The exponential leveling in elliptic singular perturbation problems with complicated attractors
type: Item
instance of: scholarly article
title: The exponential leveling in elliptic singular perturbation problems with complicated attractors
zbMATH Open document ID: 0726.60058
DOI: 10.1007/BF02789203
author: Alexander Eizenberg
published in: Journal d'Analyse Mathématique
publication date: 1990-01-01T00:00:00Z
review text: The elliptic differential operator \[ L^{\epsilon}u=\epsilon \sum^{h}_{i,j=1}a_{ij}(x)u_{x_ i}u_{x_ j}+(b(x),\nabla u),\quad \epsilon >0, \] is considered in an open bounded domain \(D\subset {\mathbb{R}}^ n\) with \(C^ 2\)-smooth boundary \(\partial D\). It is proved that the solutions of the equation \(L^{\epsilon}u^{\epsilon}=0\), where \(u^{\epsilon}\in C(\bar D)\cap C^ 2(D)\), satisfy the exponential leveling property \[ \sup_{x,y\in K}| u^{\epsilon}(x)-u^{\epsilon}(y)| \leq \max_{\partial D}| u^{\epsilon}| \exp (-(V(K)-\delta)\epsilon^{-1}) \] for \(0<\epsilon \leq \epsilon (\delta)\), \(\delta >0\) and for any compact \(K\subset D\). Moreover, the constant V(K) is described in terms of the Venttsel'-Frejdlin action functional.
Mathematics Subject Classification ID: 60H10
Mathematics Subject Classification ID: 35B25
zbMATH DE Number: 4199185
zbMATH Keywords: singular perturbations
zbMATH Keywords: elliptic differential operator
zbMATH Keywords: exponential leveling property
zbMATH Keywords: Venttsel'-Frejdlin action functional
MaRDI profile type: Publication
cites work: Recent progress on the small parameter exit problem†
cites work: Exponential Leveling for Stochastically Perturbed Dynamical Systems
cites work: The exponential leveling and the Ventcel-Freidlin ``minimal action'' function
cites work: A PDE approach to some asymptotic problems concerning random differential equations with small noise intensities
cites work: Q3672830
cites work: Q4086524
cites work: Q4741050
cites work: Q3881612
full work available at URL: https://doi.org/10.1007/bf02789203
OpenAlex ID: W2012057125
description: scientific article; zbMATH DE number 4199185