An ergodic control problem arising from the principal eigenfunction of an elliptic operator
type: Item
instance of: scholarly article
title: An ergodic control problem arising from the principal eigenfunction of an elliptic operator
zbMATH Open document ID: 0746.35004
author: Alain Bensoussan
author: Hideo Nagai
published in: Journal of the Mathematical Society of Japan
publication date: 1992-06-25T00:00:00Z
review text: The paper considers the equation (*) \(-(1/2)\Delta v+(1/2)|\nabla v|^ 2+\alpha v-V(x)=0\) in the whole space \(\mathbb{R}^ n\). It is assumed that \(\alpha>0\) and that the function \(V\) is smooth nonnegative and \(V(x)\to\infty\) as \(| x|\to\infty\). By means of special transformation \(v=-\log u\) and the method of super- and subsolutions there are established the existence of nonnegative solutions of (*) and the close relationship between (*) and the eigenvalue problem of the Schrödinger operator \(-(1/2)\Delta+V\). Particulary, it is shown that the limit of solutions of (*) as \(\alpha\to 0\) can be expressed in terms of the principal eigenvalue and eigenfunction of the Schrödinger operator.
Mathematics Subject Classification ID: 35B37
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 93E20
Mathematics Subject Classification ID: 35B30
Mathematics Subject Classification ID: 35J10
zbMATH DE Number: 9372
zbMATH Keywords: method of super- and subsolutions
zbMATH Keywords: existence of nonnegative solutions
reviewed by: Uldis Raitums
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04310049
OpenAlex ID: W2002159744
DOI: 10.2969/JMSJ/04310049
description: scientific article; zbMATH DE number 9372