Optimal and Semi group Estimates Equivalent to Logarithmic Sobolev Inequalities

Abstract

We show the exponential integrability and transportation cost related to logarithmic Sobolev inequalities .We investigate the continuous rearrangement and symmetry of solutions of elliptic problems . The symmetry of solutions of p-Laplace equations in the Euclidean space is considered . We study the generalization of an inequality by Talagrand and Links' with the logarithmic Sobolev inequality . We also show the optimal Euclidean Sobolev logarithmic inequality and the Hardy type integral inequality associated with the generalized translation We give the log- Sobolev inequalities and regions with exterior exponential cusps and spectral gabs .