*Published Paper*

**Inserted:** 13 sep 2016

**Last Updated:** 13 sep 2016

**Journal:** Mathematical Methods in the Applied Sciences

**Pages:** on line first

**Year:** 2016

**Doi:** 10.1002/mma.4072

**Abstract:**

We consider the problem $\min\int_\mathbb R \frac{1}{2}

\dot{\gamma}

^2+W(\gamma) 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 $\min\int_0^1 K(\gamma)\vert\dot{\gamma}\vert d t$ 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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