Metric methods for heteroclinic connections

TL;DR

This paper introduces a metric approach to finding heteroclinic connections by reducing a variational problem to a geodesic problem with a specific metric, proving existence of minimizers under minimal assumptions.

Contribution

It develops a new metric-based method for heteroclinic connection problems, establishing existence results with minimal assumptions on the potential function.

Findings

Existence of minimizers for the geodesic problem under minimal conditions.

Reduction of the variational problem to a geodesic problem with a specific metric.

Potential applicability to PDE problems due to robustness of the method.

Abstract

We consider the problem $\min\int_{\mathbb{R}} \frac{1}{2}|\dot{\gamma}|^2+W(\gamma)\mathop{}\mathopen{}\mathrm{d} t $ among curves connecting two given wells of $W\geq 0$ and we reduce it, following a standard method, to a geodesic problem of the form $\min\int_0^1 K(\gamma)|\dot{\gamma}|\mathop{}\mathopen{}\mathrm{d} t$ with $K=\sqrt{2W}$. 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.