Existence of minimizers for a class of anisotropic free discontinuity problems
type: Item
instance of: scholarly article
title: Existence of minimizers for a class of anisotropic free discontinuity problems
zbMATH Open document ID: 0963.49012
DOI: 10.1007/BF02505913
author: Cristina Trombetti
published in: Annali di Matematica Pura ed Applicata. Serie Quarta
publication date: 2001-06-22T00:00:00Z
review text: The paper is concerned with the existence of minimizing pairs \((u,K)\) for the generalized Mumford--Shah functional \[ \int_{\Omega\setminus K}f(x,\nabla u)+\alpha(u-g)^2 dx +\beta{\mathcal H}^{n-1}(K\cap\Omega). \] under uniform continuity assumptions on \(f(\cdot,p)\). The idea of the proof is to estabilish a density lower bound \[ {\mathcal H}^{n-1}(S_u\cap B_\rho(x))\geq\theta\rho^{n-1} \qquad\forall x\in \overline{S}_u, B_\rho(x)\subset \Omega \] for minimizers \(u\) of the weak formulation of the problem in the space of special functions with bounded variation. The density lower bound implies that \({\mathcal H}^{n-1}(\Omega\cap\overline{S}_u\setminus S_u)=0\), and hence that \((u,\overline{S}_u)\) is a minimizing pair. The proof of the density lower bound, following the original approach of De Giorgi, Carriero and Leaci, is related to the rate of decay in \(\rho\) of the energy \[ \int_{B_\rho(x)}f(x,\nabla v(x)) dx \] for local minimizers \(v\) of \(v\mapsto\int f(x,\nabla v)\) in a Sobolev space. In this particular situation a recent regularity result of Cupini, Fusco and Petti leads to the decay estimate.
reviewed by: Luigi Ambrosio
Mathematics Subject Classification ID: 49J45
Mathematics Subject Classification ID: 49N60
Mathematics Subject Classification ID: 49Q20
Mathematics Subject Classification ID: 68U10
Mathematics Subject Classification ID: 49J10
zbMATH DE Number: 1472054
zbMATH Keywords: Mumford-Shah functional
zbMATH Keywords: image segmentation
zbMATH Keywords: special functions of bounded variation
zbMATH Keywords: SBV
zbMATH Keywords: regularity
MaRDI profile type: Publication
cites work: Rank one property for derivatives of functions with bounded variation
cites work: Q4710887
cites work: A new proof of the SBV compactness theorem
cites work: Existence theory for a new class of variational problems
cites work: A General Chain Rule for Distributional Derivatives
cites work: Q4364772
cites work: Q4364771
cites work: Q4007942
cites work: Hölder continuity of local minimizers
cites work: Q3973627
cites work: Q3202752
cites work: Existence theorem for a minimum problem with free discontinuity set
cites work: Q5565773
cites work: Relaxation in \(BV\) versus quasiconvexification in \(W^{1,p}\); a model for the interaction between fracture and damage
cites work: Q4385189
cites work: Optimal approximations by piecewise smooth functions and associated variational problems
description: scientific article; zbMATH DE number 1472054