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.

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