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arXiv · 2210.12488

Symmetry breaking and weighted Euclidean logarithmic Sobolev inequalities

Jean Dolbeault·Andres Zuniga
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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.

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Jean Dolbeault, Andres Zuniga. 2022-10-22. Symmetry breaking and weighted Euclidean logarithmic Sobolev inequalities. https://arxiv.org/abs/2210.12488

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