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Monotonicity formulas for obstacle problems with Lipschitz coefficients

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Abstract

We prove quasi-monotonicity formulas for classical obstacle-type problems with energies being the sum of a quadratic form with Lipschitz coefficients, and a Hölder continuous linear term. With the help of those formulas we are able to carry out the full analysis of the regularity of free-boundary points following the approaches by Caffarelli (J Fourier Anal Appl 4(4–5), 383–402, 1998), Monneau (J Geom Anal 13(2), 359–389, 2003), and Weiss (Invent Math 138(1), 23–50, 1999).

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Acknowledgments

The first two authors have been supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

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Authors and Affiliations

  1. DiMaI “U. Dini”, Università di Firenze, V.le Morgagni 67/A, 50134, Florence, Italy

    M. Focardi

  2. Dipartimento di Matematica, Università di Pisa, L.go Pontecorvo 5, 56127, Pisa, Italy

    M. S. Gelli

  3. Max-Planck-Institut für Mathematik in den Naturwissenschaften, Inselstrasse 22, 04103, Leipzig, Germany

    E. Spadaro

Authors
  1. M. Focardi
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  2. M. S. Gelli
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  3. E. Spadaro
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Correspondence to M. Focardi.

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Communicated by L. Ambrosio.

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Focardi, M., Gelli, M.S. & Spadaro, E. Monotonicity formulas for obstacle problems with Lipschitz coefficients. Calc. Var. 54, 1547–1573 (2015). https://doi.org/10.1007/s00526-015-0835-0

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