Journal
JOURNAL OF GEOMETRIC ANALYSIS
Volume 28, Issue 3, Pages 2225-2253Publisher
SPRINGER
DOI: 10.1007/s12220-017-9902-4
Keywords
Degenerate metric; Isoperimetric problem; Geodesics
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Funding
- National Science Foundation [D.M.S. 1362879]
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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 Alama et al. (Commun Pure Appl Math 70:340-377, 2017) where the metric 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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