Two examples of nonconvex self-similar solution curves for a crystalline curvature flow
type: Item
instance of: scholarly article
title: Two examples of nonconvex self-similar solution curves for a crystalline curvature flow
zbMATH Open document ID: 1077.53054
author: Tetsuya Ishiwata
author: Hiroki Yagisita
author: S. Yazaki
author: Takeo K. Ushijima
published in: Proceedings of the Japan Academy. Series A
publication date: 2005-04-04T00:00:00Z
review text: The main goal of the present paper consists in the construction of examples of nonconvex self-similar solutions for a crystalline curvature flow with an interfacial energy of which the Wulff shape is a regular triangle or a square. Let \(\mathcal{P}\) a \(K\)-admissible curve. For each edge \(S_k\), a crystalline curvature \(H_k\) is defined in terms of the length of \(S_k\) and the length of the \(n\)-th edge of the Wulff shape. Under a crystalline curvature flow, each edge \(S_k\) keeps the same direction but moves in the outward normal direction with the velocity \(V_k\) determined by a homogeneous function of some degree \(\alpha>0\) in the crystalline curvature \(H_k\): \[ V_k=\sigma(\theta_k)| H_k| ^{\alpha-1}H_k, \] where \(\sigma\) defines the interfacial energy density and \(\theta_k\) is the exterior normal angle of the \(k\)-th edge \(S_k\). Among other results, the authors show that, for \(\alpha\in(0,1)\), a nonconvex self-similar solution exists even in the case of free-orientation motions, i.e., \(\sigma(\theta+\pi)=\sigma(\theta)\). This result is in contrast with that obtained in the case of motion by smooth interfacial density, where the curve becomes convex in a finite time.
reviewed by: Cezar Oniciuc
Mathematics Subject Classification ID: 53C44
Mathematics Subject Classification ID: 34A34
Mathematics Subject Classification ID: 74N05
zbMATH DE Number: 2151799
zbMATH Keywords: crystalline curvature
zbMATH Keywords: nonconvex self-similar shrinking curve
MaRDI profile type: Publication
full work available at URL: pjaa.80.151
OpenAlex ID: W2076668886
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DOI: 10.3792/PJAA.80.151
description: scientific article; zbMATH DE number 2151799