Asymmetric heteroclinic double layers
Michelle Schatzman · ESAIM Control Optimisation and Calculus of Variations · 2002
Let W be a non-negative function of class C3 from to , 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 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 to itself which satisfies the equation and the boundary conditions The above convergences are exponentially fast; the numbers m+ and m- are unknowns of the problem.