Curvature-dependent energies: the elastic case
type: Item
instance of: scholarly article
title: Curvature-dependent energies: the elastic case
zbMATH Open document ID: 1361.53007
author: Emilio Acerbi
author: Domenico Mucci
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2017-03-17T00:00:00Z
review text: The authors continue the analysis begun in [the authors, ``Curvature dependent energies: a geometric and analytical approach'', Proc. R. Soc. Edinb., Sect. A, Math. (to appear), \url{doi:10.1017/S0308210516000202}]. In the present paper the authors study the curvature functional for non-smooth Cartesian curves in codimension higher than one. They use the Gauss map \(\tau_u:I\to \mathbb{S}^N\) defined as \(\tau_u=\dot c_u/|\dot c_u|, \;\dot c_u=(1,\dot u^1,\dots \dot u^N)\), the total curvature \(TC(c)\) of a curve considered by Milnor (in the case of Cartesian curves we have \(TC(c_u)=\int_{c_u}k_{c_u}d\mathcal{H}^1=\int_I|\tau_u|dt\)). The authors define the \(p\)-curvature functional of smooth Cartesian curves as \(TCP_p(c_u)=\int |\dot c_u|^{1-p}|\dot \tau_u|^pdt, \;p>1\). They extend the definition to the non-smooth case of continuous functions \(u\) with relaxed energy and with no corner points. After introducing the energy functional in Section 2, they recall the definition of Gauss graph of Cartesian curves. Next they deal, in Section 3, with the energy functional \(\mathcal{E}_p^0\) on currents, following an approach by \textit{M. Giaquinta} et al. [Cartesian currents in the calculus of variations I. Cartesian currents. Berlin: Springer (1998; Zbl 0914.49001); Cartesian currents in the calculus of variations II. Variational integrals. Berlin: Springer (1998; Zbl 0914.49002)]. Then the authors present some structure properties of the class of currents that naturally arise in the relaxation process. Then they introduce a class of minimal currents associated to the considered relaxation problem. The following energy lower bound holds: the relaxed energy of \(u\) is greater than the energy of the corresponding minimal current with the underlying function \(u\). In Section 6, the authors outline some features concerning functions with finite relaxed energy. In particular, they prove that (in high codimenion) the set of corner points is always finite. The energy u…
reviewed by: Vasile Oproiu
Mathematics Subject Classification ID: 53A07
Mathematics Subject Classification ID: 53C43
Mathematics Subject Classification ID: 74B99
Mathematics Subject Classification ID: 49Q10
zbMATH DE Number: 6695863
zbMATH Keywords: curvature
zbMATH Keywords: Cartesian currents
zbMATH Keywords: image restoration
MaRDI profile type: Publication
full work available at URL: j.na.2016.05.012
OpenAlex ID: W2428040495
cites work: Q4954179
cites work: Singular perturbations of mean curvature flow
cites work: High-Order Total Variation-Based Image Restoration
cites work: A Higher Order Model for Image Restoration: The One-Dimensional Case
cites work: Q5565773
cites work: Normal and integral currents
cites work: Q4871701
cites work: Q4210546
cites work: On the total curvature of knots
cites work: Q3499992
DOI: 10.1016/J.NA.2016.05.012
description: scientific article; zbMATH DE number 6695863