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

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

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.

Classification : 49Q20, 49J45, 35B38, 35J60
Keywords: Mumford-Shah functional, Ambrosio-Tortorelli functional, gamma-convergence, critical points, brittle fracture

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

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

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

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

[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). | Zbl | MR

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

[7] A. Braides, $\Gamma$-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. | Zbl | MR

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

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

[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. | Zbl | MR

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

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

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

[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). | Zbl | MR

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

[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. | Zbl | MR | Numdam

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