The best constant in Sobolev's inequality, joint work with Ivan Gentil (Lyon 1) and Simon Zugmeyer (Paris 5)

Abstract: Due to its conformal invariance, the sharp Sobolev inequality takes equivalent forms on the three standard model spaces i.e. the Euclidean space, the round sphere and the hyperbolic space. By analogy, we introduce three weighted manifolds named after Caffarelli, Kohn and Nirenberg (CKN) for the following reason: the sharp Caffarelli-Kohn-Nirenberg inequality in the standard Euclidean space can be reformulated as a (sharp) Sobolev inequality written on the CKN Euclidean space. It is equivalent to similar (but new) Sobolev inequalities on the CKN sphere and the CKN hyperbolic space. In addition, the Felli-Schneider condition, that is, the region of parameters for which symmetry breaking occurs in the study of extremals, turns out to have a purely geometric interpretation as an (integrated) curvature-dimension condition. To prove these results, we shall use Bakry's generalization of the notion of scalar curvature, (a weighted version of) Otto's calculus, the reformulation of all the inequalities (and many more) as entropy-entropy production inequalities along appropriate gradient flows in Wasserstein space, and eventually elliptic PDE methods as our best tool for building rigorous and concise proofs.

analysis of PDEsdifferential geometryfunctional analysismetric geometry

Audience: researchers in the topic

( slides | video )

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Online Seminar "Geometric Analysis"

Series comments: We discuss recent trends related to geometric analysis in a broad sense. The general idea is to solve geometric problems by means of advanced tools in analysis. We will include a wide range of topics such as geometric flows, curvature functionals, discrete differential geometry, and numerical simulation.

Registration and links to videos available at blatt.sbg.ac.at/onlineseminar.php

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