| Abstract: |
| In their seminal work, Cordero-Erausquin, Nazaret and Villani [Adv. Math., 2004] proved sharp Sobolev inequalities in Euclidean spaces via Optimal Transport, raising the question whether their approach is powerful enough to produce sharp Sobolev inequalities also on Riemannian manifolds. By using $L^1$-optimal transport approach, the compact case has been successfully treated by Cavalletti and Mondino [Geom. Topol., 2017], even on metric measure spaces verifying the synthetic lower Ricci curvature bound. In the present talk we affirmatively answer the above question for noncompact Riemannian manifolds with non-negative Ricci curvature; namely, by using Optimal Transport theory with quadratic distance cost, sharp $L^p$-Sobolev and $L^p$-logarithmic Sobolev inequalities (both for $p>1$ and $p=1$) are established, where the sharp constants contain the asymptotic volume ratio arising from precise asymptotic properties of the Talentian and Gaussian bubbles, respectively. Talk based on [A].
[A] Krist\`aly Alexandru, Sharp Sobolev inequalities on noncompact Riemannian manifolds with $Ric\geq 0$ via optimal transport theory. Calc. Var. Partial Differential Equations 63 (2024), no. 8, Paper No. 200. |
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