Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones

Frank Pacard; Juncheng Wei

doi:10.1016/j.jfa.2012.03.010

Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones

Frank Pacard
, Juncheng Wei

The Chinese University of Hong Kong

Résultats de recherche: Contribution à un journal › Article › Revue par des pairs

Résumé

For all n≥1, we are interested in bounded solutions of the Allen-Cahn equation δu+u-u³=0 which are defined in all Rn+1 and whose zero set is asymptotic to a given minimal cone. In particular, in dimension n+1≥8, we prove the existence of stable solutions of the Allen-Cahn equation whose zero sets are not hyperplanes.

langue originale	Anglais
Pages (de - à)	1131-1167
Nombre de pages	37
journal	Journal of Functional Analysis
Volume	264
Numéro de publication	5
Les DOIs	https://doi.org/10.1016/j.jfa.2012.03.010
état	Publié - 1 mars 2013

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10.1016/j.jfa.2012.03.010

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title = "Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones",

abstract = "For all n≥1, we are interested in bounded solutions of the Allen-Cahn equation δu+u-u3=0 which are defined in all Rn+1 and whose zero set is asymptotic to a given minimal cone. In particular, in dimension n+1≥8, we prove the existence of stable solutions of the Allen-Cahn equation whose zero sets are not hyperplanes.",

keywords = "Allen-Cahn equation, De Giorgi conjecture, Minimal cones, Minimal hypersurfaces, Stable solutions",

author = "Frank Pacard and Juncheng Wei",

year = "2013",

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AU - Wei, Juncheng

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N2 - For all n≥1, we are interested in bounded solutions of the Allen-Cahn equation δu+u-u3=0 which are defined in all Rn+1 and whose zero set is asymptotic to a given minimal cone. In particular, in dimension n+1≥8, we prove the existence of stable solutions of the Allen-Cahn equation whose zero sets are not hyperplanes.

AB - For all n≥1, we are interested in bounded solutions of the Allen-Cahn equation δu+u-u3=0 which are defined in all Rn+1 and whose zero set is asymptotic to a given minimal cone. In particular, in dimension n+1≥8, we prove the existence of stable solutions of the Allen-Cahn equation whose zero sets are not hyperplanes.

KW - Allen-Cahn equation

KW - De Giorgi conjecture

KW - Minimal cones

KW - Minimal hypersurfaces

KW - Stable solutions

UR - https://www.scopus.com/pages/publications/84872598703

U2 - 10.1016/j.jfa.2012.03.010

DO - 10.1016/j.jfa.2012.03.010

M3 - Article

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SP - 1131

EP - 1167

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

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Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones

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