Mathematics

New research on differential equations from Tohoku University summarized

  2008 NOV 3 - (VerticalNews.com) -- "We consider the dynamics of an inextensible elastic closed wire in the plane under uniform high pressure. In 1967, Tadjbakhsh and Odeh (J. Math. Anal. Appl. 18:59-74, 1967) posed a variational problem to determine the shape of a buckled elastic ring under uniform pressure," scientists writing in the journal Calculus of Variations and Partial Differential Equations report.

  "In order to comprehend a dynamics of the wire, we consider the following two mathematical questions: (i) can we construct a gradient flow for the Tadjbakhsh-Odeh functional under the inextensibility condition?; (ii) what is a behavior of the wire governed by the gradient flow near every critical point of the Tadjbakhsh-Odeh variational problem? For (i), first we derive a system of equations which governs the gradient flow, and then, give an affirmative answer to (i) by solving the system involving fourth order parabolic equations. For (ii), we first prove a stability and instability of each critical point by considering the second variation formula of the Tadjbakhsh-Odeh functional. Moreover, we give a lower bound of its Morse index," wrote S. Okabe and colleagues, Tohoku University.

  The researchers concluded: "Finally we prove a dynamical aspects of the wire near each equilibrium state."

  Okabe and colleagues published their study in Calculus of Variations and Partial Differential Equations (The dynamics of elastic closed curves under uniform high pressure. Calculus of Variations and Partial Differential Equations, 2008;33(4):493-521).

  Additional information can be obtained by contacting S. Okabe, Tohoku University, Math Institute, Sendai, Miyagi 9808578, Japan.

  The publisher of the journal Calculus of Variations and Partial Differential Equations can be contacted at: Springer, 233 Spring St., New York, NY 10013, USA.

  Keywords: Mathematics, Tohoku University.

  This article was prepared by VerticalNews Mathematics editors from staff and other reports. Copyright 2008, VerticalNews Mathematics via VerticalNews.com.