The Heteroclinic Connection Problem for General Double-Well Potentials

Abstract

Abstract. By variational methods, we provide a simple proof of existence of a heteroclinic orbit to the Hamiltonian system u ′′ = ∇W (u) that connects the two global minima of a double-well potential W . Moreover, we consider several inhomogeneous extensions.

On the existence of connecting orbits for critical values of the energy

“…In [1] existence was established under a mild monotonicity condition on U near p ± . This condition was removed in [8], see also [2]. The most general results, equivalent to the consequence of Theorem 1.1 discussed in Section 2.1, were recently obtained in [7] and in [11], see also [3].…”

“…All these papers establish existence by a variational approach. In [1], [8] and [2] by minimizing the action functional, and in [7] and [11] by minimizing the Jacobi functional. Corollary 1.4.…”

Metric methods for heteroclinic connections

“…The existence of a heteroclinic connection is a very delicate problem, because of the lack of compactness of the set H 1 (R) and of the invariance by translations of the action to be minimized. We cite [23,5,9,22,6] among the many papers dealing with this and related question. In a previous paper [18] we analyzed the same question via a purely metric method.…”

Optimal potential energy growth lower bounds for minimizing solutions to the vectorial Allen–Cahn equation in two space dimensions

“…The main difference with is that they used the ‘polar’ representation of u (in analogy to the second branch in ) in the whole domain, even when . This point, however, demanded a rather involved justification, which our argument bypasses (a related observation was made independently in very recently).Remark It is worth mentioning that the passage from to if n = 1 or m = 1 can be shown by a similar ‘local replacement’ argument, which remarkably does not require condition , see and respectively.Remark A careful examination of the proof of Theorem reveals that the assumption of the boundedness of u , as well as that of the finiteness of the number of wells, can be dropped at the expense of imposing some natural uniformity conditions on W and compromising with the softer estimate with n = 2 as the conclusion. More precisely, we further assume that holds with the same r 0 for all i = 1,⋯, and that the set shrinks uniformly to { a 1 , a 2 ,⋯} as ρ →0.…”

“…It is worth mentioning that the passage from (20) to (21) if n D 1 or m D 1 can be shown by a similar 'local replacement' argument, which remarkably does not require condition (13), see [39] and [40] respectively.…”