Asymmetric heteroclinic double layers

Schatzman, Michelle. "Asymmetric heteroclinic double layers." ESAIM: Control, Optimisation and Calculus of Variations 8 (2002): 965-1005. <http://eudml.org/doc/245851>.

@article{Schatzman2002,
abstract = {Let $W$ be a non-negative function of class $\mathrm \{C\}^\{3\}$ from $\mathbb \{R\}^2$ to $\mathbb \{R\}$, which vanishes exactly at two points $\mathbf \{a\}$ and $\mathbf \{b\}$. Let $S^1(\mathbf \{a\},\mathbf \{b\})$ be the set of functions of a real variable which tend to $\mathbf \{a\}$ at $-\infty $ and to $\mathbf \{b\}$ at $+\infty $ and whose one dimensional energy\[ E\_1(v)=\int \_\mathbb \{R\}\bigl [W(v)+\vert v^\{\prime \}\vert ^2/2\bigr ]\,\mathrm \{d\}x \]is finite. Assume that there exist two isolated minimizers $z_+$ and $z_-$ of the energy $E_1$ over $S^1(\mathbf \{a\},\mathbf \{b\})$. 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 $\mathbb \{R\}^2$ to itself which satisfies the equation\[ -\Delta u + \mathrm \{D\}W(u)^\sf T=0, \]and the boundary conditions\[ \lim \_\{x\_2\rightarrow +\infty \} u(x\_1,x\_2)=z\_+(x\_1-m\_+),\phantom\{\mathbf \{a\}\} \lim \_\{x\_2\rightarrow -\infty \} u(x\_1,x\_2)=z\_-(x\_1-m\_-), \lim \_\{x\_1\rightarrow -\infty \}u(x\_1,x\_2)=\mathbf \{a\},\phantom\{z\_+(x\_1-m\_+)\} \lim \_\{x\_1\rightarrow +\infty \}u(x\_1,x\_2)=\mathbf \{b\}. \]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; Ginzburg-Landau; non-convex optimization},
language = {eng},
pages = {965-1005},
publisher = {EDP-Sciences},
title = {Asymmetric heteroclinic double layers},
url = {http://eudml.org/doc/245851},
volume = {8},
year = {2002},
}

TY - JOUR
AU - Schatzman, Michelle
TI - Asymmetric heteroclinic double layers
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2002
PB - EDP-Sciences
VL - 8
SP - 965
EP - 1005
AB - Let $W$ be a non-negative function of class $\mathrm {C}^{3}$ from $\mathbb {R}^2$ to $\mathbb {R}$, which vanishes exactly at two points $\mathbf {a}$ and $\mathbf {b}$. Let $S^1(\mathbf {a},\mathbf {b})$ be the set of functions of a real variable which tend to $\mathbf {a}$ at $-\infty $ and to $\mathbf {b}$ at $+\infty $ and whose one dimensional energy\[ E_1(v)=\int _\mathbb {R}\bigl [W(v)+\vert v^{\prime }\vert ^2/2\bigr ]\,\mathrm {d}x \]is finite. Assume that there exist two isolated minimizers $z_+$ and $z_-$ of the energy $E_1$ over $S^1(\mathbf {a},\mathbf {b})$. 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 $\mathbb {R}^2$ to itself which satisfies the equation\[ -\Delta u + \mathrm {D}W(u)^\sf T=0, \]and the boundary conditions\[ \lim _{x_2\rightarrow +\infty } u(x_1,x_2)=z_+(x_1-m_+),\phantom{\mathbf {a}} \lim _{x_2\rightarrow -\infty } u(x_1,x_2)=z_-(x_1-m_-), \lim _{x_1\rightarrow -\infty }u(x_1,x_2)=\mathbf {a},\phantom{z_+(x_1-m_+)} \lim _{x_1\rightarrow +\infty }u(x_1,x_2)=\mathbf {b}. \]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; Ginzburg-Landau; non-convex optimization
UR - http://eudml.org/doc/245851
ER -