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A unique graph of minimal elastic energy. (English) Zbl 1110.58009

Authors’ 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.”
The results are very interesting and their proofs enjoy clarity and usefullness.

MSC:

58E25 Applications of variational problems to control theory
49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
58Z05 Applications of global analysis to the sciences
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