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: