Electron. J. Diff. Equ., Vol. 2013 (2013), No. 51, pp. 1-8.

Local estimates for gradients of solutions to elliptic equations with variable exponents

Fengping Yao

Abstract:
In this article we present local $L^\infty$ estimates for the gradient of solutions to elliptic equations with variable exponents. Under proper conditions on the coefficients, we prove that
$$ \left| \nabla u\right|\in L^{\infty}_{loc} $$
for all weak solutions of
$$ \operatorname{div} (g(|\nabla u|^2,x) \nabla u )=0\quad \hbox{in } \Omega. $$

Submitted September 3, 2012. Published February 18, 2013.
Math Subject Classifications: 35J60, 35J70.
Key Words: Regularity; divergence; nonlinear; elliptic equation; gradient; variable exponent; p(x)-Laplacian.

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Fengping Yao
Department of Mathematics, Shanghai University
Shanghai 200444, China
email: yfp@shu.edu.cn

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