We consider regular open curves in R n with fixed boundary points and moving according to the L 2 -gradient flow for a generalisation of the Helfrich functional. Natural boundary conditions are imposed along the evolution. More precisely, at the boundary the curvature vector is equal to the normal projection of a fixed given vector. A long-time existence result together with subconvergence to critical points is proven.
We present and analyze a semi-discrete finite element scheme for a system consisting of a geometric evolution equation for a curve and a parabolic equation on the evolving curve. More precisely, curve shortening flow with a forcing term that depends on a field defined on the curve is coupled with a diffusion equation for that field. The scheme is based on ideas of [12] for the curve shortening flow and [13] for the parabolic equation on the moving curve. Additional estimates are required in order to show convergence, most notably with respect to the length element: While in [12] an estimate of its error was sufficient we here also need to estimate the time derivative of the error which arises from the diffusion equation. Numerical simulation results support the theoretical findings.
In this paper we consider the elastic energy for open curves in Euclidean space subject to clamped boundary conditions and obtain the Lojasiewicz-Simon gradient inequality for this energy functional. Thanks to this inequality we can prove that a (suitably reparametrized) solution to the associated L 2 -gradient flow converges for large time to an elastica, that is to a critical point of the functional.
We study the evolution of closed inextensible planar curves under a second order flow that decreases the p-elastic energy. A short time existence result for p ∈ (1, ∞) is obtained via a minimizing movements method. For p = 2, that is in the case of the classic elastic energy, long-time existence is retrieved.
We consider closed curves on the sphere moving by the L2-gradient flow of the elastic energy both with and without penalisation of the length and show short-time and long-time existence of the flow. Moreover, when the length is penalised, we prove sub-convergence to critical points.
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