Variational mean curvatures in abstract measure spaces
type: Item
instance of: scholarly article
title: Variational mean curvatures in abstract measure spaces
zbMATH Open document ID: 1358.49042
author: Elisabetta Barozzi
author: Umberto Massari
published in: Calculus of Variations and Partial Differential Equations
publication date: 2017-01-03T00:00:00Z
review text: Variational mean curvatures were introduced in [\textit{U. Massari}, Arch. Ration. Mech. Anal. 55, 357--382 (1974; Zbl 0305.49047)] in the Euclidean setting. Later, in 1987, \textit{E. Barozzi}, et al. [Proc. Am. Math. Soc. 99, 313--316 (1987; Zbl 1358.49041)] as well as \textit{E. Barozzi} [Atti Accad. Naz. Lincei, Cl. Sci. Fis. Mat. Nat., IX. Ser., Rend. Lincei, Mat. Appl. 5, No. 2, 149--159 (1994; Zbl 0809.49038)] have proved that any finite perimeter has a sumable variational mean curvature. The initial Massari's definition of variational mean curvature is: a function \(H\in L^1(\Omega)\), where \(\Omega\) is an open set of \(\mathbb{R}^n\), is a variational mean curvature of the set \(E\subset \Omega\), if \(E\) is a local minimizer of the functional \(\mathcal{F}(F)= \int_\Omega |D\mathbf{1}_F|+ \int_{\Omega\cap F}H(x)dx\), where \(\mathbf{1}_F\) is the characteristic function of \(F\), \(F\) is an arbitary set with finite perimeter in \(\Omega\), and the distributional gradient \(D\mathbf{1}_F\) is a vector Radon measure with finite total variation. In 1987, Barozzi, E., Gonzalez, E.H.A. and Tamanini, I. proved that any finite perimeter set has variational mean curvature in \(L^1(\Omega)\). Next, the result was strongly improved by Barozzi, E. in 1994, who has shown the existence of a variational mean curvature \(H_E \in L^1(\Omega)\), for any set \(E\subset \Omega\) statisfying some appropriate conditions and which is, in a certain sense, minimal with respect to any other variational curvature \(H\) on \(\Omega\). In the present paper, the authors extend the results of Barozzi to a generic measurable space \((\Omega ,\mu)\) equipped with a \(\sigma\)-additive function \(\mu :\mathcal{M}\to (0,\infty)\), with \(\mu(\Omega)<+\infty\). After recalling some preliminary results, the authors presents the definition of a \(P\)-perimeter as a function \(P: \mathcal{M}\to [0,\infty] \), with the properties: (P1)\(P(A\cup B)+P(A\cap B)\leq P(A)+P(B)\), for all \(A,B\i…
reviewed by: Vasile Oproiu
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 49Q05
Mathematics Subject Classification ID: 53A10
zbMATH DE Number: 6670014
zbMATH Keywords: variational mean curvatures
zbMATH Keywords: measure theory
zbMATH Keywords: perimeters in Carnot-Caratheodory spaces
zbMATH Keywords: perimeters on manifolds
zbMATH Keywords: nonlocal fractional perimeters
MaRDI profile type: Publication
full work available at URL: s00526-016-1017-4
OpenAlex ID: W2432407601
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DOI: 10.1007/S00526-016-1017-4
description: scientific article; zbMATH DE number 6670014