Let W be a non-negative function of class C3 from $\xR^2$
to
$\xR$
, which vanishes exactly at two points a and b. Let
S1(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
$$
E_1(v)=\int_\xR\bigl[W(v)+\lvert v'\rvert^2/2\bigr]\,\xdif x
$$
is finite.
Assume that there exist two isolated minimizers z+ and z-
of the energy E1
over S1(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 $\xR^2$
to itself which satisfies the equation
$$
-\Delta u + \xDif W(u)^\mathsf{T}=0,
$$
and the boundary conditions
$$
\lim_{x_2\to +\infty} u(x_1,x_2)=z_+(x_1-m_+),\phantom{\mathbf{a}}
\lim_{x_2\to
-\infty} u(x_1,x_2)=z_-(x_1-m_-),
\lim_{x_1\to -\infty}u(x_1,x_2)=\mathbf{a},\phantom{z_+(x_1-m_+)}
\lim_{x_1\to+\infty}u(x_1,x_2)=\mathbf{b}.
$$
The above convergences are exponentially fast; the numbers m+
and m- are unknowns of the problem.