Variational approximation of a functional of Mumford–Shah type in codimension higher than one. (6th February 2014)

• Record Type: Journal Article
• Authors: Ghiraldin, Francesco
• Is Part Of: ESAIM. Volume 20:Number 1(2014:Jan.)
• Page Start: 190
• Page End: 221
• Publication Date: 2014-02-06
• Subjects: System analysis -- Periodicals; Calculus of variations -- Periodicals; Mathematical analysis -- Periodicals; Mathematical optimization -- Periodicals; Control theory -- Periodicals; 515.64
• DOI: 10.1051/cocv/2013061
• Languages: English
• ISSNs: 1292-8119

Abstract

<abstract abstract-type="normal" xml:lang="en"> <title> <x content-type="archive" xml:space="preserve">Abstract</x> </title> <p>In this paper we consider a new kind of Mumford–Shah functional <italic>E</italic>(<italic>u, Ω</italic>) for maps <italic>u</italic> : ℝ^{<italic>m</italic>} → ℝ^{<italic>n</italic>} with <italic>m</italic> ≥ <italic>n</italic>. The most important novelty is that the energy features a singular set <italic>S</italic>_{<italic>u</italic>} of codimension greater than one, defined through the theory of distributional jacobians. After recalling the basic definitions and some well established results, we prove an approximation property for the energy <italic>E</italic>(<italic>u, Ω</italic>) <italic>via </italic><italic>Γ</italic> −convergence, in the same spirit of the work by Ambrosio and Tortorelli [L. Ambrosio and V.M. Tortorelli, <italic>Commun. Pure Appl. Math. </italic><bold>43 </bold>(1990) 999–1036].</p> </abstract>