Non-parametric mean curvature evolution with boundary conditions
type: Item
instance of: scholarly article
title: Non-parametric mean curvature evolution with boundary conditions
zbMATH Open document ID: 0686.34013
DOI: 10.1016/0022-0396(89)90149-6
author: Gerhard Huisken
published in: Journal of Differential Equations
publication date: 1989-01-01T00:00:00Z
full work available at URL: 00-001M-0000-0013-5D38-6
review text: An evolution equation, \(\dot u+Au=0\) in \(\Omega\) \(\times [0,t]\) has previously been investigated where the surfaces, \(m_ t=graph\) u(\(\cdot,t)\) moves in the \(x^{n+1}\)-direction with speed dictated by their mean curvature, \(H=-Au\). This paper replaces this speed to be in the direction of their unit normal equal to the mean curvature. With this focus in mind the paper begins with the boundary value problem, (1) \(\dot u+(1+| Du|^ 2)^{1/2}Au=0\) in \(\Omega\) \(\times [0,\infty]\), (2) \(a^ i(Du)\cdot \gamma_ i=0\) on \(\partial \Omega \times [0,\infty]\), (3), \(u(\cdot,0)=u_ 0\). The operator A in equation (1) is the minimal surface operator given by \[ A=-D_ i(a^ i(p)),\quad a^ i=p^ i(1+| p|^ 2)^{-1\setminus 2} \] and \(\Omega\) is a bounded domain in \({\mathbb{R}}^ n\) with \(\partial \Omega\) being of class, \(C^{2,\alpha}\). The outer unit normal to \(\partial \Omega\) is denoted by \(\gamma =(\gamma_ 1,\gamma_ 2,...,\gamma_ n)\). The paper then continues to prove in a well written organized style two principal theorems. The first one requires that the boundary condition given in equation (3) satisfies \(u_ 0\in C^{2,\alpha}({\bar \Omega})\) and \(a^ i(Du_ 0)\cdot \gamma_ i=0\) on \(\partial \Omega\). Under this hypothesis it is then shown that the problem (1), (2), (3) has a smooth solution u and \(u_ t=u(\cdot,t)\) converges to a constant function as \(t\to \infty\). The proof is elegant and requires several technical details. The other theorem replaces the boundary condition given in equation (2) with \(u_ 0=\phi\) on \(\partial \Omega\) where \(\phi \in C^{2,\alpha}({\bar \Omega})\). In this situation the problem then has again a smooth solution u and \(u_ t=u(\cdot,t)\) converges to the solution of the minimal surface equation with boundary data \(\phi\) as \(t\to \infty\).
reviewed by: John Schmeelk
Mathematics Subject Classification ID: 34B10
Mathematics Subject Classification ID: 35G10
Mathematics Subject Classification ID: 35K25
zbMATH DE Number: 4124116
zbMATH Keywords: parabolic systems
zbMATH Keywords: closed convex surfaces
zbMATH Keywords: evolution equation
MaRDI profile type: Publication
OpenAlex ID: W2086928772
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description: scientific article; zbMATH DE number 4124116