*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