Existence of quadrature surfaces for positive measures with finite support
type: Item
instance of: scholarly article
title: Existence of quadrature surfaces for positive measures with finite support
zbMATH Open document ID: 0796.31004
DOI: 10.1007/BF01053435
author: Henrik Shahgholian
published in: Potential Analysis
publication date: 1994-10-05T00:00:00Z
review text: The main result is as follows. For any positive measure \(\mu\) with finite support in \(\mathbb{R}^ n\), \(n \geq 2\), there is a bounded open set \(\Omega \supset \text{supp} \mu\) such that \[ \int_{\partial \Omega} h dH^{n- 1} = \int_ \Omega h d \mu \] for all functions \(h\) harmonic in \(\overline \Omega\); here \(H^{n-1}\) denotes the \((n-1)\)-dimensional Hausdorff measure. The proof uses the free boundary problem technique of \textit{H. W. Alt} and \textit{L. A. Caffarelli} [J. Reine Angew. Math. 325, 105-144 (1981; Zbl 0449.35105)].
Mathematics Subject Classification ID: 31B20
Mathematics Subject Classification ID: 35J05
Mathematics Subject Classification ID: 35R35
zbMATH DE Number: 554328
zbMATH Keywords: harmonic function
zbMATH Keywords: free boundary problem
zbMATH Keywords: quadrature surface
MaRDI profile type: Publication
cites work: Q3895957
cites work: Quadrature surfaces as free boundaries
cites work: Q5565773
full work available at URL: https://doi.org/10.1007/bf01053435
OpenAlex ID: W1996400409
description: scientific article; zbMATH DE number 554328