Stability of weighted Laplace-Beltrami operators under \(L^ p\)-perturbation of the Riemannian metric
type: Item
instance of: scholarly article
title: Stability of weighted Laplace-Beltrami operators under \(L^ p\)-perturbation of the Riemannian metric
zbMATH Open document ID: 0868.35025
DOI: 10.1007/BF02790211
published in: Journal d'Analyse Mathématique
publication date: 1996-10-27T00:00:00Z
review text: Let \((M,G)\) be a Riemannian manifold, where \(G\) is a suitable measurable metric. The author considers operators of the form \(H:f\mapsto- \sigma^{-2}\nabla_G\cdot (\sigma^2\nabla_G f)+f\), where \(\sigma\) is a measurable weight. When \(M\) has no boundary, he shows that \(H^{-1}\) maps \(L^p\) continuously into \(W^{1,p}\) if \(p\) is close to 2; as \(G\), \(\sigma\) vary in \(L^p\), quantitative estimates for the stability of \(H^{-1}\) in trace norms are given. This result is extended to the case when \(M\) has a boundary and mixed Dirichlet-Neumann boundary conditions are imposed. Actually, this paper extends earlier results by the same author published in a previous paper [J. Differ. Equations 124, No. 2, 302-323 (1996; Zbl 0848.35082)], where \(M\) was a bounded domain with Lipschitz boundary in \(\mathbb{R}^n\) and the Dirichlet boundary condition was imposed.
Mathematics Subject Classification ID: 35J15
Mathematics Subject Classification ID: 58J32
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 927202
zbMATH Keywords: resolvents
zbMATH Keywords: mixed boundary conditions
author: Gerassimos Barbatis
reviewed by: Paul Godin
MaRDI profile type: Publication
Wikidata QID: Q115391590
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full work available at URL: https://doi.org/10.1007/bf02790211
OpenAlex ID: W2072100697
description: scientific article; zbMATH DE number 927202