A counterexample to a conjecture by De Giorgi in large dimensions

del Pino, Manuel; Kowalczyk, Michał; Wei, Juncheng

doi:10.1016/j.crma.2008.10.010

Partial Differential Equations

A counterexample to a conjecture by De Giorgi in large dimensions
[Un contre-exemple à la conjecture de De Giorgi en grandes dimensions]

Présenté par : Haïm Brezis

del Pino, Manuel ¹ ; Kowalczyk, Michał ^2,³ ; Wei, Juncheng ⁴

¹ Departamento de Ingeniería Matemática and CMM, Universidad de Chile, Casilla 170, Correo 3, Santiago, Chile
² Kent State University, Department of Mathematical Sciences, Kent, OH 44242, USA
³ Departamento de Ingeniería Matemática, Universidad de Chile, Casilla 170, Correo 3, Santiago, Chile
⁴ Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong

Comptes Rendus. Mathématique, Tome 346 (2008) no. 23-24, pp. 1261-1266.

Résumé
Abstract

Nous considérons l'équation d'Allen–Cahn :

Δ u + u (1 - u^{2}) = 0 dans R^{N} .

Une conjecture célèbre de E. De Giorgi (1978) affirme que si u est une solution bornée de ce problème telle que

\partial_{x_{N}} u > 0

, alors les ensembles de niveau

{u = λ}

λ \in R

, sont des hyperplans au moins si

N ⩽ 8

. Nous contruisons une famille de solutions qui montre que cette conjecture n'est pas vraie pour

N ⩾ 9

We consider the Allen–Cahn equation

Δ u + u (1 - u^{2}) = 0 in R^{N} .

A celebrated conjecture by E. De Giorgi (1978) states that if u is a bounded solution to this problem such that

\partial_{x_{N}} u > 0

, then the level sets

{u = λ}

λ \in R

, must be hyperplanes at least if

N ⩽ 8

. We construct a family of solutions which shows that this statement does not hold true for

N ⩾ 9

Accepté le : 2008-10-12
Publié le : 2008-11-13

DOI : 10.1016/j.crma.2008.10.010

Affiliations des auteurs :

del Pino, Manuel ¹ ; Kowalczyk, Michał ^{2, 3} ; Wei, Juncheng ⁴

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     title = {A counterexample to a conjecture by {De} {Giorgi} in large dimensions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1261--1266},
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del Pino, Manuel; Kowalczyk, Michał; Wei, Juncheng. A counterexample to a conjecture by De Giorgi in large dimensions. Comptes Rendus. Mathématique, Tome 346 (2008) no. 23-24, pp. 1261-1266. doi : 10.1016/j.crma.2008.10.010. http://www.numdam.org/articles/10.1016/j.crma.2008.10.010/

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