Metric methods for heteroclinic connections

We consider the problem min ∫ R1 2 | γ ̇| 2 + W ( γ ) d t 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 ∫ 0 1 K ( γ ) | γ ̇ | d t with K = 2 W . 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.