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Higher Integrability for Minimizers of the Mumford–Shah Functional

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Abstract

We prove higher integrability for the gradient of local minimizers of the Mumford–Shah energy functional, providing a positive answer to a conjecture of De Giorgi (Free discontinuity problems in calculus of variations. Frontiers in pure and applied mathematics, North-Holland, Amsterdam, pp 55–62, 1991).

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References

  1. Ambrosio L., Fusco N., Pallara D.: Partial regularity of free discontinuity sets. II. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 24(1), 39–62 (1997)

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

    Article  MATH  MathSciNet  Google Scholar 

  4. David, G.: Singular sets of minimizers for the Mumford–Shah functional. Progress in mathematics, Vol. 233. Birkhäuser Verlag, Basel, (2005)

  5. De Giorgi, E.: Free discontinuity problems in calculus of variations. Frontiers in pure and applied mathematics, pp. 55–62. North-Holland, Amsterdam, (1991)

  6. De Giorgi E., Carriero M., Leaci A.: Existence theorem for a minimum problem with free discontinuity set. Arch. Ration. Mech. Anal. 108(3), 195–218 (1989)

    Article  MATH  MathSciNet  Google Scholar 

  7. De Lellis, C., Focardi, M.: Higher integrability of the gradient for minimizers of the 2d Mumford–Shah energy. J. Math. Pures Appl. (to appear)

  8. Maddalena F., Solimini S.: Regularity properties of free discontinuity sets. Ann. Inst. H. Poincaré Anal. Non Linéaire 18(6), 675–685 (2001)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  9. Maddalena, F., Solimini, S.: Concentration and flatness properties of the singular set of bisected balls. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30 (2001)

  10. Maddalena, F., Solimini, S.: Concentration and flatness properties of the singular set of bisected balls. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3–4), 623–659 (2002)

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

    Article  MATH  MathSciNet  Google Scholar 

  12. Rigot, S.: Big pieces of C 1,α-graphs for minimizers of the Mumford–Shah functional. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29(2), 329–349 (2000)

    Google Scholar 

  13. Schoen R., Simon L.: A new proof of the regularity theorem for rectifiable currents which minimize parametric elliptic functionals. Indiana Univ. Math. J. 31(3), 415–434 (1982)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Hausdorff Center for Mathematics, Endenicher Allee 62, 53115, Bonn, Germany

    Guido De Philippis

  2. Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, TX, 78712, USA

    Alessio Figalli

Authors
  1. Guido De Philippis
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  2. Alessio Figalli
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Correspondence to Guido De Philippis.

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Communicated by A. Braides

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De Philippis, G., Figalli, A. Higher Integrability for Minimizers of the Mumford–Shah Functional. Arch Rational Mech Anal 213, 491–502 (2014). https://doi.org/10.1007/s00205-014-0729-x

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  • DOI: https://doi.org/10.1007/s00205-014-0729-x

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