Parametric and non-parametric minima
type: Item
instance of: scholarly article
title: Parametric and non-parametric minima
zbMATH Open document ID: 0564.49026
DOI: 10.1007/BF01169003
published in: Manuscripta Mathematica
publication date: 1984-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155021
review text: Let \(u\in BV(G)\) be a local minimum for the non-parametric area functional \(\int \sqrt{1+| Du|^ 2}\). It is well known that the hypograph of u, i.e., \(E=\{x,t)\in G\times {\mathbb{R}}:\quad t<u(x)\}\) is a Caccioppoli set in \(G\times R\) of least perimeter with respect to variations with compact support in \(G\times R\). If \(u\in C^ 2\) this property of E was proved by \textit{W. H. Fleming} [Rend. Circ. Mat. Palermo, II. Ser. 11, 69-90 (1962; Zbl 0107.313)] by applying the divergence theorem to a suitable vector field. Nevertheless, the Fleming argument does not seem to work, since the function is only in BV in this case; in fact, the vector field is not regular enough. The author proves the same result, also extended for more general functionals, by applying a special form of divergence theorem in BV [the author, Traces of bounded vector fields and divergence theorems (to appear)].
reviewed by: Elvira Mascolo
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 26B30
Mathematics Subject Classification ID: 57R25
Mathematics Subject Classification ID: 46E99
zbMATH DE Number: 3899650
zbMATH Keywords: local minimum
zbMATH Keywords: non-parametric area functional
zbMATH Keywords: hypograph
zbMATH Keywords: Caccioppoli set
zbMATH Keywords: divergence theorem
author: Gabriele Anzellotti
MaRDI profile type: Publication
cites work: Pairings between measures and bounded functions and compensated compactness
cites work: The Euler Equation for Functionals with Linear Growth
cites work: On the minima of functionals with linear growth
cites work: Integral representation on \(BV(\Omega )\) of \(\Gamma\)-limits of variational integrals
cites work: Nuovi teoremi relativi alle misure \((r - 1)\)-dimensionali in uno spazio ad \(r\) dimensioni
cites work: On the oriented Plateau problem
cites work: Q4196944
cites work: Q5508508
description: scientific article; zbMATH DE number 3899650