BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Louis Dupaigne (Université Claude Bernard Lyon 1) DTSTART:20210928T170000Z DTEND:20210928T180000Z DTSTAMP:20260423T021015Z UID:OSGA/81 DESCRIPTION:Title: Th e best constant in Sobolev's inequality\, joint work with Ivan Gentil (Lyo n 1) and Simon Zugmeyer (Paris 5)\nby Louis Dupaigne (Université Clau de Bernard Lyon 1) as part of Online Seminar "Geometric Analysis"\n\n\nAbs tract\nDue to its conformal invariance\, the sharp Sobolev inequality take s\nequivalent forms on the three standard model spaces i.e. the Euclidean\ nspace\, the round sphere and the hyperbolic space. By analogy\, we introd uce\nthree weighted manifolds named after Caffarelli\, Kohn and Nirenberg (CKN)\nfor the following reason: the sharp Caffarelli-Kohn-Nirenberg inequ ality in\nthe standard Euclidean space can be reformulated as a (sharp) So bolev\ninequality written on the CKN Euclidean space. It is equivalent to similar\n(but new) Sobolev inequalities on the CKN sphere and the CKN hype rbolic\nspace. In addition\, the Felli-Schneider condition\, that is\, the region of\nparameters for which symmetry breaking occurs in the study of extremals\,\nturns out to have a purely geometric interpretation as an (in tegrated)\ncurvature-dimension condition. To prove these results\, we shal l use Bakry's\ngeneralization of the notion of scalar curvature\, (a weigh ted version of)\nOtto's calculus\, the reformulation of all the inequaliti es (and many more)\nas entropy-entropy production inequalities along appro priate gradient flows\nin Wasserstein space\, and eventually elliptic PDE methods as our best tool\nfor building rigorous and concise proofs.\n LOCATION:https://researchseminars.org/talk/OSGA/81/ END:VEVENT END:VCALENDAR