# Entire solutions in ${\mathbb{R}}^{2}$ 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

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topAlessio, 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 $\varepsilon >0$, $a:\mathbb \{R\}\rightarrow \mathbb \{R\}$ 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 $\varepsilon >0$, $a:\mathbb {R}\rightarrow \mathbb {R}$ 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 -

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