Long time behavior of Riemannian mean curvature flow of graphs
type: Item
instance of: scholarly article
title: Long time behavior of Riemannian mean curvature flow of graphs
zbMATH Open document ID: 1025.53039
DOI: 10.1016/S0022-247X(02)00233-0
author: Giovanna Citti
author: Maria Manfredini
published in: Journal of Mathematical Analysis and Applications
publication date: 2003-02-11T00:00:00Z
review text: The authors consider the mean curvature flow of graphs \(u\) over convex domains \(\Omega\) in \({\mathbb{R}}^n\) with respect to the conformal metric \(h \delta_{ij}\) where \(h \in {\mathcal{C}}^2(\Omega) \) such that \(h \geq {\text{constant}} >0\). Imposing a zero boundary condition, they prove that the solutions tend to \(0\) exponentially fast as \(t \rightarrow \infty\). The problem has been used to model image recognition. An initial function \(u_0\), which contains the dependence of the observer for a given image \(I_0\), is evolved by the mean curvature flow with respect to a Riemannian metric \(h\delta_{ij}\) induced by \(I_0\). The reconstructed image is the normalization \(u / \sup u\) of the solution. It is further proved in this paper that the quotient \(u / \sup u\) tends to the first eigenfunction of the associated linearized problem.
reviewed by: Alina Stancu
Mathematics Subject Classification ID: 53C44
Mathematics Subject Classification ID: 35K60
zbMATH DE Number: 1866568
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: image recognition
zbMATH Keywords: mean curvature flow
zbMATH Keywords: convex domains
zbMATH Keywords: conformal metric
zbMATH Keywords: eigenfunction
Wikidata QID: Q115340221
MaRDI profile type: Publication
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full work available at URL: s0022-247x(02)00233-0
OpenAlex ID: W2120147116
description: scientific article; zbMATH DE number 1866568