Entire solutions in ℝ 2 for a class of Allen-Cahn equations

Francesca Alessio; Piero Montecchiari

ESAIM: Control, Optimisation and Calculus of Variations (2005)

  • Volume: 11, Issue: 4, page 633-672
  • ISSN: 1292-8119

Abstract

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We consider a class of semilinear elliptic equations of the form - ε 2 Δ u ( x , y ) + a ( x ) W ' ( u ( x , y ) ) = 0 , ( x , y ) ∈ ℝ 2 where ε > 0 , a : ℝ → ℝ is a periodic, positive function and W : ℝ → ℝ is modeled on the classical two well Ginzburg-Landau potential W ( s ) = ( s 2 - 1 ) 2 . We look for solutions to (1) which verify the asymptotic conditions u ( x , y ) → ± 1 as x → ± ∞ uniformly with respect to y ∈ ℝ . We show via variational methods that if ε is sufficiently small and a is not constant, then (1) admits infinitely many of such solutions, distinct up to translations, which do not exhibit one dimensional symmetries.

How to cite

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Alessio, Francesca, and Montecchiari, Piero. "Entire solutions in $\mathbb {R}^{2}$ for a class of Allen-Cahn equations." ESAIM: Control, Optimisation and Calculus of Variations 11.4 (2005): 633-672. <http://eudml.org/doc/245111>.

@article{Alessio2005,
abstract = {We consider a class of semilinear elliptic equations of the form\[ -\varepsilon ^\{2\}\Delta u(x,y)+a(x)W^\{\prime \}(u(x,y))=0,\quad (x,y)\in \mathbb \{R\}^\{2\} \]where $\varepsilon &gt;0$, $a:\mathbb \{R\}\rightarrow \mathbb \{R\}$ is a periodic, positive function and $W:\mathbb \{R\}\rightarrow \mathbb \{R\}$ is modeled on the classical two well Ginzburg-Landau potential $W(s)=(s^\{2\}-1)^\{2\}$. We look for solutions to (1) which verify the asymptotic conditions $u(x,y)\rightarrow \pm 1$ as $x\rightarrow \pm \infty $ uniformly with respect to $y\in \mathbb \{R\}$. We show via variational methods that if $\varepsilon $ is sufficiently small and $a$ is not constant, then (1) admits infinitely many of such solutions, distinct up to translations, which do not exhibit one dimensional symmetries.},
author = {Alessio, Francesca, Montecchiari, Piero},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {heteroclinic solutions; elliptic equations; variational methods; variational methods.},
language = {eng},
number = {4},
pages = {633-672},
publisher = {EDP-Sciences},
title = {Entire solutions in $\mathbb \{R\}^\{2\}$ for a class of Allen-Cahn equations},
url = {http://eudml.org/doc/245111},
volume = {11},
year = {2005},
}

TY - JOUR
AU - Alessio, Francesca
AU - Montecchiari, Piero
TI - Entire solutions in $\mathbb {R}^{2}$ for a class of Allen-Cahn equations
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2005
PB - EDP-Sciences
VL - 11
IS - 4
SP - 633
EP - 672
AB - We consider a class of semilinear elliptic equations of the form\[ -\varepsilon ^{2}\Delta u(x,y)+a(x)W^{\prime }(u(x,y))=0,\quad (x,y)\in \mathbb {R}^{2} \]where $\varepsilon &gt;0$, $a:\mathbb {R}\rightarrow \mathbb {R}$ is a periodic, positive function and $W:\mathbb {R}\rightarrow \mathbb {R}$ is modeled on the classical two well Ginzburg-Landau potential $W(s)=(s^{2}-1)^{2}$. We look for solutions to (1) which verify the asymptotic conditions $u(x,y)\rightarrow \pm 1$ as $x\rightarrow \pm \infty $ uniformly with respect to $y\in \mathbb {R}$. We show via variational methods that if $\varepsilon $ is sufficiently small and $a$ is not constant, then (1) admits infinitely many of such solutions, distinct up to translations, which do not exhibit one dimensional symmetries.
LA - eng
KW - heteroclinic solutions; elliptic equations; variational methods; variational methods.
UR - http://eudml.org/doc/245111
ER -

References

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