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A scenario for symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities

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Date
2012
Link to item file
http://hal.archives-ouvertes.fr/hal-00695542
Dewey
Analyse
Sujet
Freefem++; finite element method; Fixed point theory; Bifurcation theory; Fixed point; self-adaptive mesh; Roothan method; symmetry breaking; radial symmetry; Caffarelli-Kohn-Nirenberg inequality; Schrödinger operator
Journal issue
Journal of Numerical Mathematics
Volume
20
Number
3-4
Publication date
2012
Article pages
233-250
Publisher
De Gruyter
DOI
http://dx.doi.org/10.1515/jnum-2012-0012
URI
https://basepub.dauphine.fr/handle/123456789/9251
Collections
  • CEREMADE : Publications
Metadata
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Author
Esteban, Maria J.
Dolbeault, Jean
Type
Article accepté pour publication ou publié
Abstract (EN)
The purpose of this paper is to explain the phenomenon of symmetry breaking for optimal functions in functional inequalities by the numerical computations of some well chosen solutions of the corresponding Euler-Lagrange equations. For many of those inequalities it was believed that the only source of symmetry breaking would be the instability of the symmetric optimizer in the class of all admissible functions. But recently, it was shown by an indirect argument that for some Caffarelli-Kohn-Nirenberg inequalities this conjecture was not true. In order to understand this new symmetry breaking mechanism we have computed the branch of minimal solutions for a simple problem. A reparametrization of this branch allows us to build a scenario for the new phenomenon of symmetry breaking. The computations have been performed using Freefem++.

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