Nonlocal geometric expansion of convex planar curves
type: Item
instance of: scholarly article
title: Nonlocal geometric expansion of convex planar curves
zbMATH Open document ID: 1029.49038
author: Ben Andrews
author: Mikhail Feldman
published in: Journal of Differential Equations
publication date: 2002-09-08T00:00:00Z
review text: The expansion is described by the equation \[ V = F(t, \kappa, \gamma) \quad \text{on} \;\Gamma_t , \tag{1} \] where \(\Omega_t\) is a time dependent bounded convex set in two dimensional space; \(\Gamma_t = \partial \Omega_t,\) \(V = V_{\Gamma_t}(y)\) is the outer normal velocity of \(\Gamma_t\) at \(y \in \Gamma_t,\) \(\gamma = \gamma_{\Omega_t}(t)\) is the radius of the largest disk in \(\overline \Omega_t\) that touches \(\Gamma_t\) at \(y,\) and \(\kappa = \kappa_{\Gamma_t}(y)\) is the curvature of \(\Gamma_t\) at \(y.\) Due to the definitions \(\kappa \gamma \leq 1,\) thus the function \(F(t, \kappa, \gamma)\) need only be defined for \(\kappa \gamma \leq 1.\) The functions allowed in the model include \[ F(t, \kappa, \gamma) = {\gamma \over 3(t + a)} {3 - 2 \kappa \gamma \over 2 - \kappa \gamma} , \qquad F(t, \kappa, \gamma) = \gamma \Big(1 - {\kappa \gamma \over 2} \Big) \] which arise in the modelling of collapsing sandpiles and compression molding, respectively. Under suitable hypotheses the authors show that, given an open convex and bounded set \(\Omega\) there exists a Lipschitz continuous family of open bounded convex sets \(\{\Omega_t; t \geq 0\}\) which is a viscosity solution of (1) with initial condition \(\Omega_0 = \Omega.\) The main tool is a version of Perron's method.
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 35G25
Mathematics Subject Classification ID: 35K15
zbMATH DE Number: 1797563
zbMATH Keywords: nonlocal geometric expansions
zbMATH Keywords: Monge-Kantorovich mass transfer
zbMATH Keywords: convex solutions
zbMATH Keywords: viscosity solutions
zbMATH Keywords: Perron's method
reviewed by: Hector O. Fattorini
MaRDI profile type: Publication
full work available at URL: jdeq.2001.4107
OpenAlex ID: W2093848199
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DOI: 10.1006/JDEQ.2001.4107
description: scientific article; zbMATH DE number 1797563