Jean DOLBEAULT

CEREMADE UMR CNRS n°7534
Office B514 ter
Université Paris Dauphine
Place du Maréchal de Lattre de Tassigny
F-75775 Paris Cedex 16 - France
Tel. (33) 1 44 05 47 68
Fax. (33) 1 44 05 45 99
e-mail: dolbeaul@ceremade.dauphine.fr
web: https://ceremade.dauphine.fr/~dolbeaul

Some links, news, etc.

Stability results for Sobolev's inequality goes back to a celebrated result of Bianchi and Egnell in 1991, but no estimate of the constant was known so far. See this paper on the Sharp stability for Sobolev and log-Sobolev inequalities, with optimal dimensional dependence, for a first answer to this long-standing question. A shorter version is availab le here
Nonlinear (fast diffusion) flows coupled with entropy methods are a remarkable tool not only for proving optimal functional inequalities but also to obtain stability estimates: see these three papers interpolation on the sphere, Gaussian interpolation and logarithmic Sobolev inequalities
A result on stability in Gagliardo-Nirenberg-Sobolev inequalities (to appear as a Memoir of the AMS) which includes the case of Sobolev's inequality with constructive estimates, based on the carré du champ method and new regularity results, with slides here. Also see the One World PDE Seminar (July 7, 2020) in the subcritical case, with slides here
A review on Functional inequalities: nonlinear flows and entropy methods as a tool for obtaining sharp and constructive results can be found here. Also see the slides
In Fourier based L² hypocoercivity methods, how sharp are the decay estimates ? See this paper on the Goldstein-Taylor model and for simple collision operators. Also see this classification of hypocoercivity rates in terms of the underlying functional inequalities
Korn and related inequalities are well known tools for mechanics in bounded domains, but also play a role in kinetic equations. Weighted Korn and Poincaré-Korn inequalities also make sense in the Euclidean space for bounded measures and there are interesting consequences on the convergence to special macroscopic modes for some kinetic equations as well as for rates of convergence
Critical magnetic field for 2d magnetic Dirac-Coulomb operators and Hardy inequalities can be found here. You can also have a look at a new Min-max principle (to appear in J. Spectral Th.) for Dirac type operators with applications to quantum chemistry, and Keller and Lieb-Thirring estimates of the eigenvalues in the gap of Dirac operators (to appear in Rev. Mat. Iberoamericana) which play the role of Sobolev type inequalities for Dirac operators

An introduction to some papers on magnetic interpolation inequalities, magnetic nonlinear Schrödinger equations and related symmetry issues
An introduction to some recent advances on Phase transitions and symmetry in PDEs
A short text of presentation of Symmetry and symmetry breaking issues by nonlinear flow methods
A short text of presentation of Hypocoercivity methods

A summer course on Stability in functional inequalities, (19-24/8/2024)
Bibliographical references for the course on Stability in functional inequalities (CMM-DIM, Santiago, Chili, March-April 2024)
Lecture notes on Entropy methods, functional inequalities and applications (Paris, January 2024)
Lecture notes on Fast diffusion, mean field drifts and reverse HLS inequalities (Cogne, 3-7 June 2019)
Notes de cours sur les Méthodes d'entropie pour les EDP (Tunis, 4-7 Février 2019)
Lecture notes on Symmetry and nonlinear diffusion flows (Prefalc project CFRRMA, Universidad de Chile, Santiago (Chile), October 10-18, 2017)

Beyond mathematics: pictures of my favourite mountains.

Fondation Sciences Mathématiques de Paris
A link to the former project EFI (Entropy, flows, inequalities) ANR-17-CE40-0030 (1/1/2018-31/12/2022)
A link to the new project Conviviality (CONVergence and Interactions VIa Analysis and probabiLITY) ANR-23-CE40-0003 (1/10/2023-30/09/2028)

2023 Thematic period on PDEs: diffusion, geometry, probability and free boundaries, at the Instituto de Ciencias Matemáticas (ICMAT), Madrid, with a workshop on the stability in functional inequalities (October 23-26th, 2023)