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 C 3 from 2 to , which vanishes exactly at two points 𝐚 and 𝐛. Let S 1 (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 / 2dx is finite. Assume that there exist two isolated minimizers z + and z - of the energy E 1 over S 1 (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 -Δu+DW(u) 𝖳 =0, and the boundary conditions lim x 2 + u(x 1 ,x 2 )=z + (x 1 -m + ),𝐚lim 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.

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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