Integral representation results for functionals defined on \(\text{SBV}(\Omega;\mathbb{R}^ m)\)
type: Item
instance of: scholarly article
title: Integral representation results for functionals defined on \(\text{SBV}(\Omega;\mathbb{R}^ m)\)
zbMATH Open document ID: 0880.49010
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 1998-01-13T00:00:00Z
review text: The main result of the paper concerns the integral representation (with Carathéodory integrands) of functionals of the form \[ F: \text{SBV}(\Omega;{\mathbb{R}}^m) \times {\mathcal B}(\Omega) \longrightarrow [0,+\infty], \] satisfying some properties like ``locality'', ``lower semicontinuity'', ``growth conditions'', and so on\dots. Above \(\text{SBV}(\Omega,{\mathbb{R}}^m)\) denotes the Ambrosio-De Giorgi space of special functions of bounded variation defined on an open set \(\Omega \in {\mathbb{R}}^n\) and with values in \({\mathbb{R}}^m\), while \({\mathcal B}(\Omega)\) stands for the family of Borel subsets of \({\Omega}\). The theorem presented in the paper generalizes an analogous result of Buttazzo and Dal Maso proved for functionals defined on Sobolev spaces. The proof of the theorem involves blow-up techniques and approximation results for functions of bounded variation. Some applications to the relaxation in image segmentation problems, as well as in fracture mechanics (more precisely to the relaxation of nonlinear elasticity energies for brittle fracture problem) are also given.
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 74B20
Mathematics Subject Classification ID: 74R99
Mathematics Subject Classification ID: 49J10
zbMATH DE Number: 1010953
zbMATH Keywords: SBV functions
zbMATH Keywords: Caccioppoli partitions
zbMATH Keywords: integral representations of functionals on SBV
zbMATH Keywords: relaxation
zbMATH Keywords: image segmentation
zbMATH Keywords: brittle fracture problems
author: Andrea Braides
author: V. Chiadò Piat
reviewed by: Zdzisław Denkowski
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1010953