A mass-transportation approach to sharp Sobolev and Gagliardo–Nirenberg inequalities

Cordero–Erausquin, Dario; Nazaret, Bruno; Villani, Cédric

doi:10.1016/s0001-8708(03)00080-x

Cited by 293 publications

(365 citation statements)

References 20 publications

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“…Since then, the Brenier map has found many interesting geometric applications. For instance, it has been used to derive Aleksandrov-Fenchel inequalities by Alesker, Dar, and Milman [1], and to derive sharp Sobolev and related inequalities on R n [15] and on bounded domains [26] by Cordero-Nazaret-Villani [15] and Maggi-Villani [26]. One can consult Villani's book [43] for more details and background on optimal mass transportation theory.…”

Section: Theorem 1 (Prékopa-leindler Inequality)mentioning

confidence: 99%

“…To elucidate the connection of displacement semiconvexity with logarithmic Sobolev and transportation inequalities, let us rewrite the absolute entropy (15) . If µ is a probability measure, then E(u) = Ent µ (f ) is nothing but the relative entropy (25), with f = u/µ the Radon-Nikodym derivative of u with respect to the reference measure µ.…”

Section: The Displacement Convexity Point Of Viewmentioning

confidence: 99%

See 1 more Smart Citation

Prékopa–Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport

Cordero–Erausquin¹,

McCann

Schmuckenschläger

2006

Ann. Fac. Sci. Toulouse Math

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We investigate Prékopa-Leindler type inequalities on a Riemannian manifold M equipped with a measure with density e −V where the potential V and the Ricci curvature satisfy Hess x V + Ric x ≥ λ I for all x ∈ M , with some λ ∈ R. As in our earlier work [14], the argument uses optimal mass transport on M , but here, with a special emphasis on its connection with Jacobi fields. A key role will be played by the differential equation satisfied by the determinant of a matrix of Jacobi fields. We also present applications of the method to logarithmic Sobolev inequalities (the Bakry-Emery criterion will be recovered) and to transport inequalities. A study of the displacement convexity of the entropy functional completes the exposition. RésuméNousétudions l'extension d'inégalités de type Prékopa-Leindler au cas d'une variété riemannienne Méquipée d'une mesure ayant une densité e −V où le potentiel V et la courbure de Ricci vérifient Hess x V + Ric x ≥ λ I (∀x ∈ M ), pour un certain λ ∈ R. Nous ferons appel, comme dans notre travail précédent [14], au transport optimal de mesure. Mais nous exploiterons plus encore son lien avec les champs de Jacobi, ce qui permettra de ramener la discussionà l'étude du déterminant d'une matrice de champs de Jacobi. Nous présentonségalement d'autres applications de la méthode, en particulier aux inégalités de Sobolev logarithmiques (critère de BakryEmery) età l'étude de la convexité de déplacement de la fonctionnelle entropie.

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Section: Theorem 1 (Prékopa-leindler Inequality)mentioning

confidence: 99%

Section: The Displacement Convexity Point Of Viewmentioning

confidence: 99%

Prékopa–Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport

Cordero–Erausquin¹,

McCann

Schmuckenschläger

2006

Ann. Fac. Sci. Toulouse Math

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“…The first problem has become classical in the calculus of variations; it has deep connections to analysis [30,51,73,74,98], geometry [29,63,70,71,79,83,95,104], dynamics [7,10,59,82] and nonlinear partial differential equations [18,20,21,36,42,72,102], as well as applications in physics [38,75,97], statistics [88], engineering [15,16,57,86,107], atmospheric modeling [33][34][35]87], and economics [24,25,27,41,49]. The second is a problem in functional analysis, at the junction between measure theory and convex geometry.…”

Section: Introductionmentioning

confidence: 99%

Optimal transportation, topology and uniqueness

2011

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The Monge-Kantorovich transportation problem involves optimizing with respect to a given a cost function. Uniqueness is a fundamental open question about which little is known when the cost function is smooth and the landscapes Communicated by N. Trudinger. Formerly titled Extremal doubly stochastic measures and optimal transportation.It is a pleasure to thank Nassif Ghoussoub and Herbert Kellerer, who provided early encouragement in this direction, and Pierre-Andre Chiappori, Ivar Ekeland, and Lars Nesheim, whose interest in economic applications fortified our resolve to persist. We thank Wilfrid Gangbo, Jonathan Korman, and Robert Pego for fruitful discussions, Nathan Killoran for useful references, and programs of the Banff International Research Station (2003) containing the goods to be transported possess (non-trivial) topology. This question turns out to be closely linked to a delicate problem (# 111) of Birkhoff (Lattice Theory. Revised Edition, 1948): give a necessary and sufficient condition on the support of a joint probability to guarantee extremality among all measures which share its marginals. Fifty years of progress on Birkhoff's question culminate in Hestir and Williams' necessary condition which is nearly sufficient for extremality; we relax their subtle measurability hypotheses separating necessity from sufficiency slightly, yet demonstrate by example that to be sufficient certainly requires some measurability. Their condition amounts to the vanishing of the measure γ outside a countable alternating sequence of graphs and antigraphs in which no two graphs (or two antigraphs) have domains that overlap, and where the domain of each graph/antigraph in the sequence contains the range of the succeeding antigraph (respectively, graph). Such sequences are called numbered limb systems. We then explain how this characterization can be used to resolve the uniqueness of Kantorovich solutions for optimal transportation on a manifold with the topology of the sphere.

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“…The Euclidean Gagliardo-Nirenberg /Nash's inequality has been studied intensively and been applied in analysis and partial differential equations. See, for example, Nirenberg [39], Gagliardo [18], Cordero-Erausquin, Nazaret and Villani [11], Del Pino and Dolbeault [12]- [15] , Del Pino, Dolbeault and Gentil [16], Carlen and Loss [6], Agueh [1]- [3]. Inequalities (1.4) and (1.5) were also strengthened by the affine Moser-Trudinger inequality and affine Morrey sobolev inequality (see Cinachi, Lutwak, Yang and Zhang [10]), respectively.…”

Section: Introductionmentioning

confidence: 99%

Note on Affine Gagliardo–Nirenberg Inequalities

Zhai

2010

Potential Anal

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Abstract. This note proves sharp affine Gagliardo-Nirenberg inequalities which are stronger than all known sharp Euclidean Gagliardo-Nirenberg inequalities and imply the affine L p −Sobolev inequalities. The logarithmic version of affine L p −Sobolev inequalities is verified. Moreover, An alternative proof of the affine Moser-Trudinger and Morrey-Sobolev inequalities is given. The main tools are the equimeasurability of rearrangements and the strengthened version of the classical Pólys-Szegö principle.

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A mass-transportation approach to sharp Sobolev and Gagliardo–Nirenberg inequalities

Cited by 293 publications

References 20 publications

Prékopa–Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport

Prékopa–Leindler type inequalities on Riemannian manifolds, Jacobi fields, and optimal transport

Optimal transportation, topology and uniqueness

Note on Affine Gagliardo–Nirenberg Inequalities

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