Symmetry breaking and weighted Euclidean logarithmic Sobolev inequalities
Dolbeault, Jean; Zuniga, Andres (2022), Symmetry breaking and weighted Euclidean logarithmic Sobolev inequalities. https://basepub.dauphine.psl.eu/handle/123456789/23322
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Type
Document de travail / Working paperDate
2022Series title
Cahier de recherche CEREMADE, Université Paris Dauphine-PSLPublished in
Paris
Pages
20
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Show full item recordAuthor(s)
Dolbeault, Jean
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Zuniga, Andres

CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)
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 carré 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 carré 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.Subjects / Keywords
Logarithmic Sobolev inequality; Hardy-Sobolev inequality; Caffarelli-Kohn-Nirenberg inequality; symmetry breaking; symmetry; concentration-compactness; optimal functions; optimal constant; carré du champ methodRelated items
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