On the heteroclinic connection problem for multi-well gradient systems

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 $\sqrt{W}\mathrm{.}$ 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 in $\texttt{Vector-valued local minimizers of nonconvex}$ $\mathtt{\text{variational}}\mathtt{\text{problems}}$, and represents a more geometric alternative to the approaches for finding such connections described, for example, by N.D. Alikakos and G.Fusco in $\texttt{On the connection problem for potentials with}$ $\mathtt{\text{several}}\mathtt{\text{global}}\mathtt{\text{minima}}$, by S.V. Bolotin in $\mathtt{\text{Libration}}\mathtt{\text{motions}}\mathtt{\text{of}}\mathtt{\text{natural}}\mathtt{\text{dynamical}}\mathtt{\text{systems}}$, by J. Byeon, P. Montecchiari, and P. Rabinowitz in $\mathtt{\text{A}}\mathtt{\text{double}}\mathtt{\text{well}}\mathtt{\text{potential}}$ $\mathtt{\text{system}}$, and by P. Rabinowitz in $\texttt{Homoclinic and heteroclinic orbits for a class of Hamiltonian}$ $\mathtt{\text{systems}}$.