Approximation schemes for propagation of fronts with nonlocal velocities and Neumann boundary conditions
type: Item
instance of: scholarly article
title: Approximation schemes for propagation of fronts with nonlocal velocities and Neumann boundary conditions
zbMATH Open document ID: 1028.35068
DOI: 10.1016/S0362-546X(02)00098-6
author: Dejan Slepčev
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2003-03-11T00:00:00Z
review text: The paper is concerned with the motion of a front \((\Gamma_t)_{t\in [0,T]}\) according to the law \[ v= f(x,t,n,Dn,\Omega_t)\text{ on }\Gamma_t. \] Here, \(v\) is the normal velocity of \(\Gamma_t\), \(n\) the unit outward normal and \(\Omega_t\) the interior of \(\Gamma_t\). The front evolves in a bounded domain \(U\subset \mathbb{R}^N\) and meets \(\partial U\) orthogonally. Using the level set approach the author obtains a unique generalized solution in the form \(\Gamma_t= \{x\in\overline U\mid u(x,t)= 0\}\), where \(u\) is the viscosity solution of a nonlocal parabolic PDE with a Neumann boundary condition. Furthermore, threshold dynamics type approximation schemes for the above evolution law are introduced and their convergence to the level set solution is proved.
Mathematics Subject Classification ID: 35K55
Mathematics Subject Classification ID: 35A35
zbMATH DE Number: 1880088
zbMATH Keywords: threshold dynamics
zbMATH Keywords: nonlocal parabolic equations
zbMATH Keywords: viscosity solutions
zbMATH Keywords: level set approach
reviewed by: Klaus Deckelnick
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1880088