Best constants in the Harnack inequality for the Weinstein equation
type: Item
instance of: scholarly article
title: Best constants in the Harnack inequality for the Weinstein equation
zbMATH Open document ID: 0653.31002
DOI: 10.1007/BF01830680
author: Heinz Leutwiler
published in: Aequationes Mathematicae
publication date: 1987-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/137248
review text: Let U be the class of positive solutions of the equation \[ \sum^{n}_{1}\partial ^ 2u/\partial x^ 2_ i+(k/x_ n)\partial u/\partial x_ n+(\ell /x\quad 2_ n)u=0\quad in\quad {\mathbb{R}}\quad n_+=\{x_ n>0\} \] where \(k\in {\mathbb{R}}\) and \(-\infty <\ell \leq ((k- 1)/2)^ 2\). The author gives a representation theorem for u (modifying some known results when \(\ell =0).\) Then using a Möbius transformation \(T: B^ n\to {\mathbb{R}}\) \(n_+\), the above representation is considered in the unit ball \(B^ n\) to determine the best constants in the Harnack inequality for U. Also found here are representation theorems for positive axially symmetric solutions of certain (Grušin's and Heisenberg's) partial differential equations.
Mathematics Subject Classification ID: 31B35
zbMATH DE Number: 4065389
zbMATH Keywords: positive solutions
zbMATH Keywords: representation
zbMATH Keywords: Möbius transformation
zbMATH Keywords: Harnack inequality
zbMATH Keywords: axially symmetric
reviewed by: Victor Anandam
MaRDI profile type: Publication
cites work: Q3665386
cites work: A fundamental solution for a subelliptic operator
cites work: ON A CLASS OF HYPOELLIPTIC OPERATORS
cites work: On the uniqueness of generalized axially symmetric potentials
cites work: Die Harnacksche Metrik in der Theorie der harmonischen Funktionen
cites work: On a distance invariant under Möbius transformations in R^n
cites work: Generalized axially symmetric potential theory
description: scientific article; zbMATH DE number 4065389