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Monotonicity Formula and Regularity for General Free Discontinuity Problems

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Abstract

We give a general monotonicity formula for local minimizers of free discontinuity problems which have a critical deviation from minimality, of order d − 1. This result allows us to prove partial regularity results (that is closure and density estimates for the jump set) for a large class of free discontinuity problems involving general energies associated to the jump set, as for example free boundary problems with Robin conditions. In particular, we give a short proof to the De Giorgi–Carriero–Leaci result for the Mumford–Shah functional.

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Authors and Affiliations

  1. Laboratoire de Mathématiques, CNRS UMR 5127, Université de Savoie, Campus Scientifique, 73376, Le-Bourget-Du-Lac, France

    Dorin Bucur

  2. Universität Leipzig, Fakultät für Mathematik und Informatik, Mathematisches Institut, Postfach 100 920, 04009, Leipzig, Germany

    Stephan Luckhaus

Authors
  1. Dorin Bucur
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  2. Stephan Luckhaus
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Correspondence to Dorin Bucur.

Additional information

Communicated by G. Dal Maso

The work of D.B. was supported by the programme “Free Boundary Problems, Theory and Applications”, MSRI Berkeley, January–May 2011 and ANR-09-BLAN-0037. The work of S.L. was supported by the programme “Free Boundary Problems, Theory and Applications”, MSRI Berkeley, January–May 2011.

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Bucur, D., Luckhaus, S. Monotonicity Formula and Regularity for General Free Discontinuity Problems. Arch Rational Mech Anal 211, 489–511 (2014). https://doi.org/10.1007/s00205-013-0671-3

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  • DOI: https://doi.org/10.1007/s00205-013-0671-3

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