Conservation, Convergence, and Computation for Evolving Heterogeneous Elastic Wires

SIAM Journal on Mathematical Analysis, Volume 56, Issue 4, Page 4494-4529, August 2024.
Abstract. The elastic energy of a bending-resistant interface depends on both its geometry and its material composition. We consider such a heterogeneous interface in the plane, modeled by a curve equipped with an additional density function. The resulting energy captures the complex interplay between curvature and density effects, resembling the Canham–Helfrich functional. We describe the curve by its inclination angle, so that the equilibrium equations reduce to an elliptic system of second order. After a brief variational discussion, we investigate the associated nonlocal [math]-gradient flow evolution, a coupled quasilinear parabolic problem. We analyze the (non)preservation of quantities such as convexity, positivity, and symmetry, as well as the asymptotic behavior of the system. The results are illustrated by numerical experiments.