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Critical points of Ambrosio-Tortorelli convergeto critical points of Mumford-Shahin the one-dimensional Dirichlet case

Published online by Cambridge University Press:  24 June 2008

Gilles A. Francfort ,
Nam Q. Le  and
Sylvia Serfaty
Gilles A. Francfort
Affiliation:
LPMTM, Université Paris 13, Av. J.B. Clément, 93430 Villetaneuse, France. francfor@galilee.univ-paris13.fr
Nam Q. Le
Affiliation:
Courant Institute of Mathematical Sciences, 251 Mercer St, New York, NY 10012, USA. quangle@cims.nyu.edu
Sylvia Serfaty
Affiliation:
Courant Institute of Mathematical Sciences, 251 Mercer St, New York, NY 10012, USA. serfaty@cims.nyu.edu
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Abstract

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

Keywords

Mumford-Shah functionalAmbrosio-Tortorelli functionalGamma-convergencecritical pointsbrittle fracture
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 15 , Issue 3 , July 2009 , pp. 576 - 598
Copyright
© EDP Sciences, SMAI, 2008

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References

Ambrosio, L., Existence theory for a new class of variational problems. Arch. Ration. Mech. Anal. 111 (1990) 291–322. CrossRef
Ambrosio, L. and Tortorelli, V.M., Approximation of functionals depending on jumps by elliptic functionals via Γ -convergence. Comm. Pure Appl. Math. 43 (1990) 999–1036. CrossRef
Ambrosio, L. and Tortorelli, V.M., On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B (7) 6 (1992) 105–123.
L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2000).
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).
Bourdin, B., Numerical implementation of the variational formulation of brittle fracture. Interfaces Free Bound. 9 (2007) 411–430. CrossRef
A. Braides, Γ-convergence for Beginners, Oxford Lecture Series in Mathematics and its Applications 22. Oxford University Press (2002).
De Giorgi, E., Carriero, M. and Leaci, A., Existence theorem for a minimum problem with free discontinuity set. Arch. Ration. Mech. Anal. 108 (1989) 195–218. CrossRef
L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. CRC Press, Boca Raton, FL (1992).
Francfort, G.A. and Marigo, J.-J., Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46 (1998) 1319–1342. CrossRef
Hutchinson, J.E. and Tonegawa, Y., Convergence of phase interfaces in the van der Waals-Cahn-Hilliard theory. Calc. Var. Partial Differential Equations 10 (2000) 49–84. CrossRef
Modica, L. and Mortola, S., 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.
P.J. Olver, Applications of Lie groups to differential equations, Graduate Texts in Mathematics 107. Springer-Verlag, New York (1986).
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).
Tonegawa, Y., Phase field model with a variable chemical potential. Proc. Roy. Soc. Edinburgh Sect. A 132 (2002) 993–1019. CrossRef
Tonegawa, Y., A diffused interface whose chemical potential lies in a Sobolev space. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (2005) 487–510.
T. Wittman, Lost in the supermarket: decoding blurry barcodes. SIAM News 37 September (2004).