Transactions of the American Mathematical Society

Author(s): Anders Linnér; Joseph W. Jerome
Journal: Trans. Amer. Math. Soc. 359 (2007), 2021-2041.
MSC (2000): Primary 58E25, 49J30; Secondary 58Z05
Posted: December 15, 2006
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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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Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 58E25, 49J30, 58Z05

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: 10.1090/S0002-9947-06-04315-7
PII: S 0002-9947(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
Posted: 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
Copyright of article: Copyright 2006, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

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