A Degenerate Isoperimetric Problem and Traveling Waves to a Bistable Hamiltonian System

Abstract

We analyze a nonstandard isoperimetric problem in the plane associated with a metric having degenerate conformal factor at two points. Under certain assumptions on the conformal factor, we establish the existence of curves of least length under a constraint associated with enclosed euclidean area. As a motivation for and application of this isoperimetric problem, we identify these isoperimetric curves, appropriately parametrized, as traveling wave solutions to a bistable Hamiltonian system of PDEs. We also determine the existence of a maximal propagation speed for these traveling waves through an explicit upper bound depending on the conformal factor.

A Degenerate Isoperimetric Problem in the Plane

“…In this article we relax the rather stringent assumption that W is a quadratic near its isolated zeros and instead simply assume that it is smooth and has positive definite Hessian matrix D 2 W at its two zeros, along with an assumption on behavior at infinity to ensure completeness of the metric. Since near either potential well W is well-approximated by its quadratic Taylor polynomial it would seem reasonable to conjecture that the same type of existence result as the one found in [1] is true in this more general setting. Surprisingly this turns out to be false.…”

“…that are Lipschitz continuous with respect to the metric d and satisfy Γ(a) = P, Γ(b) = Q and Γ (3) (t) = Γ (1) (t) Γ (2) (t) for a.e. t ∈ (a, b), thus making the metric space into a length space, cf.…”

“…Motivated by the observation that such isoperimetric curves can be used to build traveling wave solutions to certain Hamiltonian systems, the authors of [1] analyze this problem for the case where W := F 2 is a non-negative map from R 2 to R vanishing at two points, say p − and p + , and where the isoperimetric curve is required to join these two potential wells. Thus, the problem takes the form What makes this particular isoperimetric problem non-standard is both the degeneracy of the conformal factor and the fact that length is measured with respect to a metric given by F while area is measured with respect to the Euclidean metric.…”

“…Under the assumption that W is exactly equal to a nondegenerate quadratic polynomial in neighborhoods of the zeros p ± , such an isoperimetric curve joining p − to p + is shown to exist in [1] for any amount of Euclidean area A via a calibration argument. Near the wells, these minimizers are realized as integral curves of an explicit vector field and interestingly one finds that they often spiral into the zeros of F. On the other hand, if F(p)/ |p − p ± | → 0 as |p − p ± | → 0 for either zero, then minimizers do not exist for any A γ 0 ω 0 .…”

“…Section 4 contains the counter-examples, Section 5 contains the proof of the main existence result and in Section 6 we recall from [1] how under further non-degeneracy assumptions, these isoperimetric curves can serve as traveling wave solutions to the Hamiltonian system Ju t = ∆u − ∇W u (u) for u :…”