Anisotropic logarithmic Sobolev inequality with a Gaussian weight and its applications

Abstract

In this article we prove a Logarithmic Sobolev type inequality and a Poincare type inequality for functions in the anisotropic Gaussian Sobolev space. As an application we study a class of equations, whose anisotropic elliptic condition is given in term of the density of Gauss measure. Finally some extensions of the main results are given for a class of weighted (not Gaussian one) anisotropic Sobolev spaces.