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

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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 (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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