Critical points of Ambrosio-Tortorelli converge to critical points of Mumford-Shah in the one-dimensional Dirichlet case
ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 3, pp. 576-598.

Critical points of a variant of the Ambrosio-Tortorelli functional, for which non-zero Dirichlet boundary conditions replace the fidelity term, are investigated. They are shown to converge to particular critical points of the corresponding variant of the Mumford-Shah functional; those exhibit many symmetries. That Dirichlet variant is the natural functional when addressing a problem of brittle fracture in an elastic material.

DOI : 10.1051/cocv:2008041
Classification : 49Q20, 49J45, 35B38, 35J60
Mots-clés : Mumford-Shah functional, Ambrosio-Tortorelli functional, gamma-convergence, critical points, brittle fracture 
@article{COCV_2009__15_3_576_0,
     author = {Francfort, Gilles A. and Le, Nam Q. and Serfaty, Sylvia},
     title = {Critical points of {Ambrosio-Tortorelli} converge to critical points of {Mumford-Shah} in the one-dimensional {Dirichlet} case},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {576--598},
     publisher = {EDP-Sciences},
     volume = {15},
     number = {3},
     year = {2009},
     doi = {10.1051/cocv:2008041},
     mrnumber = {2542574},
     zbl = {1168.49041},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/cocv:2008041/}
}
TY  - JOUR
AU  - Francfort, Gilles A.
AU  - Le, Nam Q.
AU  - Serfaty, Sylvia
TI  - Critical points of Ambrosio-Tortorelli converge to critical points of Mumford-Shah in the one-dimensional Dirichlet case
JO  - ESAIM: Control, Optimisation and Calculus of Variations
PY  - 2009
SP  - 576
EP  - 598
VL  - 15
IS  - 3
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/cocv:2008041/
DO  - 10.1051/cocv:2008041
LA  - en
ID  - COCV_2009__15_3_576_0
ER  - 
%0 Journal Article
%A Francfort, Gilles A.
%A Le, Nam Q.
%A Serfaty, Sylvia
%T Critical points of Ambrosio-Tortorelli converge to critical points of Mumford-Shah in the one-dimensional Dirichlet case
%J ESAIM: Control, Optimisation and Calculus of Variations
%D 2009
%P 576-598
%V 15
%N 3
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/cocv:2008041/
%R 10.1051/cocv:2008041
%G en
%F COCV_2009__15_3_576_0
Francfort, Gilles A.; Le, Nam Q.; Serfaty, Sylvia. Critical points of Ambrosio-Tortorelli converge to critical points of Mumford-Shah in the one-dimensional Dirichlet case. ESAIM: Control, Optimisation and Calculus of Variations, Tome 15 (2009) no. 3, pp. 576-598. doi : 10.1051/cocv:2008041. http://archive.numdam.org/articles/10.1051/cocv:2008041/

[1] L. Ambrosio, Existence theory for a new class of variational problems. Arch. Ration. Mech. Anal. 111 (1990) 291-322. | MR | Zbl

[2] L. Ambrosio and V.M. Tortorelli, Approximation of functionals depending on jumps by elliptic functionals via Γ-convergence. Comm. Pure Appl. Math. 43 (1990) 999-1036. | MR | Zbl

[3] L. Ambrosio and V.M. Tortorelli, On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B (7) 6 (1992) 105-123. | MR | Zbl

[4] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2000). | MR | Zbl

[5] F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau vortices, Progress in Nonlinear Differential Equations and their Applications 13. Birkhäuser Boston Inc., Boston, MA (1994). | MR | Zbl

[6] B. Bourdin, Numerical implementation of the variational formulation of brittle fracture. Interfaces Free Bound. 9 (2007) 411-430. | MR | Zbl

[7] A. Braides, Γ-convergence for Beginners, Oxford Lecture Series in Mathematics and its Applications 22. Oxford University Press (2002). | MR

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

[9] L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. CRC Press, Boca Raton, FL (1992). | MR | Zbl

[10] G.A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46 (1998) 1319-1342. | MR | Zbl

[11] J.E. Hutchinson and Y. Tonegawa, Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory. Calc. Var. Partial Differential Equations 10 (2000) 49-84. | MR | Zbl

[12] L. Modica and S. Mortola, Il limite nella Γ-convergenza di una famiglia di funzionali ellittici. Boll. Un. Mat. Ital. A (5) 14 (1977) 526-529. | MR | Zbl

[13] D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. XLII (1989) 577-685. | MR | Zbl

[14] P.J. Olver, Applications of Lie groups to differential equations, Graduate Texts in Mathematics 107. Springer-Verlag, New York (1986). | MR | Zbl

[15] E. Sandier and S. Serfaty, Vortices in the magnetic Ginzburg-Landau model, Progress in Nonlinear Differential Equations and their Applications 70. Birkhäuser Boston Inc., Boston, MA (2007). | MR | Zbl

[16] Y. Tonegawa, Phase field model with a variable chemical potential. Proc. Roy. Soc. Edinburgh Sect. A 132 (2002) 993-1019. | MR | Zbl

[17] Y. Tonegawa, A diffused interface whose chemical potential lies in a Sobolev space. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (2005) 487-510. | Numdam | MR | Zbl

[18] T. Wittman, Lost in the supermarket: decoding blurry barcodes. SIAM News 37 September (2004).

Cité par Sources :