A counterexample to a conjecture by De Giorgi in large dimensions

Manuel del Pino; Michał Kowalczyk; Juncheng Wei

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

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

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

@article{CRMATH_2008__346_23-24_1261_0,
     author = {Manuel del Pino and Micha{\l} Kowalczyk and Juncheng Wei},
     title = {A counterexample to a conjecture by {De} {Giorgi} in large dimensions},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1261--1266},
     publisher = {Elsevier},
     volume = {346},
     number = {23-24},
     year = {2008},
     doi = {10.1016/j.crma.2008.10.010},
     language = {en},
}

TY  - JOUR
AU  - Manuel del Pino
AU  - Michał Kowalczyk
AU  - Juncheng Wei
TI  - A counterexample to a conjecture by De Giorgi in large dimensions
JO  - Comptes Rendus. Mathématique
PY  - 2008
SP  - 1261
EP  - 1266
VL  - 346
IS  - 23-24
PB  - Elsevier
DO  - 10.1016/j.crma.2008.10.010
LA  - en
ID  - CRMATH_2008__346_23-24_1261_0
ER  -

%0 Journal Article
%A Manuel del Pino
%A Michał Kowalczyk
%A Juncheng Wei
%T A counterexample to a conjecture by De Giorgi in large dimensions
%J Comptes Rendus. Mathématique
%D 2008
%P 1261-1266
%V 346
%N 23-24
%I Elsevier
%R 10.1016/j.crma.2008.10.010
%G en
%F CRMATH_2008__346_23-24_1261_0

Manuel del Pino; Michał Kowalczyk; Juncheng Wei. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.10.010/

Bibliographie
Cité par

[1] G. Alberti; L. Ambrosio; X. Cabré On a long-standing conjecture of E. De Giorgi: symmetry in 3D for general nonlinearities and a local minimality property, Acta Appl. Math., Volume 65 (2001) no. 1–3, pp. 9-33 (Special issue dedicated to Antonio Avantaggiati on the occasion of his 70th birthday)

[2] L. Ambrosio; X. Cabré Entire solutions of semilinear elliptic equations in $R^{3}$ and a conjecture of De Giorgi, J. Amer. Math. Soc., Volume 13 (2000), pp. 725-739

[3] M.T. Barlow; R.F. Bass; C. Gui The Liouville property and a conjecture of De Giorgi, Comm. Pure Appl. Math., Volume 53 (2000) no. 8, pp. 1007-1038

[4] H. Berestycki; F. Hamel; R. Monneau One-dimensional symmetry of bounded entire solutions of some elliptic equations, Duke Math. J., Volume 103 (2000), pp. 375-396

[5] E. Bombieri; E. De Giorgi; E. Giusti Minimal cones and the Bernstein problem, Invent. Math., Volume 7 (1969), pp. 243-268

[6] X. Cabré, J. Terra, Saddle-shaped solutions of bistable diffusion equations in all of $R^{2 m}$ , Preprint, 2008

[7] L.A. Caffarelli; A. Cordoba Phase transitions: uniform regularity of the intermediate layers, J. Reine Angew. Math., Volume 593 (2006), pp. 209-235

[8] E. De Giorgi Convergence problems for functionals and operators, Rome, 1978, Pitagora, Bologna (1979), pp. 131-188

[9] M. del Pino; M. Kowalczyk; J. Wei Concentration on curves for nonlinear Schrödinger equations, Comm. Pure Appl. Math., Volume 60 (2007) no. 1, pp. 113-146

[10] M. del Pino; M. Kowalczyk; J. Wei The Toda system and clustering interfaces in the Allen–Cahn equation, Arch. Rational Mech. Anal., Volume 190 (2008), pp. 141-187

[11] M. del Pino, M. Kowalczyk, F. Pacard, J. Wei, The Toda system and multiple-end solutions of autonomous planar elliptic problems, Preprint, 2007

[12] M. del Pino, M. Kowalczyk, J. Wei, On De Giorgi conjecture in dimension $N ⩾ 9$ , Preprint, 2008

[13] A. Farina Symmetry for solutions of semilinear elliptic equations in $R^{N}$ and related conjectures, Ricerche Mat., Volume 48 (1999) no. suppl., pp. 129-154

[14] N. Ghoussoub; C. Gui On a conjecture of De Giorgi and some related problems, Math. Ann., Volume 311 (1998), pp. 481-491

[15] D. Jerison; R. Monneau Towards a counter-example to a conjecture of De Giorgi in high dimensions, Ann. Mat. Pura Appl., Volume 183 (2004), pp. 439-467

[16] R.V. Kohn; P. Sternberg Local minimizers and singular perturbations, Proc. Roy. Soc. Edinburgh Sect. A, Volume 11 (1989), pp. 69-84

[17] L. Modica Convergence to minimal surfaces problem and global solutions of $Δ u = 2 (u^{3} - u)$ , Rome, 1978, Pitagora, Bologna (1979), pp. 223-244

[18] L. Modica The gradient theory of phase transitions and the minimal interface criterion, Arch. Rational Mech. Anal., Volume 98 (1987) no. 2, pp. 123-142

[19] L. Modica; S. Mortola Un esempio di Γ-convergenza, Boll. Unione Mat. Ital. Sez. B, Volume 14 (1977), pp. 285-299

[20] F. Pacard; M. Ritoré From the constant mean curvature hypersurfaces to the gradient theory of phase transitions, J. Differential Geom., Volume 64 (2003) no. 3, pp. 356-423

[21] O. Savin, Regularity of flat level sets in phase transitions, Ann. of Math., in press

[22] L. Simon Entire solutions of the minimal surface equation, J. Differential Geometry, Volume 30 (1989), pp. 643-688

[23] J. Simons Minimal varieties in riemannian manifolds, Ann. of Math. (2), Volume 88 (1968), pp. 62-105

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Nonlinear Schrödinger equations: concentration on weighted geodesics in the semi-classical limit

Manuel del Pino; Michał Kowalczyk; Juncheng Wei

C. R. Math (2005)

Existence et propriétés qualitatives de fronts coniques bistables en dimension 2 d'espace

François Hamel; Régis Monneau; Jean-Michel Roquejoffre

C. R. Math (2004)

High frequency periodic solutions of semilinear equations

Geneviève Allain; Anne Beaulieu

C. R. Math (2007)