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 (2009)

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

Abstract

top
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.

How to cite

top

Francfort, 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 (2009): 576-598. <http://eudml.org/doc/245594>.

@article{Francfort2009,
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},
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/245594},
volume = {15},
year = {2009},
}

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
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/245594
ER -

References

top
  1. [1] L. Ambrosio, Existence theory for a new class of variational problems. Arch. Ration. Mech. Anal. 111 (1990) 291–322. Zbl0711.49064MR1068374
  2. [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. Zbl0722.49020MR1075076
  3. [3] L. Ambrosio and V.M. Tortorelli, On the approximation of free discontinuity problems. Boll. Un. Mat. Ital. B (7) 6 (1992) 105–123. Zbl0776.49029MR1164940
  4. [4] L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press (2000). Zbl0957.49001MR1857292
  5. [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). Zbl0802.35142MR1269538
  6. [6] B. Bourdin, Numerical implementation of the variational formulation of brittle fracture. Interfaces Free Bound. 9 (2007) 411–430. Zbl1130.74040MR2341850
  7. [7] A. Braides, Γ -convergence for Beginners, Oxford Lecture Series in Mathematics and its Applications 22. Oxford University Press (2002). Zbl1198.49001MR1968440
  8. [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. Zbl0682.49002MR1012174
  9. [9] L.C. Evans and R.F. Gariepy, Measure theory and fine properties of functions. CRC Press, Boca Raton, FL (1992). Zbl0804.28001MR1158660
  10. [10] G.A. Francfort and J.-J. Marigo, Revisiting brittle fracture as an energy minimization problem. J. Mech. Phys. Solids 46 (1998) 1319–1342. Zbl0966.74060MR1633984
  11. [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. Zbl1070.49026MR1803974
  12. [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. Zbl0364.49006MR473971
  13. [13] D. Mumford and J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. XLII (1989) 577–685. Zbl0691.49036MR997568
  14. [14] P.J. Olver, Applications of Lie groups to differential equations, Graduate Texts in Mathematics 107. Springer-Verlag, New York (1986). Zbl0588.22001MR836734
  15. [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). Zbl1112.35002MR2279839
  16. [16] Y. Tonegawa, Phase field model with a variable chemical potential. Proc. Roy. Soc. Edinburgh Sect. A 132 (2002) 993–1019. Zbl1013.35070MR1926927
  17. [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. Zbl1170.35416MR2185866
  18. [18] T. Wittman, Lost in the supermarket: decoding blurry barcodes. SIAM News 37 September (2004). 

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.