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
Abstract:
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.
\[
D\psi
^{p(\cdot)},
\partial_t\psi
^{\gamma_1'}\in L^q_\text{loc}~~~\Rightarrow~~~
Du
^{p(\cdot)}\in L^q_\text{loc},~~~\text{for any}~q>1,
\]
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