*Accepted Paper*

**Inserted:** 5 sep 2017

**Last Updated:** 20 mar 2018

**Journal:** Indiana Univ. Math. J.

**Year:** 2018

**Abstract:**

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

\dot{\gamma}

^2+W(\gamma))d t $ among curves lying in a non-locally-compact metric space and connecting two given zeros of $W\geq 0$. For this problem, the optimal curves are usually called heteroclinic connections.
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 under some suitable compactness assumptions on $K$, which are minimal. The method allows to solve some PDE problems in unbounded domains, in particular in two variables $x,y$, when $y=t$ and when the metric space is an $L^2$ space in the first variable $x$, and the potential $W$ includes a Dirichlet energy in the same variable. We then apply this technique to the problem of connecting, in a functional space, two different heteroclinic connections between two points of the Euclidean space, as it was previously studied by Alama-Bronsard-Gui and by Schatzman more than fifteen years ago. With a very different technique, we are able to recover the same results, and to weaken some assumptions.

**Keywords:**
Geodesics, double-well potentials, lack of compactness, non-linear elliptic PDEs

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