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arXiv · 1607.08463

A Degenerate Isoperimetric Problem in the Plane

Jiri Dadok·Peter Sternberg
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Abstract

We establish sufficient conditions for existence of curves minimizing length as measured with respect to a degenerate metric on the plane while enclosing a specified amount of Euclidean area. Non-existence of minimizers can occur and examples are provided. This continues the investigation begun in [ABCDS] where the metric ds^2 near the singularities equals a quadratic polynomial times the standard metric. Here we allow the conformal factor to be any smooth non-negative potential vanishing at isolated points provided the Hessian at these points is positive definite. These isoperimetric curves, appropriately parametrized, arise as traveling wave solutions to a bi-stable Hamiltonian system.

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Jiri Dadok, Peter Sternberg. 2016-07-28. A Degenerate Isoperimetric Problem in the Plane. https://arxiv.org/abs/1607.08463

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