Symmetry breaking and weighted Euclidean logarithmic Sobolev inequalities

TL;DR

This paper establishes weighted logarithmic Sobolev inequalities on Euclidean space, characterizes symmetry and symmetry breaking ranges, and applies advanced mathematical methods to improve understanding of related inequalities and their implications for weighted diffusion flows.

Contribution

It introduces new weighted logarithmic Sobolev inequalities, characterizes symmetry properties, and extends the carré du champ method to nonlinear elliptic equations and entropy estimates.

Findings

Identified symmetry and symmetry breaking ranges for optimal functions.

Applied the carré du champ method to nonlinear elliptic equations and entropy estimates.

Improved understanding of weighted diffusion flows and related inequalities.

Abstract

On the Euclidean space, we establish some Weighted Logarithmic Sobolev (WLS) inequalities. We characterize a symmetry range in which optimal functions are radially symmetric, and a symmetry breaking range. (WLS) inequalities are a limit case for a family of subcritical Caffarelli-Kohn-Nirenberg (CKN) inequalities with similar symmetry properties. A generalized carr\'e du champ method applies not only to the optimal solution of the nonlinear elliptic Euler-Lagrange equation and proves a rigidity result as for (CKN) inequalities, but also to entropy type estimates, with the full strength of the carr\'e du champ method in a parabolic setting. This is a significant improvement on known results for (CKN). Finally, we briefly sketch some consequences of our results for the weighted diffusion flow.