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2017-08-25 - Colloque/Présentation - communication orale - Anglais - 22 page(s)

Troestler Christophe , "A computer assisted proof of the symmetry of solutions to a PDE" in Computers in Scientific Discovery 8, Mons, Belgium, 2017

  • Codes CREF : Calcul des variations (DI1128), Mathématiques (DI1100), Analyse fonctionnelle (DI1122), Analyse mathématique (DI1120), Analyse numérique (DI1123), Equations différentielles et aux dérivées partielles (DI1127)
  • Unités de recherche UMONS : Analyse numérique (S835)
  • Instituts UMONS : Institut de Recherche sur les Systèmes Complexes (Complexys)
  • Centres UMONS : Modélisation mathématique et informatique (CREMMI)
Texte intégral :

Abstract(s) :

(Anglais) The Curie symmetry principle, which asserts that the symmetries of the cause must also be contained in the effects, no longer holds in general for non-linear partial differential equations (PDE) with possibly infinitely many solutions. In this context, one has to restrict to “low energy” solutions. Solutions with the lowest energy (ground states) usually — though not always — possess all the symmetries of the problem. For solutions with the lowest energy among sign-changing ones, only part of the symmetries is preserved. In this talk, I will consider the simple problem -Δu = |u|ᵖ⁻²u with zero Dirichlet boundary conditions on a square domain and show how numerical computations help to discover the residual symmetry of the least-energy sign-changing solutions as well as to prove that what the simulation tells is rigorously valid.

Mots-clés :
  • (Anglais) computer assisted proof
  • (Anglais) symmetry
  • (Anglais) PDE