Chun Chi Lin

On the geometric flow of kirchhoff elastic rods

Recently, rod theory has been applied to the mathematical modeling of bacterial fibers and biopolymers (e.g., DNA) to study their mechanical properties and shapes (e.g., supercoiling). In static rod theory, an elastic rod in equilibrium is the critical point of an elastic energy. This induces a natural question of how to find elasticae. In this paper, we focus on how to find the critical points by means of gradient flows. We relate a geometric function of curves to the isotropic Kirchhoff elastic energy of rods so that the generalized elastic curves are the centerlines of elastic rods in equilibrium. Thus, the variational problem for rods is formulated in curve geometry. This problem turns out to be a generalization of curve-straightening flows, which induce nonlinear fourth-order evolution equations. We establish the long time existence of length-preserving gradient flow for the geometric energy. Furthermore, by studying the asymptotic behavior, we show that the limit curves are the centerlines of the Ki...

Evolution of open elastic curves in ℝn subject to fixed length and natural boundary conditions

Abstract We consider regular open curves in ℝn with fixed boundary points, curvature equal to zero at the boundary, subject to a fixed length constraint and moving according to the L2-gradient flow of the elastic energy. For this flow we prove a long-time existence result and subconvergence to critical points.

Two-by-two upper triangular matrices and Morrey’s conjecture

It is shown that every homogeneous gradient Young measure supported on matrices of the form

L 2-flow of elastic curves with clamped boundary conditions

\begin{pmatrix} a_{1,1} &{} \cdots &{} a_{1,n-1} &{} a_{1,n} \\ 0 &{} \cdots &{} 0 &{} a_{2,n} \end{pmatrix}

Interior Continuity of Two-Dimensional Weakly Stationary-Harmonic Multiple-Valued Functions

a1,1⋯a1,n-1a1,n0⋯0a2,n is a laminate. This is used to prove the same result on the 3-dimensional nonlinear submanifold of

L 2-flow of elastic curves with knot points and clamped ends

\mathbb {M}^{2 \times 2}