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A Unique Graph of minimal Elastic Energy

#### By: Anders Linner and Joseph W. Jerome

Nonlinear functionals that appear as a product of two integrals are here
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 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. An appendix is included, which
characterizes extremals so that the Pontrjagin maximum principle may be
applied.

This paper will appear in Trans. Amer. Math. Soc. 359 (2007), 2021--2041, and,
together with its
appendix,
can be viewed sequentially: