Bounded variation and relaxed curvature of surfaces
type: Item
instance of: scholarly article
title: Bounded variation and relaxed curvature of surfaces
zbMATH Open document ID: 1445.53005
author: Domenico Mucci
author: Alberto Saracco
published in: Milan Journal of Mathematics
publication date: 2020-06-30T00:00:00Z
full work available at URL: https://arxiv.org/abs/1807.09576
review text: In the non-parametric case, the surface \(\Sigma \) is assumed to be the graph \(G_{u}=\left\{ \left( x,u\left( x\right) \right) \left\vert x\in Q\right. \right\} \ \)of a continuous and real-valued function \(u\) defined on a closed and bounded domain \(Q\subset \mathbb{R}^{2}\), e.g., \(Q=\left[ 0,1\right] ^{2}\), the unit square. In his celebrated paper of [Atti Accad. Naz. Lincei, Rend., VI. Ser. 3, 357--362 (1926; JFM 52.0251.01)], \textit{L. Tonelli} showed that the graph surface \(\Sigma \) has finite relaxed area in Lebesgue's sense if and only if \(u\) is a function of bounded variation. The aim of this paper is to extend (at least partially) Tonelli's result concerning the area to a similar notion of total mean and Gauss curvature. In correspondence to a relaxed formula that takes into account both area and curvatures, one expects that if \(u\) has finite relaxed energy, then both \(u\) and the outward unit normal \(v_{u}\) are function of bounded variation. Moreover, the non-smooth counterpart of the density of the total mean and Gauss curvature energy of smooth functions \(u\), suggests that suitable distributions (depending on the approximate derivative of \(u\), and of the unit normal \(v_{u}\)) are expected to be measures with finite total variation, too.
reviewed by: Ergin Bayram
Mathematics Subject Classification ID: 53A05
Mathematics Subject Classification ID: 26B30
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 53A70
zbMATH DE Number: 7216257
zbMATH Keywords: curvature of surfaces
zbMATH Keywords: polyhedral surfaces
zbMATH Keywords: bounded variation
MaRDI profile type: Publication
OpenAlex ID: W3102009869
arXiv ID: 1807.09576
cites work: Curvature-dependent energies: a geometric and analytical approach
cites work: Q4954179
cites work: Q3970542
cites work: Funzioni BV e tracce
cites work: Q5749875
cites work: Bounded variation and differentiability of functions
cites work: Q4210546
cites work: On the convergence of metric and geometric properties of polyhedral surfaces
cites work: On the total curvature of knots
cites work: Q3499992
cites work: Curvatures of Smooth and Discrete Surfaces
DOI: 10.1007/S00032-020-00311-W
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description: scientific article