Boundary behavior of positive solutions of \(\Delta u=Pu\) on a Riemann surface
type: Item
instance of: scholarly article
title: Boundary behavior of positive solutions of \(\Delta u=Pu\) on a Riemann surface
zbMATH Open document ID: 0940.31003
author: Takeyoshi Satō
published in: Journal of the Mathematical Society of Japan
publication date: 1999-09-23T00:00:00Z
review text: On a hyperbolic Riemann surface \(R\), a \(P\)-solution is a \(C^2\)-function satisfying the Schrödinger equation \(\Delta u=Pu\) where \(P dx dy\) is a nonnegative Hölder continuous 2-form on \(R\). Let \(\Delta_1\) be the minimal Martin boundary and for \(b\in \Delta_1\) let \({\mathcal G}(b)\) be the associated filter; let \(\chi\) be the canonical measure of the constant function 1. The author proves a Fatou-type limit theorem for positive \(P\)-solutions: There is a set of positive \(\chi\)-measure \(\Delta_{HP}^0 \subset \Delta_1\) such that a positive continuous \(P\)-supersolution on \(R\) has a limit following the filter \({\mathcal G}(b)\) on \(\Delta_{HP}^0\), \(\chi\) a.e.; under a restriction on \(P\), it is shown that \(\Delta_{HP}^0\) is almost as large as \(\Delta_1\).
Mathematics Subject Classification ID: 31A35
Mathematics Subject Classification ID: 31C35
zbMATH DE Number: 1336913
zbMATH Keywords: Schrödinger equation on a hyperbolic Riemann surface
zbMATH Keywords: fine limit
zbMATH Keywords: Martin boundary
reviewed by: Victor Anandam
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/05110167
OpenAlex ID: W2165458104
DOI: 10.2969/JMSJ/05110167
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description: scientific article; zbMATH DE number 1336913