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Manuscripta Mathematica

, Volume 142, Issue 1–2, pp 215–232 | Cite as

Density lower bound estimates for local minimizers of the 2d Mumford–Shah energy

  • Camillo De Lellis
  • Matteo FocardiEmail author
Article

Abstract

We prove, using direct variational arguments, an explicit energy-treshold criterion for regular points of 2-dimensional Mumford-Shah energy minimizers. From this we infer an explicit constant for the density lower bound of De Giorgi, Carriero and Leaci.

Keywords

Radon Measure Admissible Function Monotonicity Formula Perforated Domain Dirichlet Energy 
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References

  1. 1.
    Ambrosio L., Caselles V., Masnou S., Morel J.M.: Connected components of sets of finite perimeter and applications to image processing.. J. Eur. Math. Soc. 3, 39–92 (2001)MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Ambrosio, L., Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York (2000)Google Scholar
  3. 3.
    Bonnet A.: On the regularity of edges in image segmentation. Ann. Inst. H. Poincaré, Analyse Non Linéaire 13(4), 485–528 (1996)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Bucur, D., Luckhaus, S.: Monotonicity formula and regularity for general free discontinuity problems (2012, in press)Google Scholar
  5. 5.
    Carriero M., Leaci A.: Existence theorem for a Dirichlet problem with free discontinuity set. Nonlinear Anal. 15(7), 661–677 (1990)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Dal Maso G., Morel J.M., Solimini S.: A variational method in image segmentation: existence and approximation results. Acta Math. 168(1–2), 89–151 (1992)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    David G.: C 1-arcs for minimizers of the Mumford–Shah functional. SIAM J. Appl. Math. 56(3), 783–888 (1996)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    David, G.: Singular sets of minimizers for the Mumford–Shah functional. Progress in Mathematics, 233. Birkhäuser Verlag, Basel, xiv+581 pp. ISBN: 978-3-7643-7182-1; 3-7643-7182-X (2005)Google Scholar
  9. 9.
    David G., Léger J.C.: Monotonicity and separation for the Mumford–Shah problem. Ann. Inst. H. Poincaré Anal. Non Linéaire 19(5), 631–682 (2002)zbMATHCrossRefGoogle Scholar
  10. 10.
    De Giorgi E., Carriero M., Leaci A.: Existence theorem for a minimum problem with free discontinuity set. Arch. Ration. Mech. Anal. 108, 195–218 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    De Lellis, C.; Focardi, M.: Higher integrability of the gradient for minimizers of the 2d Mumford–Shah energy. J. Math. Pures Appl. (2012). http://dx.doi.org/10.1016/j.matpur.2013.01.006
  12. 12.
    Focardi M., Gelli M.S., Ponsiglione M.: Fracture mechanics in perforated domains: a variational model for brittle porous media. Math. Models Methods Appl. Sci. 19, 2065–2100 (2009)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Maddalena, F., Solimini, S.: Blow-up techniques and regularity near the boundary for free discontinuity problems. Adv. Nonlinear Stud. 1(2) (2001)Google Scholar
  14. 14.
    Mumford D., Shah J.: Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42(5), 577–685 (1989)MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Institut für MathematikUniversität ZürichZürichSwitzerland
  2. 2.Dipartimento di Matematica “U. Dini”Università di FirenzeFirenzeItaly

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