A derivation theorem for capacities with respect to a Radon measure
type: Item
instance of: scholarly article
title: A derivation theorem for capacities with respect to a Radon measure
zbMATH Open document ID: 0622.28006
DOI: 10.1016/0022-1236(87)90004-8
author: G. Dal Maso
author: Umberto Mosco
author: Giuseppe Buttazzo
published in: Journal of Functional Analysis
publication date: 1987-01-01T00:00:00Z
full work available at URL: http://purl.umn.edu/4109
review text: Let \(\mu\) be a Borel measure and let \(\nu\) be a Radon measure such that \(\mu\) is absolutely continuous with respect to \(\nu\). Then it is a classical result that \(\frac{d\mu}{d\nu}(x)=\lim_{r\to 0}\frac{\mu (B_ r(x))}{\nu (B_ r(x))}\nu\)-almost everywhere; i.e. the Radon-Nikodým derivative can be identified by the values of the two measures on balls. However the values of \(\mu\) on balls are not available in some problems of Calculus of Variations where there are only informations of variational nature on the measure \(\mu\). In this paper the authors prove an alternative result on the computation of \(d\mu /d\nu\) which makes use of the concept of \(\mu\)-capacity of a set and therefore it can be easily employed in the above mentioned situations. In fact they prove that, if \(\mu\) and \(\nu\) are zero on every set of capacity zero, then \[ \frac{d\mu}{d\nu}(x)=\lim_{r\to 0}\frac{cap_{\mu}(B_ r(x),B_{2r}(x))}{\nu (B_ r(x))} \] \(\nu\)-almost everywhere, where \[ cap_{\mu}(B_ r(x),B_{2r}(x))=\min \{\int_{B_{2r}}| Du|^ 2dx+\int_{B_ r}u^ 2d\mu: u-1\in H^ 1_ 0(B_{2r})\}. \] Moreover they use this result in order to characterize the limit of a variational convergence of measure introduced in \textit{G. Dal Maso} and \textit{U. Mosco} [Appl. Math. Optimization 15, 15-64 (1987)].
Mathematics Subject Classification ID: 28A12
Mathematics Subject Classification ID: 28A15
Mathematics Subject Classification ID: 28C05
Mathematics Subject Classification ID: 49J45
zbMATH DE Number: 4008728
zbMATH Keywords: derivation theorems on balls
zbMATH Keywords: Radon measure
zbMATH Keywords: Radon-Nikodým derivative
zbMATH Keywords: capacity
zbMATH Keywords: limit of a variational convergence of measure
reviewed by: Anna Salvadori
MaRDI profile type: Publication
OpenAlex ID: W2014277083
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description: scientific article; zbMATH DE number 4008728