A second order local minimality criterion for the triple junction singularity of the Mumford-Shah functional
ESAIM: Control, Optimisation and Calculus of Variations, Volume 24 (2018) no. 1, pp. 401-435.

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.

Received:
Accepted:
DOI: 10.1051/cocv/2017010
Classification: 49K10, 49Q20
Keywords: Calculus of variations, local minimality, mumford-Shah functional, free discontinuity problems, second variation
Cristoferi, Riccardo 1

1
@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, Volume 24 (2018) no. 1, pp. 401-435. doi : 10.1051/cocv/2017010. http://archive.numdam.org/articles/10.1051/cocv/2017010/

[1] G. Alberti, G. Bouchitté and G. Dal Maso 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] L. Ambrosio, N. Fusco and D. Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (2000) | MR | Zbl

[3] M. Bonacini and M. Morini, 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] F. Cagnetti, M.G. Mora and M. Morini, A second order minimality condition for the Mumford-Shah functional. Calc. Var. Partial Differ. Equ. 33 (2008) 37–74 | DOI | MR | Zbl

[5] M. Cicalese, G.P. Leonardi and F. Maggi, Improved convergence theorems for bubble clusters. I. The planar case. Indiana Univ. Mat. J. 65 (2016) 1979–2050 | DOI | MR | Zbl

[6] M. Cicalese, G.P. Leonardi and F. Maggi, Sharp stability inequalities for planar double bubbles. Preprint (2015) | arXiv | MR

[7] G. Dal Maso, J.M. Morel and S. Solimini, A variational method in image segmentation: existence and approximation results. Acta Math. 168 (1992) 89–151 | DOI | MR | Zbl

[8] E. De Giorgi, M. Carriero and A. Leaci, Existence theorem for a minimum problem with free discontinuity set. Arch. Rational Mech. Anal. 108 (1989) 195–218 | DOI | MR | Zbl

[9] P. Grisvard, Elliptic problems in nonsmooth domains, Vol. 24 of Monographs and Studies in Mathematics. Pitman (Advanced Publishing Program), Boston, MA (1985) | MR | Zbl

[10] H. Koch, G. Leoni and M. Morini, 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] G. Lazzaroni and R. Toader, Energy release rate and stress intensity factor in antiplane elasticity. J. Math. Pures Appl. 95 (2011) 565–584 | DOI | MR | Zbl

[12] G.P. Leonardi and F. Maggi, Improved convergence theorems for bubble clusters. II. The three-dimensional case. Preprint (2015) | arXiv | MR

[13] F. Maddalena and S. Solimini, Lower semicontinuity for functionals with free discontinuities. Arch. Rational Mech. Anal. 159 (2001) 273–294 | DOI | MR | Zbl

[14] M.G. Mora, Local calibrations for minimizers of the Mumford-Shah functional with a triple junction. Commun. Contemporary Math. 4 (2002) 297–326 | DOI | MR | Zbl

[15] M.G. Mora and M. Morini, 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] J.M. Morel and S. Solimini, Variational models in image segmentation. Birkhäuser (1994)

[17] M. Morini, Global calibrations for the non-homogeneous Mumford-Shah functional. Ann. Sc. Norm. Super. Pisa Cl. Sci. 1 (2002) 603–648 | Numdam | MR | Zbl

[18] D. Mumford and J. Shah, Boundary detection by minimizing functionals, I. Proc. IEEE Conf. on Comouter Vision and Pattern recognition 42 (1989) 577–685

[19] D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42 (1989) 577–685 | DOI | MR | Zbl

[20] L. Simon, 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

Cited by Sources: