# Critical points of Ambrosio-Tortorelli converge to critical points of Mumford-Shah in the one-dimensional Dirichlet case

Gilles A. Francfort; Nam Q. Le; Sylvia Serfaty

ESAIM: Control, Optimisation and Calculus of Variations (2008)

- Volume: 15, Issue: 3, page 576-598
- ISSN: 1292-8119

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topFrancfort, Gilles A., Le, Nam Q., and 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 15.3 (2008): 576-598. <http://eudml.org/doc/90928>.

@article{Francfort2008,

abstract = {
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.
},

author = {Francfort, Gilles A., Le, Nam Q., Serfaty, Sylvia},

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

keywords = {Mumford-Shah functional; Ambrosio-Tortorelli functional; Gamma-convergence; critical points; brittle fracture; -convergence},

language = {eng},

month = {6},

number = {3},

pages = {576-598},

publisher = {EDP Sciences},

title = {Critical points of Ambrosio-Tortorelli converge to critical points of Mumford-Shah in the one-dimensional Dirichlet case},

url = {http://eudml.org/doc/90928},

volume = {15},

year = {2008},

}

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

DA - 2008/6//

PB - EDP Sciences

VL - 15

IS - 3

SP - 576

EP - 598

AB -
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.

LA - eng

KW - Mumford-Shah functional; Ambrosio-Tortorelli functional; Gamma-convergence; critical points; brittle fracture; -convergence

UR - http://eudml.org/doc/90928

ER -

## References

top- L. Ambrosio, Existence theory for a new class of variational problems. Arch. Ration. Mech. Anal.111 (1990) 291–322. Zbl0711.49064
- 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. Zbl0722.49020
- L. Ambrosio and V.M. Tortorelli, On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B (7)6 (1992) 105–123. Zbl0776.49029
- L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2000). Zbl0957.49001
- F. Bethuel, H. Brezis and F. Hélein, Ginzburg-Landau vortices, Progress in Nonlinear Differential Equations and their Applications13. Birkhäuser Boston Inc., Boston, MA (1994).
- B. Bourdin, Numerical implementation of the variational formulation of brittle fracture. Interfaces Free Bound.9 (2007) 411–430. Zbl1130.74040
- A. Braides, Γ-convergence for Beginners, Oxford Lecture Series in Mathematics and its Applications22. Oxford University Press (2002).
- 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. Zbl0682.49002
- L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. CRC Press, Boca Raton, FL (1992). Zbl0804.28001
- G.A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids46 (1998) 1319–1342. Zbl0966.74060
- J.E. Hutchinson and Y. Tonegawa, Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory. Calc. Var. Partial Differential Equations10 (2000) 49–84. Zbl1070.49026
- 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.
- D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math.XLII (1989) 577–685. Zbl0691.49036
- P.J. Olver, Applications of Lie groups to differential equations, Graduate Texts in Mathematics107. Springer-Verlag, New York (1986). Zbl0588.22001
- E. Sandier and S. Serfaty, Vortices in the magnetic Ginzburg-Landau model, Progress in Nonlinear Differential Equations and their Applications70. Birkhäuser Boston Inc., Boston, MA (2007). Zbl1112.35002
- Y. Tonegawa, Phase field model with a variable chemical potential. Proc. Roy. Soc. Edinburgh Sect. A132 (2002) 993–1019. Zbl1013.35070
- 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. Zbl1170.35416
- T. Wittman, Lost in the supermarket: decoding blurry barcodes. SIAM News37 September (2004).

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