This paper is the first part of an ongoing project aimed at providing a local minimality criterion, based on a second variation approach, for the triple point configurations of the Mumford-Shah functional.
Accepté le :
DOI : 10.1051/cocv/2017010
Mots clés : Calculus of variations, local minimality, mumford-Shah functional, free discontinuity problems, second variation
@article{COCV_2018__24_1_401_0,
author = {Cristoferi, Riccardo},
title = {A second order local minimality criterion for the triple junction singularity of the {Mumford-Shah} functional},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {401--435},
publisher = {EDP-Sciences},
volume = {24},
number = {1},
year = {2018},
doi = {10.1051/cocv/2017010},
mrnumber = {3843190},
zbl = {1401.49019},
language = {en},
url = {http://archive.numdam.org/articles/10.1051/cocv/2017010/}
}
TY - JOUR AU - Cristoferi, Riccardo TI - A second order local minimality criterion for the triple junction singularity of the Mumford-Shah functional JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 401 EP - 435 VL - 24 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/cocv/2017010/ DO - 10.1051/cocv/2017010 LA - en ID - COCV_2018__24_1_401_0 ER -
%0 Journal Article %A Cristoferi, Riccardo %T A second order local minimality criterion for the triple junction singularity of the Mumford-Shah functional %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 401-435 %V 24 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/cocv/2017010/ %R 10.1051/cocv/2017010 %G en %F COCV_2018__24_1_401_0
Cristoferi, Riccardo. A second order local minimality criterion for the triple junction singularity of the Mumford-Shah functional. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 1, pp. 401-435. doi : 10.1051/cocv/2017010. http://archive.numdam.org/articles/10.1051/cocv/2017010/
[1] , and The calibration method for the Mumford-Shah functional and free-discontinuity problems. Calc. Var. Partial Differ. Equ. 16 (2003) 299–333 | DOI | MR | Zbl
[2] , and , Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (2000) | MR | Zbl
[3] and , Stable regular critical points of the Mumford-Shah functional are local minimizers. Ann. Inst. Henri Poincaré Anal. Non Linéaire 32 (2015) 533–570 | DOI | Numdam | MR | Zbl
[4] , and , A second order minimality condition for the Mumford-Shah functional. Calc. Var. Partial Differ. Equ. 33 (2008) 37–74 | DOI | MR | Zbl
[5] , and , Improved convergence theorems for bubble clusters. I. The planar case. Indiana Univ. Mat. J. 65 (2016) 1979–2050 | DOI | MR | Zbl
[6] , and , Sharp stability inequalities for planar double bubbles. Preprint (2015) | arXiv | MR
[7] , and , A variational method in image segmentation: existence and approximation results. Acta Math. 168 (1992) 89–151 | DOI | MR | Zbl
[8] , and , Existence theorem for a minimum problem with free discontinuity set. Arch. Rational Mech. Anal. 108 (1989) 195–218 | DOI | MR | Zbl
[9] , Elliptic problems in nonsmooth domains, Vol. 24 of Monographs and Studies in Mathematics. Pitman (Advanced Publishing Program), Boston, MA (1985) | MR | Zbl
[10] , and , On optimal regularity of free boundary problems and a conjecture of De Giorgi. Commun. Pure Appl. Math. 58 (2005) 1051–1076 | DOI | MR | Zbl
[11] and , Energy release rate and stress intensity factor in antiplane elasticity. J. Math. Pures Appl. 95 (2011) 565–584 | DOI | MR | Zbl
[12] and , Improved convergence theorems for bubble clusters. II. The three-dimensional case. Preprint (2015) | arXiv | MR
[13] and , Lower semicontinuity for functionals with free discontinuities. Arch. Rational Mech. Anal. 159 (2001) 273–294 | DOI | MR | Zbl
[14] , Local calibrations for minimizers of the Mumford-Shah functional with a triple junction. Commun. Contemporary Math. 4 (2002) 297–326 | DOI | MR | Zbl
[15] and , Local calibrations for minimizers of the Mumford-Shah functional with a regular discontinuity set. Ann. Inst. Henri Poincaré Anal. Non Linéaire 18 (2001) 403–436 | DOI | Numdam | MR | Zbl
[16] and , Variational models in image segmentation. Birkhäuser (1994)
[17] , Global calibrations for the non-homogeneous Mumford-Shah functional. Ann. Sc. Norm. Super. Pisa Cl. Sci. 1 (2002) 603–648 | Numdam | MR | Zbl
[18] and , Boundary detection by minimizing functionals, I. Proc. IEEE Conf. on Comouter Vision and Pattern recognition 42 (1989) 577–685
[19] and , Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42 (1989) 577–685 | DOI | MR | Zbl
[20] , Lectures on geometric measure theory, Vol. 3 of Proceedings of the Centre for Mathematical Analysis. Australian National University, Australian National University Centre for Mathematical Analysis, Canberra (1983) | MR | Zbl
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