End-to-end construction for the Allen–Cahn equation in the plane

Abstract

Abstract. In this paper, we construct a wealth of bounded, entire solutions of the Allen-Cahn equation in the plane. The asymptotic behavior at infinity of these solutions is determined by 2L half affine lines, in the sense that, along each of these half affine lines, the solution is close to a suitable translated and rotated copy of a one dimensional heteroclinic solution. The solutions we construct belong to a smooth 2L-dimensional family of bounded, entire solutions of the Allen-Cahn equation, in agreement with the result of [3] and, in some sense, they provide a description of a collar neighborhood of part of the compactification of the moduli space of 2L-ended solutions for the Allen-Cahn equation. Our construction is inspired by a construction of minimal surfaces by M. Traizet [12].

Finite Morse Index Implies Finite Ends

“…Near each end the solution approaches to the one-dimensional profile exponentially; see Del Pino-Kowalczyk-Pacard [21], Gui [39], and Kowalczyk-Liu-Pacard [48][49][50]. The existence of multiple-ended solutions and infinite-ended solutions to the Allen-Cahn equation in R 2 have been constructed in [2,22,51,52]. The structure and classification of four end solutions have been studied extensively in [21,40,[48][49][50].…”

Singly Periodic Solutions of the Allen-Cahn Equation and the Toda Lattice

“…However, due to the fact that the level set of u consists of curves whose mutual distances at infinity are constant there is no hope for an estimate of the error in a norm that takes into account decay both in the direction orthogonal and parallel to the level set. This makes our present situation different than that of the multiple end solutions considered in [4] or [16]. As we will see later on in order to deal with this difficulty we take advantage of the fact that at infinity our equation reduces essentially to a one dimensional periodic problem.…”

“…The Toda system is a classical example of an integrable system describing the dynamics of n particles moving along a straight line and whose interaction with their closest neighbour is given by the exponential of the negative of their distance. Another construction of 2n-end solutions was given in [16] (see also [1], [3]) . A particular example of 2n solutions is the set of four-end solutions.…”

“…The planar solutions are special examples of the so-called multiple-end solutions, which by definition are solutions whose nodal curves are asymptotic at infinity to finitely many copies of half planar solutions (see [5], [4], [13], [14], [15], [16] where the subject was treated extensively). Any planar solution has an important property of being monotone in one direction.…”

A Harnack inequality for the parabolic Allen–Cahn equation

“…The conjecture was proved for n = 2 by Ghoussoub-Gui [21], for n = 3 by Ambrosio-Cabre [3] and for 4 ≤ n ≤ 8 by Savin, under an extra assumption (see [35,33,34]). The equation has been deeply investigated, see for example: [1,9,10,11,12,13,14,15,18,19,22,23,24,25,27,28], but still offers many interesting questions. Note that De Giorgi's conjecture is false for n ≥ 9 (see, for example, [17]).…”