Symmetry breaking and weighted Euclidean logarithmic Sobolev 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.
Keywords
Cite
@article{arxiv.2210.12488,
title = {Symmetry breaking and weighted Euclidean logarithmic Sobolev inequalities},
author = {Jean Dolbeault and Andres Zuniga},
journal= {arXiv preprint arXiv:2210.12488},
year = {2022}
}