Calculus of Variations and Geometric Measure Theory

A. H. Erhardt

Calderón-Zygmund theory for parabolic obstacle problems with nonstandard growth

created by erhardt on 27 Jan 2014
modified on 05 Mar 2014


Published Paper

Inserted: 27 jan 2014
Last Updated: 5 mar 2014

Journal: Adv. Nonlinear Anal.
Volume: 3
Number: 1
Pages: 15-44
Year: 2014
Doi: 10.1515/anona-2013-0024


We establish local Calderón-Zygmund estimates for solutions to certain parabolic problems with irregular obstacles and nonstandard $p(x,t)$-growth. More precisely, we will show that the spatial gradient $Du$ of the solution to the obstacle problem is as integrable as the obstacle $\psi$, i.e. \[
^{\gamma_1'}\in L^q_\text{loc}~~~\Rightarrow~~~
^{p(\cdot)}\in L^q_\text{loc},~~~\text{for any}~q>1, \] where $\gamma_1'=\frac{\gamma_1}{\gamma_1-1}$ and $\gamma_1$ is the lower bound for $p(\cdot)$.

Keywords: Nonstandard growth, Calderón-Zygmund estimates, Variational inequality, nonlinear parabolic obstacle problems, localizable solutions