In this paper we consider a new kind of Mumford-Shah functional E(u, Ω) for maps u : ℝm → ℝn with m ≥ n. The most important novelty is that the energy features a singular set Su 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 E(u, Ω) via Γ -convergence, in the same spirit of the work by Ambrosio and Tortorelli [L. Ambrosio and V.M. Tortorelli, Commun. Pure Appl. Math. 43 (1990) 999-1036].
Mots clés : jacobian, Γ-convergence, higher codimension, Mumford-Shah, Ginzburg-Landau, phase transition
@article{COCV_2014__20_1_190_0,
author = {Ghiraldin, Francesco},
title = {Variational approximation of a functional of {Mumford-Shah} type in codimension higher than one},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {190--221},
publisher = {EDP-Sciences},
volume = {20},
number = {1},
year = {2014},
doi = {10.1051/cocv/2013061},
mrnumber = {3182697},
zbl = {1286.49054},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2013061/}
}
TY - JOUR AU - Ghiraldin, Francesco TI - Variational approximation of a functional of Mumford-Shah type in codimension higher than one JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 190 EP - 221 VL - 20 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2013061/ DO - 10.1051/cocv/2013061 LA - en ID - COCV_2014__20_1_190_0 ER -
%0 Journal Article %A Ghiraldin, Francesco %T Variational approximation of a functional of Mumford-Shah type in codimension higher than one %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 190-221 %V 20 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2013061/ %R 10.1051/cocv/2013061 %G en %F COCV_2014__20_1_190_0
Ghiraldin, Francesco. Variational approximation of a functional of Mumford-Shah type in codimension higher than one. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 1, pp. 190-221. doi : 10.1051/cocv/2013061. http://archive.numdam.org/articles/10.1051/cocv/2013061/
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