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

Diening L, Schwarzacher S (2014)
Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal 106: 70-85.

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Journal Article | Published | English

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We consider Calder\'on-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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Diening L, Schwarzacher S. Global gradient estimates for the $p(\cdot)$-Laplacian. Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. 2014;106:70-85.
Diening, L., & Schwarzacher, S. (2014). Global gradient estimates for the $p(\cdot)$-Laplacian. Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal, 106, 70-85. doi:10.1016/j.na.2014.04.006
Diening, L., and Schwarzacher, S. (2014). Global gradient estimates for the $p(\cdot)$-Laplacian. Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal 106, 70-85.
Diening, L., & Schwarzacher, S., 2014. Global gradient estimates for the $p(\cdot)$-Laplacian. Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal, 106, p 70-85.
L. Diening and S. Schwarzacher, “Global gradient estimates for the $p(\cdot)$-Laplacian”, Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal, vol. 106, 2014, pp. 70-85.
Diening, L., Schwarzacher, S.: Global gradient estimates for the $p(\cdot)$-Laplacian. Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. 106, 70-85 (2014).
Diening, Lars, and Schwarzacher, Sebastian. “Global gradient estimates for the $p(\cdot)$-Laplacian”. Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal 106 (2014): 70-85.
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arXiv 1312.5570