On the Loss of Compactness in the Vectorial Heteroclinic Connection Problem

TL;DR

This paper presents a new variational proof for the existence of heteroclinic solutions in a Hamiltonian ODE system related to phase transitions, analyzing the loss of compactness in minimising sequences.

Contribution

It introduces a novel variational approach that differs from previous methods, providing new assumptions and a priori estimates for solutions.

Findings

Existence of heteroclinic solutions established

Analysis of loss of compactness in minimising sequences

New assumptions yield a priori estimates

Abstract

We give an alternative proof of the theorem of Alikakos-Fusco [AF] concerning existence of heteroclinic solutions to a Hamiltonian ODE system on the whole real line which arises in the theory of phase transitions. Our method is variational but differs from the original artificial constraint method of [AF] and establishes existence by analysing the loss of compactness in minimising sequences of the action in the appropriate functional space. Our assumptions are slightly different from those considered previously and also imply a priori estimates for the solution.