Metric methods for heteroclinic connections

We consider the problem min∫R12|γ̇|2+W(γ)dt among curves connecting two given wells of W≥0, and we reduce it, following a standard method, to a geodesic problem of the form min∫01K(γ)|γ̇|dt with K=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. Copyright © 2016 John Wiley & Sons, Ltd.