Metric methods for heteroclinic connections

Abstract

We consider the problem min∫double-struckR12|trueγ̇|2+W(γ)normaldt among curves connecting two given wells of W≥0, and we reduce it, following a standard method, to a geodesic problem of the form min∫01K(γ)|trueγ̇|normaldt with K=2W. We then prove existence of curves minimizing this new action just by proving that the distance induced by K is proper (i.e., its closed balls are compact). The assumptions on W are minimal, and the method seems robust enough to be applied in the future to some PDE problems. Copyright © 2016 John Wiley & Sons, Ltd.

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

“…Condition (1.1) was first introduced in [7]. A sufficient condition for (1.1) is that lim inf |x|→∞ U (x) > 0.…”

“…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.…”

“…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.…”

On the existence of heteroclinic connections

“…Remark 1. The assumptions on W imply (see for example [14], [22] and [12]) the existence of a Lipschitz continuous map u H : R → R m that satisfies lim t→±∞ u(t) = a ± ,…”

Prescribed energy connecting orbits for gradient systems

“…The issue of existence of heteroclinic solutions connecting the equilibria of multi-well potentials has been quite explored in the literature; the interested reader is referred to [8], [11], [19], [23], [28], and [26,31] for different approaches on the subject.…”