Partial Differential Equations
A counterexample to a conjecture by De Giorgi in large dimensions
Comptes Rendus. Mathématique, Volume 346 (2008) no. 23-24, pp. 1261-1266.

We consider the Allen–Cahn equation

Δu+u(1u2)=0in RN.
A celebrated conjecture by E. De Giorgi (1978) states that if u is a bounded solution to this problem such that xNu>0, then the level sets {u=λ}, λR, must be hyperplanes at least if N8. We construct a family of solutions which shows that this statement does not hold true for N9.

Nous considérons l'équation d'Allen–Cahn :

Δu+u(1u2)=0dans RN.
Une conjecture célèbre de E. De Giorgi (1978) affirme que si u est une solution bornée de ce problème telle que xNu>0, alors les ensembles de niveau {u=λ}, λR, sont des hyperplans au moins si N8. Nous contruisons une famille de solutions qui montre que cette conjecture n'est pas vraie pour N9.

Accepted:
Published online:
DOI: 10.1016/j.crma.2008.10.010
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
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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, Volume 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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