# Asymmetric heteroclinic double layers

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

- Volume: 8, page 965-1005
- ISSN: 1292-8119

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topSchatzman, Michelle. "Asymmetric heteroclinic double layers." ESAIM: Control, Optimisation and Calculus of Variations 8 (2010): 965-1005. <http://eudml.org/doc/90681>.

@article{Schatzman2010,

abstract = {
Let W be a non-negative function of class C3 from $\xR^2$ to
$\xR$, which vanishes exactly at two points a and b. Let
S1(a, b) be the set of functions of a real variable which tend
to a at -∞
and to b at +∞ and whose one dimensional energy
$$
E\_1(v)=\int\_\xR\bigl[W(v)+\lvert v'\rvert^2/2\bigr]\,\xdif x
$$
is finite.
Assume that there exist two isolated minimizers z+ and z-
of the energy E1
over S1(a, b). Under a mild coercivity condition on the
potential W and a generic spectral condition on the linearization
of the
one-dimensional Euler–Lagrange operator at z+ and z-, it is
possible to prove that there exists a function u
from $\xR^2$ to itself which satisfies the equation
$$
-\Delta u + \xDif W(u)^\mathsf\{T\}=0,
$$
and the boundary conditions
$$
\lim\_\{x\_2\to +\infty\} u(x\_1,x\_2)=z\_+(x\_1-m\_+),\phantom\{\mathbf\{a\}\}
\lim\_\{x\_2\to
-\infty\} u(x\_1,x\_2)=z\_-(x\_1-m\_-),
\lim\_\{x\_1\to -\infty\}u(x\_1,x\_2)=\mathbf\{a\},\phantom\{z\_+(x\_1-m\_+)\}
\lim\_\{x\_1\to+\infty\}u(x\_1,x\_2)=\mathbf\{b\}.
$$
The above convergences are exponentially fast; the numbers m+
and m- are unknowns of the problem.
},

author = {Schatzman, Michelle},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Heteroclinic connections; Ginzburg–Landau; elliptic systems
in unbounded domains; non convex optimization.; heteroclinic connections; Ginzburg-Landau; elliptic systems in unbounded domains; non-convex optimization},

language = {eng},

month = {3},

pages = {965-1005},

publisher = {EDP Sciences},

title = {Asymmetric heteroclinic double layers},

url = {http://eudml.org/doc/90681},

volume = {8},

year = {2010},

}

TY - JOUR

AU - Schatzman, Michelle

TI - Asymmetric heteroclinic double layers

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2010/3//

PB - EDP Sciences

VL - 8

SP - 965

EP - 1005

AB -
Let W be a non-negative function of class C3 from $\xR^2$ to
$\xR$, which vanishes exactly at two points a and b. Let
S1(a, b) be the set of functions of a real variable which tend
to a at -∞
and to b at +∞ and whose one dimensional energy
$$
E_1(v)=\int_\xR\bigl[W(v)+\lvert v'\rvert^2/2\bigr]\,\xdif x
$$
is finite.
Assume that there exist two isolated minimizers z+ and z-
of the energy E1
over S1(a, b). Under a mild coercivity condition on the
potential W and a generic spectral condition on the linearization
of the
one-dimensional Euler–Lagrange operator at z+ and z-, it is
possible to prove that there exists a function u
from $\xR^2$ to itself which satisfies the equation
$$
-\Delta u + \xDif W(u)^\mathsf{T}=0,
$$
and the boundary conditions
$$
\lim_{x_2\to +\infty} u(x_1,x_2)=z_+(x_1-m_+),\phantom{\mathbf{a}}
\lim_{x_2\to
-\infty} u(x_1,x_2)=z_-(x_1-m_-),
\lim_{x_1\to -\infty}u(x_1,x_2)=\mathbf{a},\phantom{z_+(x_1-m_+)}
\lim_{x_1\to+\infty}u(x_1,x_2)=\mathbf{b}.
$$
The above convergences are exponentially fast; the numbers m+
and m- are unknowns of the problem.

LA - eng

KW - Heteroclinic connections; Ginzburg–Landau; elliptic systems
in unbounded domains; non convex optimization.; heteroclinic connections; Ginzburg-Landau; elliptic systems in unbounded domains; non-convex optimization

UR - http://eudml.org/doc/90681

ER -

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