Asymmetric heteroclinic double layers
ESAIM: Control, Optimisation and Calculus of Variations, Volume 8  (2002), p. 965-1005

Let $W$ be a non-negative function of class ${\mathrm{C}}^{3}$ from ${ℝ}^{2}$ to $ℝ$, which vanishes exactly at two points $𝐚$ and $𝐛$. Let ${S}^{1}\left(𝐚,𝐛\right)$ be the set of functions of a real variable which tend to $𝐚$ at $-\infty$ and to $𝐛$ at $+\infty$ and whose one dimensional energy ${E}_{1}\left(v\right)={\int }_{ℝ}\left[W\left(v\right)+|{v}^{\text{'}}{|}^{2}/2\right]\phantom{\rule{0.166667em}{0ex}}\mathrm{d}x$ is finite. Assume that there exist two isolated minimizers ${z}_{+}$ and ${z}_{-}$ of the energy ${E}_{1}$ over ${S}^{1}\left(𝐚,𝐛\right)$. 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 ${ℝ}^{2}$ to itself which satisfies the equation $-\Delta u+\mathrm{D}W{\left(u\right)}^{𝖳}=\mathsf{0},$ and the boundary conditions $\underset{{x}_{2}\to +\infty }{lim}u\left({x}_{1},{x}_{2}\right)={z}_{+}\left({x}_{1}-{m}_{+}\right),\phantom{𝐚}\underset{{x}_{2}\to -\infty }{lim}u\left({x}_{1},{x}_{2}\right)={z}_{-}\left({x}_{1}-{m}_{-}\right),\underset{{x}_{1}\to -\infty }{lim}u\left({x}_{1},{x}_{2}\right)=𝐚,\phantom{{z}_{+}\left({x}_{1}-{m}_{+}\right)}\underset{{x}_{1}\to +\infty }{lim}u\left({x}_{1},{x}_{2}\right)=𝐛.$ The above convergences are exponentially fast; the numbers ${m}_{+}$ and ${m}_{-}$ are unknowns of the problem.

DOI : https://doi.org/10.1051/cocv:2002039
Classification:  35J50,  35J60,  35B40,  35A15,  35Q99
Keywords: heteroclinic connections, Ginzburg-Landau, elliptic systems in unbounded domains, non convex optimization
@article{COCV_2002__8__965_0,
author = {Schatzman, Michelle},
title = {Asymmetric heteroclinic double layers},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
publisher = {EDP-Sciences},
volume = {8},
year = {2002},
pages = {965-1005},
doi = {10.1051/cocv:2002039},
zbl = {1092.35030},
mrnumber = {1932983},
language = {en},
url = {http://www.numdam.org/item/COCV_2002__8__965_0}
}

Schatzman, Michelle. Asymmetric heteroclinic double layers. ESAIM: Control, Optimisation and Calculus of Variations, Volume 8 (2002) , pp. 965-1005. doi : 10.1051/cocv:2002039. http://www.numdam.org/item/COCV_2002__8__965_0/

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