Kristály, A. (2024) Calculus of Variations and Partial Differential Equations [Matematică, Q1]

Publicat: 20 August 2024

Kristály, A. (2024) Sharp Sobolev inequalities on noncompact Riemannian manifolds with Ric >= 0 via optimal transport theory. Calculus of Variations and Partial Differential Equations, 63, 200.

DOI: https://doi.org/10.1007/s00526-024-02810-9

✓ Publisher: Springer
✓ Categories: Mathematics, Applied; Mathematics
✓ Article Influence Score (AIS): 1.736 (2023) / Q1 in all categories.

Abstract: In their seminal work, Cordero-Erausquin, Nazaret and Villani (Adv Math 182(2):307-332, 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 21:603-645, 2017), even on metric measure spaces verifying the synthetic lower Ricci curvature bound. In the present paper 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. As a byproduct, we give an alternative, elementary proof to the main result of do Carmo and Xia (Math 140:818-826, 2004) and subsequent results, concerning the quantitative volume non-collapsing estimates on Riemannian manifolds with non-negative Ricci curvature that support Sobolev inequalities.

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