Willmore flow of planar networks

Abstract

Geometric gradient flows for elastic energies of Willmore type play an important role in mathematics and in many applications. The evolution of elastic curves has been studied in detail both for closed as well as for open curves. Although elastic flows for networks also have many interesting features, they have not been studied so far from the point of view of mathematical analysis. So far it was not even clear what are appropriate boundary conditions at junctions. In this paper we give a well-posedness result for Willmore flow of networks in different geometric settings and hence lay a foundation for further mathematical analysis. A main point in the proof is to check whether different proposed boundary conditions lead to a well posed problem. In this context one has to check the Lopatinskii-Shapiro condition in order to apply the Solonnikov theory for linear parabolic systems in Hölder spaces which is needed in a fixed point argument. We also show that the solution we get is unique in a purely geometric sense.Mathematics Subject Classification (2010): 35K52, 53C44 (primary); 35K61, 35K41 (secondary). *

A Second Order Gradient Flow of p-Elastic Planar Networks

“…If p = 2, noting that θ i ss N i = (∂ s κ i )N i = ∇ s κ i we see that a stationary network satisfies the natural boundary conditions at the triple junctions derived for the L 2 -gradient flow of elastic networks in [8,15,14].…”

A Survey of the Elastic Flow of Curves and Networks

“…For the proof see [23,Lemma 3.31]. Moreover by inspecting the proof of Theorem 3.32 in [23] we see that the following holds.…”

“…[40] [ 40] [ 40,41] Open curves, clamped b.c. [52] [ 29] [ 18,41] Non compact curves [40] [ 40] Networks [15,22,23] [ 14,22] [ 14] We refer also to the two recent PhD theses [38,48]. The aim of this expository paper is to arrange (most of) this material in a unitary form, proving in full detail the results for the elastic flow of closed curves and underlying the differences with the other cases.…”

“…To get uniqueness when we let evolve an open curve or a network, one has to use Proposition 2.16. We refer to [15,23,38,52] for a complete proof.…”

Degenerate Elastic Networks

“…The main reason for which we got interested in this problem is a previous study of its dynamical counterpart (see [13,14]). The study of the static problem has often revealed to be useful for the analysis of the asymptotic behavior of the solutions of the associated gradient flow and of the singularities that can appear during the evolution.…”