A GENERALIZED DISTRIBUTION INEQUALITY AND APPLICATIONS

Abstract

The distribution inequality was recently proposed by Tran & Nguyen to investigate gradient estimates for solutions of partial differential equations. Specifically, the authors have proposed several sufficient conditions for two mesurable functions to obtain comparisons with norm between those two functions in general Lebesgue spaces. The results are then applied to some classes of p-Laplace type problems. In this paper, we extend this inequality for applying to many other classes of equations. More precisely, the distribution function inequality we propose can be applied to the p(x)-Laplace equation which is known as the typical version of quasi-linear elliptic equations with variable exponents.