is a given obstacle function, we show estimates of Calderón-Zygmund type, i.e. \begin{equation}
D\psi
{p(\cdot)} \in L{q} \Longrightarrow
Du
{p(\cdot)} \in Lq, \end{equation
} for any $q > 1$, provided that the modulus of continuity $\omega$ of the exponent function $p$ satisfies the condition \begin{equation} \omega(\rho)\log(1\rho) \to 0 \quad \mbox{ as } \quad \rho \to 0. \end{equation}