Boundary-clustered interfaces for the Allen-Cahn equation

Andrea Malchiodi; Wei Ming Ni; Juncheng Wei

doi:10.2140/pjm.2007.229.447

Boundary-clustered interfaces for the Allen-Cahn equation

Andrea Malchiodi
, Wei Ming Ni
, Juncheng Wei

School of Mathematics

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

24 Scopus citations

Abstract

We consider the Allen-Cahn equation where Ω = B₁(0) is the unit ball in ℝⁿ and ε > 0 is a small parameter. We prove the existence SN of a radial solution u_ε having N interfaces {u_ε(r)=0} = {n-ary union}^N_j=1, where 1> r^ε₁ > r^ε₂ >...> r^ε_N are such that 1-r^ε₁ ~_ε log(1/^ε) and r^ε_j-1 - r^ε_j ~ ε log(1/ε) for j = 2,..., N. Moreover, the Morse index of u_ε in H¹_r (Ω_ε) is exactly N.

Original language	English (US)
Pages (from-to)	447-468
Number of pages	22
Journal	Pacific Journal of Mathematics
Volume	229
Issue number	2
DOIs	https://doi.org/10.2140/pjm.2007.229.447
State	Published - Feb 2007

Keywords

Allen-Cahn equation
Boundary clustered interfaces

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10.2140/pjm.2007.229.447

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title = "Boundary-clustered interfaces for the Allen-Cahn equation",

abstract = "We consider the Allen-Cahn equation where Ω = B1(0) is the unit ball in ℝn and ε > 0 is a small parameter. We prove the existence SN of a radial solution uε having N interfaces \{uε(r)=0\} = \{n-ary union\}Nj=1, where 1> rε1 > rε2 >...> rεN are such that 1-rε1 \textasciitilde{}ε log(1/ε) and rεj-1 - rεj \textasciitilde{} ε log(1/ε) for j = 2,..., N. Moreover, the Morse index of uε in H1r (Ωε) is exactly N.",

keywords = "Allen-Cahn equation, Boundary clustered interfaces",

author = "Andrea Malchiodi and Ni, \{Wei Ming\} and Juncheng Wei",

year = "2007",

month = feb,

doi = "10.2140/pjm.2007.229.447",

language = "English (US)",

volume = "229",

pages = "447--468",

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T1 - Boundary-clustered interfaces for the Allen-Cahn equation

AU - Malchiodi, Andrea

AU - Ni, Wei Ming

AU - Wei, Juncheng

PY - 2007/2

Y1 - 2007/2

N2 - We consider the Allen-Cahn equation where Ω = B1(0) is the unit ball in ℝn and ε > 0 is a small parameter. We prove the existence SN of a radial solution uε having N interfaces {uε(r)=0} = {n-ary union}Nj=1, where 1> rε1 > rε2 >...> rεN are such that 1-rε1 ~ε log(1/ε) and rεj-1 - rεj ~ ε log(1/ε) for j = 2,..., N. Moreover, the Morse index of uε in H1r (Ωε) is exactly N.

AB - We consider the Allen-Cahn equation where Ω = B1(0) is the unit ball in ℝn and ε > 0 is a small parameter. We prove the existence SN of a radial solution uε having N interfaces {uε(r)=0} = {n-ary union}Nj=1, where 1> rε1 > rε2 >...> rεN are such that 1-rε1 ~ε log(1/ε) and rεj-1 - rεj ~ ε log(1/ε) for j = 2,..., N. Moreover, the Morse index of uε in H1r (Ωε) is exactly N.

KW - Allen-Cahn equation

KW - Boundary clustered interfaces

UR - https://www.scopus.com/pages/publications/42449155618

UR - https://www.scopus.com/pages/publications/42449155618#tab=citedBy

U2 - 10.2140/pjm.2007.229.447

DO - 10.2140/pjm.2007.229.447

M3 - Article

AN - SCOPUS:42449155618

SN - 0030-8730

VL - 229

SP - 447

EP - 468

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

IS - 2

ER -

Boundary-clustered interfaces for the Allen-Cahn equation

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