A Degenerate Isoperimetric Problem and Traveling Waves to a Bi-stable Hamiltonian System

TL;DR

This paper investigates a specialized isoperimetric problem with degenerate metrics and connects its solutions to traveling wave solutions in a bi-stable Hamiltonian PDE system, establishing existence and speed bounds.

Contribution

It introduces a novel isoperimetric problem with degenerate conformal metrics and links its solutions to traveling waves in a bi-stable Hamiltonian system, providing existence and speed bounds.

Findings

Existence of least-length curves under area constraints in degenerate metrics.

Identification of these curves as traveling wave solutions.

Explicit upper bound on the maximal propagation speed.

Abstract

We analyze a non-standard 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 bi-stable Hamiltonian system of PDE's. We also determine the existence of a maximal propagation speed for these traveling waves through an explicit upper bound depending on the conformal factor.