On probabilistic approach to the eigenvalue problem for maximal elliptic operator
type: Item
instance of: scholarly article
title: On probabilistic approach to the eigenvalue problem for maximal elliptic operator
zbMATH Open document ID: 0963.35147
author: Masatoshi Fujisaki
published in: Osaka Journal of Mathematics
publication date: 2001-06-28T00:00:00Z
review text: Let \(D\) be a bounded domain in \(\mathbb{R}^d\) with smooth boundary \(\partial D\). Let \(L^\alpha\), \(\alpha\in A\) (\(A\) is a given set), be nondegenerate second-order linear differential operators with parameter \(\alpha\), let also \(L= \sup_{\alpha\in A}L^\alpha\). We consider the Dirichlet eigenvalue problem with respect to \(L\) on \(D\): \[ Lu+\lambda u= 0\quad\text{on }{\mathcal D},\quad u>0\quad\text{on }{\mathcal D},\quad u/\partial{\mathcal D}= 0.\tag{1} \] It is supposed that there exist an eigenvalue \(\lambda\) and a corresponding (smooth) eigenfunction \(u\) satisfying (1). Various properties of \(\lambda\) are discussed. An probabilistic representation for \(\lambda\) is obtained.
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 60H30
zbMATH DE Number: 1455446
zbMATH Keywords: elliptic operator
zbMATH Keywords: eigenvalue problem
zbMATH Keywords: stochastic differential equation
reviewed by: Alexandra Rodkina
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1455446