Asymmetric heteroclinic double layers
ESAIM: Control, Optimisation and Calculus of Variations, Volume 8 (2002), pp. 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: 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},
     pages = {965--1005},
     publisher = {EDP-Sciences},
     volume = {8},
     year = {2002},
     doi = {10.1051/cocv:2002039},
     mrnumber = {1932983},
     zbl = {1092.35030},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2002039/}
}
TY  - JOUR
AU  - Schatzman, Michelle
TI  - Asymmetric heteroclinic double layers
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2002
SP  - 965
EP  - 1005
VL  - 8
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/cocv:2002039/
DO  - 10.1051/cocv:2002039
LA  - en
ID  - COCV_2002__8__965_0
ER  - 
%0 Journal Article
%A Schatzman, Michelle
%T Asymmetric heteroclinic double layers
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2002
%P 965-1005
%V 8
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/cocv:2002039/
%R 10.1051/cocv:2002039
%G en
%F 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/articles/10.1051/cocv:2002039/

[1] S. Alama, L. Bronsard and C. Gui, Stationary layered solutions in 2 for an Allen-Cahn system with multiple well potential. Calc. Var. Partial Diff. Eqs. 5 (1997) 359-390. | Zbl

[2] I. Fonseca and L. Tartar, The gradient theory of phase transitions for systems with two potential wells. Proc. Roy. Soc. Edinburgh Sect. A 111 (1989) 89-102. | MR | Zbl

[3] K. Ishige, The gradient theory of the phase transitions in Cahn-Hilliard fluids with Dirichlet boundary conditions. SIAM J. Math. Anal. 27 (1996) 620-637. | Zbl

[4] T. Kato, Perturbation theory for linear operators. Springer-Verlag, Berlin (1995). Reprint of the 1980 edition. | MR | Zbl

[5] L.D. Landau and E.M. Lifchitz, Physique statistique. Ellipses (1994).

[6] J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications, Vol. 1. Dunod, Paris (1968). Travaux et Recherches Mathématiques, No. 17. | MR | Zbl

[7] L. Modica, The gradient theory of phase transitions and the minimal interface criterion. Arch. Rational Mech. Anal. 98 (1987) 123-142. | MR | Zbl

[8] L. Modica, Gradient theory of phase transitions with boundary contact energy. Ann. Inst. H. Poincaré Anal. Non Linéaire 4 (1987) 487-512. | Numdam | MR | Zbl

[9] N.C. Owen, J. Rubinstein and P. Sternberg, Minimizers and gradient flows for singularly perturbed bi-stable potentials with a Dirichlet condition. Proc. Roy. Soc. London Ser. A 429 (1990) 505-532. | MR | Zbl

[10] M. Reed and B. Simon, Methods of modern mathematical physics. I. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, Second Edition (1980). Functional analysis. | MR | Zbl

[11] P. Sternberg, Vector-valued local minimizers of nonconvex variational problems. Rocky Mountain J. Math. 21 (1991) 799-807. Current directions in nonlinear partial differential equations. Provo, UT (1987). | MR | Zbl

[12] A.I. Volpert, V.A. Volpert and V.A. Volpert, Traveling wave solutions of parabolic systems. American Mathematical Society, Providence, RI, 1994. Translated from the Russian manuscript by James F. Heyda. | MR | Zbl

Cited by Sources: