Evolving a convex closed curve to another one via a length-preserving linear flow
type: Item
instance of: scholarly article
title: Evolving a convex closed curve to another one via a length-preserving linear flow
zbMATH Open document ID: 1187.35119
author: Yu-Chu Lin
author: Dong-Ho Tsai
published in: Journal of Differential Equations
publication date: 2009-11-18T00:00:00Z
review text: Recently S.-T.~Yau proposed the curvature difference flow \[ {{\partial X}\over{\partial t}}(\varphi,t) = \left[ k_1(\varphi,t)-k_2(\varphi)\right] N_1(\varphi,t), \qquad X(\varphi,0)=X_1(\varphi), \qquad \varphi \in {\mathbf S}^1, \leqno(1) \] for closed plane curves \(\gamma_1,\gamma_2\) in \({\mathbb R}^2\) parametrized by smooth embeddings \(X_1,X_2:{\mathbf S^1}\rightarrow{\mathbb R}^2\). Here \(k_1(\varphi,t)\) and \(k_2(\varphi)\) denote the curvatures of the evolving curve \(\gamma_{1,t}\) and of the stationary curve \(\gamma_2\), and \(N_1\) is the outer unit normal vector field to \(\gamma_{1,t}\). Under some appropriate conditions, this flow is expected to evolve \(\gamma_1\) to \(\gamma_2\). Motivated by this and so-called expansion flows, the authors consider the related flow for closed convex curves given by \[ {{\partial X}\over{\partial t}}(\varphi,t) = \left[ {1\over{k_1(\varphi,t)}}-{1\over{k_2(\varphi,t)}} \right] N_1(\varphi,t), \qquad X(\varphi,0)=X_1(\varphi), \qquad \varphi \in {\mathbf S}^1, \leqno(2) \] with notation as before, but where \(k_2(\varphi,t)\) is now the curvature of \(\gamma_2\) at the point at which its outer normal vector is \(N_1(\varphi,t)\). A natural way of studying this flow is to parametrize the two curves by their outer unit normals and to express the flow in terms of the support functions \(u_1\) and \(u_2\) of \(\gamma_1\) and \(\gamma_2\). This leads to the linear evolution problem \[ {{\partial u_1}\over{\partial t}}(\theta,t) = \left[ (u_1)_{\theta\theta}(\theta,t) + u_1(\theta,t) \right] - \left[ (u_2)_{\theta\theta}(\theta,t) + u_2(\theta) \right]. \leqno(3) \] All geometric quantities associated to \(\gamma_1\) can be expressed in terms of \(u_1\). Assuming the condition \(\int_{{\mathbf S}^1} u_1(\theta)\,d\theta = \int_{{\mathbf S}^1} u_2(\theta)\,d\theta\), which implies that the lengths of \(\gamma_1\) and \(\gamma_2\) are equal, the authors prove the following: either the flow (2) exists on some maximal ti…
Mathematics Subject Classification ID: 35K15
Mathematics Subject Classification ID: 53C44
Mathematics Subject Classification ID: 53A04
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 5634874
zbMATH Keywords: curvature difference flow
zbMATH Keywords: expansion flows
zbMATH Keywords: support function
zbMATH Keywords: linear parabolic equations
reviewed by: John Urbas
MaRDI profile type: Publication
full work available at URL: j.jde.2009.07.024
OpenAlex ID: W2050232379
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DOI: 10.1016/J.JDE.2009.07.024
description: scientific article; zbMATH DE number 5634874