A discussion about the homogenization of moving interfaces
type: Item
instance of: scholarly article
title: A discussion about the homogenization of moving interfaces
zbMATH Open document ID: 1180.35070
author: Pierre Cardaliaguet
author: P.-L. Lions
author: Panagiotis E. Souganidis
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2009-05-12T00:00:00Z
review text: The authors present a number of new results concerning the behavior of moving interfaces in periodic environments and analyze in detail their averaging behavior. They concentrate on fronts moving by either first-order oscillatory velocity or mean curvature coupled with an oscillating forcing. Under sharp assumptions, they show that the fronts either homogenize or get trapped or oscillate. Several concrete examples are also discussed. There has been considerable interest lately in the homogenization theory for first- and second-order partial differential equations in periodic/almost-periodic and stationary, ergodic, random environments. Of special interest is the study of the averaged behavior of moving interfaces in such environments. In this note the authors expand considerably this investigation in the context of periodic media. Although it is important both for applications and interesting mathematically to consider random media, very little is known for moving interfaces in this setting except for some first-order motions. The general framework concerns the behavior as \(\varepsilon\to 0\) of the solution \(u^\varepsilon\) of the initial value problem \[ \begin{cases} u^\varepsilon_t+F(\varepsilon D^2u^\varepsilon,Du^\varepsilon,\frac{x} {\varepsilon})=0&\text{in }\mathbb{R}^N\times(0,\infty),\\ u^\varepsilon=u_0 &\text{on }\mathbb{R}^N \times\{0\},\end{cases} \] where \(u_0\in UC(\mathbb{R}^N)\), the space of uniformly continuous functions on \(\mathbb{R}^N\), and \(F\) is (degenerate) elliptic, geometric and periodic. In the random setting, the homogenization of this problem is still an open problem, except for completely degenerate (first-order) and (degenerate) semilinear equations with either convex or concave gradient dependence.
reviewed by: Vasile Iftode
Mathematics Subject Classification ID: 35B27
Mathematics Subject Classification ID: 35F21
zbMATH DE Number: 5552978
zbMATH Keywords: viscosity solution
zbMATH Keywords: periodic environments
zbMATH Keywords: averaging behavior
MaRDI profile type: Publication
full work available at URL: j.matpur.2009.01.014
OpenAlex ID: W2014882983
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DOI: 10.1016/J.MATPUR.2009.01.014
description: scientific article; zbMATH DE number 5552978