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

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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.

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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  3. Bonnet A.: On the regularity of edges in image segmentation. Ann. Inst. H. Poincaré, Analyse Non Linéaire 13(4), 485–528 (1996)

    MathSciNet  MATH  Google Scholar 

  4. Bucur, D., Luckhaus, S.: Monotonicity formula and regularity for general free discontinuity problems (2012, in press)

  5. Carriero M., Leaci A.: Existence theorem for a Dirichlet problem with free discontinuity set. Nonlinear Anal. 15(7), 661–677 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  7. David G.: C 1-arcs for minimizers of the Mumford–Shah functional. SIAM J. Appl. Math. 56(3), 783–888 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

    Article  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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. 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)

    Article  MathSciNet  MATH  Google Scholar 

  13. Maddalena, F., Solimini, S.: Blow-up techniques and regularity near the boundary for free discontinuity problems. Adv. Nonlinear Stud. 1(2) (2001)

  14. Mumford D., Shah J.: Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42(5), 577–685 (1989)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, 8057, Zürich, Switzerland

    Camillo De Lellis

  2. Dipartimento di Matematica “U. Dini”, Università di Firenze, V.le Morgagni 67/A, 50134, Firenze, Italy

    Matteo Focardi

Authors
  1. Camillo De Lellis
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  2. Matteo Focardi
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Correspondence to Matteo Focardi.

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Lellis, C.D., Focardi, M. Density lower bound estimates for local minimizers of the 2d Mumford–Shah energy. manuscripta math. 142, 215–232 (2013). https://doi.org/10.1007/s00229-012-0601-y

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  • DOI: https://doi.org/10.1007/s00229-012-0601-y

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