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

Research output: Contribution to journal › Article › peer-review

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.

Original language	English
Pages (from-to)	1131-1167
Number of pages	37
Journal	Journal of Functional Analysis
Volume	264
Issue number	5
DOIs	https://doi.org/10.1016/j.jfa.2012.03.010
Publication status	Published - 1 Mar 2013

Keywords

Allen-Cahn equation
De Giorgi conjecture
Minimal cones
Minimal hypersurfaces
Stable solutions

Access to Document

10.1016/j.jfa.2012.03.010

Cite this

@article{d60008b0b268485bad9d7b1592b7a42a,

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",

month = mar,

day = "1",

doi = "10.1016/j.jfa.2012.03.010",

language = "English",

volume = "264",

pages = "1131--1167",

journal = "Journal of Functional Analysis",

issn = "0022-1236",

number = "5",

}

TY - JOUR

T1 - Stable solutions of the Allen-Cahn equation in dimension 8 and minimal cones

AU - Pacard, Frank

AU - Wei, Juncheng

PY - 2013/3/1

Y1 - 2013/3/1

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

AN - SCOPUS:84872598703

SN - 0022-1236

VL - 264

SP - 1131

EP - 1167

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 5

ER -

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

Abstract

Keywords

Access to Document

Other files and links

Fingerprint

Cite this