Reverse Hardy-Littlewood-Sobolev inequalities

Creators: Carrillo, José A.; Delgadino, Matías G.; Dolbeault, Jean; Frank, Rupert L.; Hoffmann, Franca

Abstract

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.

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.

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