Global gradient estimates for the $p(\cdot)$-Laplacian

We consider Calderón-Zygmund type estimates for the non-homogeneous $p(\cdot)$-Laplacian system $-\text{div}(|D u|^{p(\cdot)-2} Du) = -\text{div}(|G|^{p(\cdot)-2} G),$ where $p$ is a variable exponent. We show that $|G|^{p(\cdot)} \in L^q(\mathbb{R}^n)$ implies $|D u|^{p(\cdot)} \in L^q(\mathbb{R}^n)$ for any $q \geq 1$. We also prove local estimates independent of the size of the domain and introduce new techniques to variable analysis.

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