On construction of Martin boundaries for second order elliptic equations
type: Item
instance of: scholarly article
title: On construction of Martin boundaries for second order elliptic equations
zbMATH Open document ID: 0726.31009
author: Minoru Murata
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1990-01-01T00:00:00Z
review text: A generalized version of the classical Martin theorem says that any positive solution of a second order linear elliptic equation in a domain of a Riemannian manifold is represented uniquely by an integral of the Martin kernel over the Martin boundary with respect to a finite positive Borel measure which is zero at the non-minimal Martin boundary. This paper is concerned with positive solutions of a second order elliptic equation \[ (1)\quad \sum^{n}_{i,j=1}- \frac{1}{\sqrt{g}}\frac{\partial}{\partial x_ i}(\sqrt{g}a^{ij}\frac{\partial u}{\partial x_ j})+ \] \[ +\sum^{n}_{i=1}(b^ i\frac{\partial u}{\partial x_ i}- \frac{1}{\sqrt{g}}\frac{\partial}{\partial x_ i}(\sqrt{g}d^ iu))+cu=0 \] in a domain of a non-compact Riemannian manifold, where \(g=\det (g_{ij})\) with \((g_{ij})\) being the Riemannian metric, \((a^{ij})\) is positive definite and locally bounded, \((b^ i)\) and \((d^ i)\) are locally p-th integrable for some \(p>n\), and c is locally max(p/2,1)-th integrable. The aim of this paper is to establish two methods (decomposition methods) to construct the Martin boundary and Martin kernel for such an equation as (1), and to explicitly construct Martin boundaries and Martin kernels for equations in unbounded domains of \(R^ n\) by applying the decomposition methods to the equations. The first method (tensor product decomposition method) is concerned with decomposition of an operator and a domain into a sum of tensor products of operators and a direct product of domains; and the second one (direct sum decomposition method) is concerned with decomposition of a domain into the union of a finite number of ends and a relatively compact set.
Mathematics Subject Classification ID: 31C35
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35J67
zbMATH DE Number: 4198435
zbMATH Keywords: Martin boundary
zbMATH Keywords: positive solutions
zbMATH Keywords: decomposition methods
zbMATH Keywords: Martin kernel
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195170848
OpenAlex ID: W2000421044
cites work: Q3727040
cites work: Q3678061
cites work: On the Martin boundary of Lipschitz strips
cites work: Criticality and the \(\lambda\)-property for elliptic equations
cites work: Positive harmonic functions on complete manifolds of negative curvature
cites work: Q4751075
cites work: Q3315861
cites work: Estimates of harmonic measure
cites work: Q3342857
cites work: Positive Harmonic Functions on Lipschitz Domains
cites work: Picard principle for linear elliptic differential operators
cites work: On existence of Green function and positive superharmonic functions for linear elliptic operators of second order
cites work: Boundary behavior of harmonic functions in non-tangentially accessible domains
cites work: Q5548784
cites work: POSITIVE SOLUTIONS OF SECOND ORDER ELLIPTIC EQUATIONS IN UNBOUNDED DOMAINS
cites work: Positive harmonic functions on complete manifolds with non-negative curvature outside a compact set
cites work: An axiomatic treatment of pairs of elliptic differential equations
cites work: Minimal Positive Harmonic Functions
cites work: Structure of positive solutions to \((-\Delta +V)u=0\) in \(R^ n\)
cites work: Isolated singularities and positive solutions of elliptic equations in \(R^ n\)
cites work: Martin boundary over an isolated singularity of rotation free density
cites work: Extreme nonmonotoneity of the Picard principle
cites work: Q5559870
cites work: Q3732129
cites work: Q3829874
cites work: On positive solutions of second-order elliptic equations, stability results, and classification
cites work: Le problème de Dirichlet pour les équations elliptiques du second ordre à coefficients discontinus
cites work: Martin boundary for \(\Delta\)-P
cites work: The Martin boundary of the half disk with rotation free densities
cites work: The Martin boundaries of equivalent sheaves
cites work: On the Martin compactification of a bounded Lipschitz domain in a Riemannian manifold
cites work: Comparisons of Kernel functions, boundary Harnack principle and relative Fatou theorem on Lipschitz domains
DOI: 10.2977/PRIMS/1195170848
description: scientific article; zbMATH DE number 4198435