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Reverse Hardy-Littlewood-Sobolev inequalities

Carrillo, José A. and Delgadino, Matías G. and Dolbeault, Jean and Frank, Rupert L. and Hoffmann, Franca (2019) Reverse Hardy-Littlewood-Sobolev inequalities. Journal de Mathématiques Pures et Appliquées, 132 . pp. 133-165. ISSN 0021-7824.

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This paper is devoted to a new family of reverse Hardy–Littlewood–Sobolev inequalities which involve a power law kernel with positive exponent. We investigate the range of the admissible parameters and the properties of the optimal functions. A striking open question is the possibility of concentration which is analyzed and related with free energy functionals and nonlinear diffusion equations involving mean field drifts.

Item Type:Article
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URLURL TypeDescription Paper
Frank, Rupert L.0000-0001-7973-4688
Additional Information:Crown Copyright © 2019 Published by Elsevier Masson SAS. Received 12 July 2018, Available online 10 September 2019. This research has been partially supported by the projects EFI, contract ANR-17-CE40-0030 (J.D.) and Kibord, contract ANR-13-BS01-0004 (J.D., F.H.) of the French National Research Agency (ANR), and by the U.S. National Science Foundation through grant DMS-1363432 (R.L.F.). The research stay of F.H. in Paris in December 2017 was partially supported by the Simons Foundation and by Mathematisches Forschungsinstitut Oberwolfach. Some of the preliminary investigations were done at the Institute Mittag-Leffler during the fall program Interactions between Partial Differential Equations & Functional Inequalities. The authors thank J.A. Carrillo for preliminary discussions which took place there and R.L.F. thanks the University Paris-Dauphine for hospitality in February 2018.
Funding AgencyGrant Number
Agence Nationale pour la Recherche (ANR)ANR-17-CE40-0030
Agence Nationale pour la Recherche (ANR)ANR-13-BS01-0004
Simons FoundationUNSPECIFIED
Mathematisches Forschungsinstitut OberwolfachUNSPECIFIED
Subject Keywords:Reverse Hardy–Littlewood–Sobolev inequalities; Interpolation; Symmetrization; Concentration; Minimizer; Existence of optimal functions; Regularity; Uniqueness; Euler–Lagrange equations; Free energy; Nonlinear diffusion; Mean field equations; Nonlinear springs; Measure valued solutions
Classification Code:2010 Mathematics Subject Classification. Primary: 35A23; Secondary: 26D15, 35K55, 46E35, 49J40
Record Number:CaltechAUTHORS:20180604-111058055
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Official Citation:José A. Carrillo, Matías G. Delgadino, Jean Dolbeault, Rupert L. Frank, Franca Hoffmann, Reverse Hardy–Littlewood–Sobolev inequalities, Journal de Mathématiques Pures et Appliquées, Volume 132, 2019, Pages 133-165, ISSN 0021-7824, (
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:86785
Deposited By: Tony Diaz
Deposited On:04 Jun 2018 18:16
Last Modified:15 Nov 2019 18:49

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