Metric methods for heteroclinic connections

Abstract

We consider the problem $\min {\int }_{\mathbb{R}}\frac{1}{2}|\dot{\gamma }{|}^2+W(\gamma )\mathrm{d}t$ among curves connecting two given wells of $W\ge 0$ and we reduce it, following a standard method, to a geodesic problem of the form $\min {\int }_0^{1}K(\gamma )|\dot{\gamma }|\mathrm{d}t$ 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.

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