Asymmetric heteroclinic double layers

Michelle Schatzman

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

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

Abstract

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Let W be a non-negative function of class C3 from 2 to , 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 ) = W ( v ) + | v ' | 2 / 2 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 2 to itself which satisfies the equation - Δ u + W ( u ) 𝖳 = 0 , and the boundary conditions lim x 2 + u ( x 1 , x 2 ) = z + ( x 1 - m + ) , 𝐚 lim x 2 - u ( x 1 , x 2 ) = z - ( x 1 - m - ) , lim x 1 - u ( x 1 , x 2 ) = 𝐚 , z + ( x 1 - m + ) lim x 1 + u ( x 1 , x 2 ) = 𝐛 . The above convergences are exponentially fast; the numbers m+ and m- are unknowns of the problem.

How to cite

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Schatzman, 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 -

References

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