Logarithmic Sobolev Trace Inequalities

TL;DR

This paper establishes a new logarithmic Sobolev trace inequality in Gaussian spaces, analyzes the trace operator in weighted Sobolev spaces, and explores applications to partial differential equations, demonstrating the sharpness of the results.

Contribution

It introduces a novel logarithmic Sobolev trace inequality in Gaussian spaces and examines the trace operator in weighted Sobolev spaces with applications to PDEs.

Findings

Proved a sharp logarithmic Sobolev trace inequality in Gaussian space.

Analyzed the trace operator in weighted Sobolev spaces.

Applied results to partial differential equations.

Abstract

We prove a logarithmic Sobolev trace inequality in a gaussian space and we study the trace operator in the weighted Sobolev space W^{1,p}(\Omega,\gamma) for sufficiently regular domain. We exhibit examples to show the sharpness of the results. Applications to PDE are also considered.