Moduli space theory for the Allen-Cahn equation in the plane

Abstract

In this paper we study entire solutions of the Allen-Cahn equation Δu − F (u) = 0, where F is an even, bistable function. We are particularly interested in the description of the moduli space of solutions which have some special structure at infinity. The solutions we are interested in have their zero set asymptotic to 2k, k ≥ 2 oriented affine half-lines at infinity and, along each of these affine half-lines, the solutions are asymptotic to the one-dimensional heteroclinic solution: such solutions are called multiple-end solutions, and their set is denoted by M 2k. The main result of our paper states that if u ∈ M 2k is nondegenerate, then locally near u the set of solutions is a smooth manifold of dimension 2k. This paper is part of a program whose aim is to classify all 2k-ended solutions of the Allen-Cahn equation in dimension 2, for k ≥ 2.

Finite Morse Index Implies Finite Ends

“…(In fact, more restrictions are needed; see Del Pino, Kowalczyk, and Pacard [21], Gui [40].) Finite Morse index is a spectrum condition that is hard to use to obtain an energy estimate.…”

Solutions of the Allen-Cahn equation which are invariant under screw-motion

“…As far as the existence of solutions is concerned, there are two completely different approaches: the use of the variational structure of (1.1) as in [5] or [1] ; or perturbation results based on the implementation of an infinite dimensional Liapunov-Schmidt reduction argument as in [7], [8], [9] or [10]. The former is usually simpler and takes advantage of the symmetries of the solutions constructed while the latter produces solutions with less (or even without any) symmetry but is technically more involved.…”

Untitled

“…It may also happen that L 0 has just one eigenvalue, µ 0 = 0 and continuous spectrum, in which case we will set µ 1 = α 2 0 . Next, we recall some facts about the moduli space theory developed in [del Pino et al 2013]. We will mostly use this theory in the case of four-end solutions.…”

“…We will now briefly describe the balancing formulas for four-end solutions in the form they were introduced in [del Pino et al 2013]. Assume that u is a solution of (1-1) which is defined in ‫ޒ‬ 2 .…”

“…In order to proceed with the statement of our results, we will define the family of four-end solutions of (1-1), which is a particular example of a more general family of 2k-end solutions [del Pino et al 2013]. Intuitively, a four-end solution u is characterized by the fact that its nodal set N (u) is asymptotic at infinity to four half lines, and along each of these half lines it looks locally like the heteroclinic solution.…”