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

Francesca Alessio; Piero Montecchiari

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • 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 15.7cm - 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 ([see full textsee full text]) 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 ([see full textsee full text]) 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 (2010): 633-672. <http://eudml.org/doc/90781>.

@article{Alessio2010,
abstract = { We consider a class of semilinear elliptic equations of the form 15.7cm -$\varepsilon^\{2\}\Delta u(x,y)+a(x)W'(u(x,y))=0,\quad (x,y)\in\{\mathbb\{R\}\}^\{2\}$ where $\varepsilon>0$, $a:\{\mathbb\{R\}\}\to\{\mathbb\{R\}\}$ is a periodic, positive function and $W:\{\mathbb\{R\}\}\to\{\mathbb\{R\}\}$ is modeled on the classical two well Ginzburg-Landau potential $W(s)=(s^\{2\}-1)^\{2\}$. We look for solutions to ([see full textsee full text]) which verify the asymptotic conditions $u(x,y)\to\pm 1$ as $x\to\pm\infty$ uniformly with respect to $y\in\{\mathbb\{R\}\}$. We show via variational methods that if ε is sufficiently small and a is not constant, then ([see full textsee full text]) 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.; heteroclinic solutions; variational methods.},
language = {eng},
month = {3},
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/90781},
volume = {11},
year = {2010},
}

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
DA - 2010/3//
PB - EDP Sciences
VL - 11
IS - 4
SP - 633
EP - 672
AB - We consider a class of semilinear elliptic equations of the form 15.7cm -$\varepsilon^{2}\Delta u(x,y)+a(x)W'(u(x,y))=0,\quad (x,y)\in{\mathbb{R}}^{2}$ where $\varepsilon>0$, $a:{\mathbb{R}}\to{\mathbb{R}}$ is a periodic, positive function and $W:{\mathbb{R}}\to{\mathbb{R}}$ is modeled on the classical two well Ginzburg-Landau potential $W(s)=(s^{2}-1)^{2}$. We look for solutions to ([see full textsee full text]) which verify the asymptotic conditions $u(x,y)\to\pm 1$ as $x\to\pm\infty$ uniformly with respect to $y\in{\mathbb{R}}$. We show via variational methods that if ε is sufficiently small and a is not constant, then ([see full textsee full text]) 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.; heteroclinic solutions; variational methods.
UR - http://eudml.org/doc/90781
ER -

References

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