Two examples of fattening for the curvature flow with a driving force
type: Item
instance of: scholarly article
title: Two examples of fattening for the curvature flow with a driving force
zbMATH Open document ID: 0826.35051
published in: Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Serie IX. Rendiconti Lincei. Matematica e Applicazioni
publication date: 1995-11-23T00:00:00Z
full work available at URL: https://eudml.org/doc/244251
review text: The authors consider the (mean) curvature flow with a driving force \(g(t)\) for curves in the plane. They give two examples of a bounded Lipschitz function \(u_0 : \mathbb{R}^2 \to \mathbb{R}\) whose level curves evolve in time, in the generalized viscosity sense, such that a certain regular initial level curve develops, after a positive finite time, a full two-dimensional interior part. This behavior is called fattening, and can be interpreted as a phenomenon of non-uniqueness of the level-set flow. For \(g \equiv 0\) it is known that all smooth compact level curves of \(u_0\) never develop an interior. One of the examples is \(g(t) = 1 - t\) where the initial level curve is the union of two disjoint circles having the same radius. The second example is \(g \equiv 1\). Here, the initial level curve is the union of two disjoint circles having different radii.
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 58J35
zbMATH DE Number: 722299
zbMATH Keywords: mean curvature flow
zbMATH Keywords: viscosity solutions
zbMATH Keywords: fattening
author: Giovanni Bellettini
author: Maurizio Paolini
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 722299