Asymmetric heteroclinic double layers

Schatzman, Michelle

ESAIM: Control, Optimisation and Calculus of Variations, Tome 8 (2002), pp. 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 $S^1(𝐚,𝐛)$ 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

$$E_1\left(v\right)={\int }_{ℝ}\left[W\left(v\right)+|v^{\text{'}}{|}^2/2\right]\mathrm{d}x$$

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 ${ℝ}^2$ to itself which satisfies the equation

$$-\Delta u+\mathrm{D}W{\left(u\right)}^{𝖳}=\mathsf{0},$$

and the boundary conditions

$$\underset{x_2\to +\infty }{lim}u(x_1,x_2)=z_{+}(x_1-m_{+}),\phantom{𝐚}\underset{x_2\to -\infty }{lim}u(x_1,x_2)=z_{-}(x_1-m_{-}),\underset{x_1\to -\infty }{lim}u(x_1,x_2)=𝐚,\phantom{z_{+}(x_1-m_{+})}\underset{x_1\to +\infty }{lim}u(x_1,x_2)=𝐛.$$

The above convergences are exponentially fast; the numbers $m_{+}$ and $m_{-}$ are unknowns of the problem.

Classification : 35J50, 35J60, 35B40, 35A15, 35Q99
Mots clés : heteroclinic connections, Ginzburg-Landau, elliptic systems in unbounded domains, non convex optimization

[1] S. Alama, L. Bronsard and C. Gui, Stationary layered solutions in ${ℝ}^2$ 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