# Numerical study of a new global minimizer for the Mumford-Shah functional in R3

ESAIM: Control, Optimisation and Calculus of Variations (2007)

- Volume: 13, Issue: 3, page 553-569
- ISSN: 1292-8119

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topMerlet, Benoît. "Numerical study of a new global minimizer for the Mumford-Shah functional in R3." ESAIM: Control, Optimisation and Calculus of Variations 13.3 (2007): 553-569. <http://eudml.org/doc/250007>.

@article{Merlet2007,

abstract = {
In [Progress Math.233 (2005)], David suggested the existence of a new type of global minimizers for the Mumford-Shah functional in $\mathbf\{R\}^3$. The singular set of such a new minimizer belongs to a three parameters family of sets $(0<\delta_1,\delta_2,\delta_3<\pi)$.
We first derive necessary conditions satisfied by global minimizers of this family. Then we are led to study the first eigenvectors of the Laplace-Beltrami operator with Neumann boundary conditions on subdomains of $\mathbf\{S\}^2$ with three reentrant corners. The necessary conditions are constraints on the eigenvalue and on the ratios between the three singular coefficients of the associated eigenvector.
We use numerical methods (Singular Functions Method and Moussaoui's extraction formula) to compute the eigenvalues and the singular coefficients. We conclude that there is no $(\delta_1,\delta_2,\delta_3)$ for which the necessary conditions are satisfied and this shows that the hypothesis was wrong.
},

author = {Merlet, Benoît},

journal = {ESAIM: Control, Optimisation and Calculus of Variations},

keywords = {Mumford-Shah functional; numerical analysis; boundary value problems for second-order; elliptic equations in domains with corners},

language = {eng},

month = {6},

number = {3},

pages = {553-569},

publisher = {EDP Sciences},

title = {Numerical study of a new global minimizer for the Mumford-Shah functional in R3},

url = {http://eudml.org/doc/250007},

volume = {13},

year = {2007},

}

TY - JOUR

AU - Merlet, Benoît

TI - Numerical study of a new global minimizer for the Mumford-Shah functional in R3

JO - ESAIM: Control, Optimisation and Calculus of Variations

DA - 2007/6//

PB - EDP Sciences

VL - 13

IS - 3

SP - 553

EP - 569

AB -
In [Progress Math.233 (2005)], David suggested the existence of a new type of global minimizers for the Mumford-Shah functional in $\mathbf{R}^3$. The singular set of such a new minimizer belongs to a three parameters family of sets $(0<\delta_1,\delta_2,\delta_3<\pi)$.
We first derive necessary conditions satisfied by global minimizers of this family. Then we are led to study the first eigenvectors of the Laplace-Beltrami operator with Neumann boundary conditions on subdomains of $\mathbf{S}^2$ with three reentrant corners. The necessary conditions are constraints on the eigenvalue and on the ratios between the three singular coefficients of the associated eigenvector.
We use numerical methods (Singular Functions Method and Moussaoui's extraction formula) to compute the eigenvalues and the singular coefficients. We conclude that there is no $(\delta_1,\delta_2,\delta_3)$ for which the necessary conditions are satisfied and this shows that the hypothesis was wrong.

LA - eng

KW - Mumford-Shah functional; numerical analysis; boundary value problems for second-order; elliptic equations in domains with corners

UR - http://eudml.org/doc/250007

ER -

## References

top- M. Amara and M.-A. Moussaoui, Approximation of solutions and singularities coefficients for an elliptic problem in a plane polygonal domain. Note Technique, E.N.S. Lyon (1989).
- A. Bonnet, On the regularity of edges in image segmentation. Ann. Inst. H. Poincaré Anal. Non Linéaire13 (1996) 485–528. Zbl0883.49004
- M. Bourlard, M. Dauge, M.S Lubuma and S. Nicaise, Coefficients of the singularities for elliptic boundary value problems on domains with conical points. III. Finite element methods on polygonal domains. SIAM J. Numer. Anal.29 (1992) 136–155. Zbl0794.35015
- P. Ciarlet, Jr. and J. He, The singular complement method for 2d scalar problems. C. R. Math. Acad. Sci. Paris336 (2003) 353–358. Zbl1028.65118
- M. Dauge, Elliptic boundary value problems on corner domains, Lect. Notes Math.1341. Smoothness and asymptotics of solutions. Springer-Verlag, Berlin (1988).
- M. Dauge, S. Nicaise, M. Bourlard and M.S. Lubuma, Coefficients des singularités pour des problèmes aux limites elliptiques sur un domaine à points coniques. I. Résultats généraux pour le problème de Dirichlet. RAIRO Modél. Math. Anal. Numér.24 (1990) 27–52. Zbl0691.35023
- M. Dauge, S. Nicaise, M. Bourlard and M.S. Lubuma, Coefficients des singularités pour des problèmes aux limites elliptiques sur un domaine à points coniques. II. Quelques opérateurs particuliers. RAIRO Modél. Math. Anal. Numér.24 (1990) 343–367. Zbl0723.35035
- G. David, Singular sets of minimizers for the Mumford-Shah functional. Progress Math.233, Birkhäuser Verlag, Basel (2005). Zbl1086.49030
- 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. Zbl0682.49002
- A. Ern and J.-L Guermond, Éléments finis: théorie, applications, mise en œ uvre. Math. Appl. 36, Springer-Verlag, Berlin (2002). Zbl0993.65123
- P. Grisvard, Singularities in boundary value problems, Recherches Math. Appl. 22. Masson, Paris (1992). Zbl0766.35001
- V.A. Kondrat'ev, Boundary value problems for elliptic equations in domains with conical or angular points. Trudy Moskov. Mat. Obšč.16 (1967) 209–292.
- M.-A. Moussaoui, Sur l'approximation des solutions du problème de Dirichlet dans un ouvert avec coins, in Singularities and constructive methods for their treatment (Oberwolfach, 1983). Lect. Notes Math.1121, Springer, Berlin (1985) 199–206.
- D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math.42 (1989) 577–685. Zbl0691.49036

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