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Symmetry and symmetry breaking: rigidity and flows in elliptic PDEs

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proc-ICM2018-6.pdf (337.4Kb)
Date
2018
Dewey
Analyse
Sujet
Symmetry; symmetry breaking; interpolation inequalities; Caffarelli-Kohn-Niren-berg inequalities; optimal constants; rigidity results; fast diffusion equation; carré du champ; bifurcation; instability.
DOI
http://dx.doi.org/10.1142/9789813272880_0138
Conference name
International Congress of Mathematics - ICM 2018
Conference date
05-2018
Conference city
Rio de Janeiro
Conference country
Brazil
Book title
International Congress of Mathematicians 2018
Publisher
World Scientific
Publisher city
Singapore
Pages number
13
ISBN
978-981-327-287-3;978-981-327-288-0
URI
https://basepub.dauphine.fr/handle/123456789/20630
Collections
  • CEREMADE : Publications
Metadata
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Author
Dolbeault, Jean
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Esteban, Maria J.
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Loss, Michael
7772 School of Mathematics - Georgia Institute of Technology
Type
Communication / Conférence
Item number of pages
2261-2285
Abstract (EN)
The issue of symmetry and symmetry breaking is fundamental in all areas of science. Symmetry is often assimilated to order and beauty while symmetry breaking is the source of many interesting phenomena such as phase transitions, instabilities, segregation, self-organization, etc. In this contribution we review a series of sharp results of symmetry of nonnegative solutions of nonlinear elliptic differential equation associated with minimization problems on Euclidean spaces or manifolds. Nonnegative solutions of those equations are unique, a property that can also be interpreted as a rigidity result. The method relies on linear and nonlinear flows which reveal deep and robust properties of a large class of variational problems. Local results on linear instability leading to symmetry breaking and the bifurcation of non-symmetric branches of solutions are reinterpreted in a larger, global, variational picture in which our flows characterize directions of descent.

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