article

Title: On the connection problem for potentials with several global minima

Authors: N. D. Alikakos and G. Fusco

Issue: Volume 57 (2008), Issue 4, 1871-1906

Abstract:

$The problem considered is the existence of heteroclinic connections for Hamiltonian systems of $N$ 2nd order differential equations with potential possessing possibly more than two global minima. First restricting to potentials with exactly two global minima we give an existence theorem under very weak nondegeneracy hypotheses on the potential. Our approach is variational: we prove existence by showing that the \textit{Action} functional has a minimizer on the set of maps connecting the two minima. Next, allowing more than two minima but restricting to systems of two 2nd order equations, we analyze the phenomenon of nonexistence. In particular, by extending a result from [N. D. Ailkakos, S. I. Betel\'u, and X. Chen, Explicit stationary solutions in multiple welldynamics and non-uniqueness of interfacial energy densities, European J. Appl. Math. 17 (2006) 525-556], we conclude that generally nonexistence is robust under small analytic perturbations of the potential.$