*Published Paper*

**Inserted:** 31 jan 2019

**Last Updated:** 31 jan 2019

**Journal:** J. Differential Equations

**Volume:** 261

**Number:** 7

**Pages:** 3987-4007

**Year:** 2016

**Doi:** 10.1016/j.jde.2016.06.010

**Abstract:**

We revisit the existence problem of heteroclinic connections in $\mathbb{R}^N$ associated with Hamiltonian systems involving potentials $W:\mathbb{R}^N\to\mathbb{R}$ having several global minima. Under very mild assumptions on $W$ we present a simple variational approach to first find geodesics minimizing length of curves joining any two of the potential wells, where length is computed with respect to a degenerate metric having conformal factor $\sqrt{W}.$ Then we show that when such a minimizing geodesic avoids passing through other wells of the potential at intermediate times, it gives rise to a heteroclinic connection between the two wells. This work improves upon the approach of [P. Sternberg,*Vector-valued local minimizers of nonconvex variational problems*, Rocky Mountain J. Math., 21 (1991), pp. 799-807] and represents a more geometric alternative to the approaches of e.g. [N.D. Alikakos and G. Fusco, *On the connection problem for potentials with several global minima*, Indiana Univ. Math. J., 57 (2008), pp. 1871-1906], [S.V. Bolotin, *Libration motions of natural dynamical systems*, Vestnnik Moskov. Univ. Ser. I Mat. Mekh. (1978), pp. 72-77], or [P.H. Rabinowitz, *Homoclinic and heteroclinic orbits for a class of Hamiltonian systems*, Calc. Var. PDE, 1 (1993), 1--36], for finding such connections.

**Keywords:**
heteroclinic orbits, multi-well potentials, minimizing geodesics

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