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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

A unique graph of minimal elastic energy


Authors: Anders Linnér and Joseph W. Jerome
Journal: Trans. Amer. Math. Soc. 359 (2007), 2021-2041
MSC (2000): Primary 58E25, 49J30; Secondary 58Z05
DOI: https://doi.org/10.1090/S0002-9947-06-04315-7
Published electronically: December 15, 2006
MathSciNet review: 2276610
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Abstract: Nonlinear functionals that appear as a product of two integrals are considered in the context of elastic curves of variable length. A technique is introduced that exploits the fact that one of the integrals has an integrand independent of the derivative of the unknown. Both the linear and the nonlinear cases are illustrated. By lengthening parameterized curves it is possible to reduce the elastic energy to zero. It is shown here that for graphs this is not the case. Specifically, there is a unique graph of minimal elastic energy among all graphs that have turned 90 degrees after traversing one unit.


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Additional Information

Anders Linnér
Affiliation: Department of Mathematical Sciences, Northern Illinois University, DeKalb, Illinois 60115
Email: alinner@math.niu.edu

Joseph W. Jerome
Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208
Email: jwj@math.northwestern.edu

DOI: https://doi.org/10.1090/S0002-9947-06-04315-7
Keywords: Elastic energy, variable length, pendulum equation, Pontrjagin, maximum principle, phase constraint, optimal graph, one-dimensional Willmore equation.
Received by editor(s): January 12, 2005
Published electronically: December 15, 2006
Additional Notes: The research for this paper was essentially completed during the first author’s sabbatical year at Northwestern.
The second author’s research was supported in part by NSF grant DMS-0311263
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.