The dynamics of elastic closed curves under uniform high pressure

Okabe, Shinya
December 2008
Calculus of Variations & Partial Differential Equations;Dec2008, Vol. 33 Issue 4, p493
Academic Journal
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. 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. Finally we prove a dynamical aspects of the wire near each equilibrium state.


Related Articles

  • Regularity condition by mean oscillation to a weak solution of the 2-dimensional Harmonic heat flow into sphere. Misawa, Masashi; Ogawa, Takayoshi // Calculus of Variations & Partial Differential Equations;Dec2008, Vol. 33 Issue 4, p391 

    We show a regularity criterion to the harmonic heat flow from 2-dimensional Riemannian manifold M into a sphere. It is shown that a weak solution of the harmonic heat flow from 2-dimensional manifold into a sphere is regular under the criterion where BMO r is the space of bounded mean...

  • Compactness of sequences of two-dimensional energies with a zero-order term. Application to three-dimensional nonlocal effects. Briane, Marc; Casado�D�az, Juan // Calculus of Variations & Partial Differential Equations;Dec2008, Vol. 33 Issue 4, p463 

    In this paper we study the limit, in the sense of the G-convergence, of sequences of two-dimensional energies of the type $${\int_\Omega A_n\nabla u\cdot\nabla u\,dx+\int_\Omega u^2d\mu_n}$$ , where A n is a symmetric positive definite matrix-valued function and � n is a nonnegative...

  • A priori estimates and existence for quasi-linear elliptic equations. Zou, Heng // Calculus of Variations & Partial Differential Equations;Dec2008, Vol. 33 Issue 4, p417 

    We study the boundary value problem of quasi-linear elliptic equation where $${\Omega\subset\mathbb{R}^n}$$ ( n = 2) is a connected smooth domain, and the exponent $${m\in(1,n)}$$ is a positive number. Under appropriate conditions on the function B, a variety of results on a priori estimates,...

  • The relative isoperimetric inequality outside convex domains in R n . Choe, Jaigyoung; Ghomi, Mohammad; Ritor�, Manuel // Calculus of Variations & Partial Differential Equations;Aug2007, Vol. 29 Issue 4, p421 

    We prove that the area of a hypersurface S which traps a given volume outside a convex domain C in Euclidean space R n is bigger than or equal to the area of a hemisphere which traps the same volume on one side of a hyperplane. Further, when C has smooth boundary ? C, we show that equality...

  • A priori estimates for the scalar curvature equation on S 3. Schneider, Matthias // Calculus of Variations & Partial Differential Equations;Aug2007, Vol. 29 Issue 4, p521 

    We obtain a priori estimates for solutions to the prescribed scalar curvature equation on S 3. The usual non-degeneracy assumption on the curvature function is replaced by a new condition, which is necessary and sufficient for the existence of a priori estimates, when the curvature function is a...

  • Two surprising maximisation problems. LORD, NICK // Mathematical Gazette;Nov2013, Vol. 97 Issue 540, p535 

    The article presents two surprising problems on maximisation.

  • Interchange of infimum and integral. Omar Anza Hafsa; Jean-Philippe Mandallena // Calculus of Variations & Partial Differential Equations;Dec2003, Vol. 18 Issue 4, p433 

    We prove a new interchange theorem of infimum and integral. Its distinguishing feature is, on the one hand, to establish a general framework to deal with interchange problems for nonconvex integrands and nondecomposable sets, and, on the other hand, to link the theorems of Rockafellar and...

  • The spectral radius of matrix continuous refinement operators. Didenko, Victor; Wee Ping Yeo // Advances in Computational Mathematics;Jul2010, Vol. 33 Issue 1, p113 

    A simple analytic formula for the spectral radius of matrix continuous refinement operators is established. On the space $L_2^m({{\mathbb R}}^s)$, m = 1 and s = 1, their spectral radius is equal to the maximal eigenvalue in magnitude of a number matrix, obtained from the dilation matrix M and...

  • Droplet solutions in the diblock copolymer problem with skewed monomer composition. Ren, Xiaofeng; Wei, Juncheng // Calculus of Variations & Partial Differential Equations;Mar2006, Vol. 25 Issue 3, p333 

    A droplet solution characterizes the lamellar phase of a diblock copolymer when the two composing monomers maintain a skewed ratio. We study the threshold case where the free energy of a droplet solution is comparable to the free energy of the constant solution. Using a Lyapunov-Schmidt...


Read the Article


Sorry, but this item is not currently available from your library.

Try another library?
Sign out of this library

Other Topics