Calculus of Variations and Geometric Measure Theory

A. Zuniga - P. Sternberg

On the heteroclinic connection problem for multi-well gradient systems

created by zuniga on 31 Jan 2019


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

ArXiv: 1604.03645 PDF
Links: Journal link


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