Calculus of Variations and Geometric Measure Theory

A. Monteil - F. Santambrogio

Metric methods for heteroclinic connections

created by santambro on 13 Sep 2016


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


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