Willmore Flow of planar networks

TL;DR

This paper establishes well-posedness for the Willmore flow of planar networks, providing a mathematical foundation for analyzing elastic network evolutions with appropriate boundary conditions.

Contribution

It introduces the first rigorous mathematical analysis of Willmore flow for networks, including boundary conditions and well-posedness results.

Findings

Proved well-posedness of Willmore flow for networks.

Identified suitable boundary conditions at junctions.

Ensured uniqueness of solutions in a geometric sense.

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 H\"older spaces which is needed in a fixed point argument. We also show that the solution we get is unique in a purely geometric sense.