Metric methods for heteroclinic connections

We consider the problem $\min\int_{\mathbb{R}} \frac{1}{2}|\dotγ|^2+W(γ)\mathop{}\mathopen{}\mathrm{d} t$ 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 $\min\int_0^1 K(γ)|\dotγ|\mathop{}\mathopen{}\mathrm{d} t$ 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.

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