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On the Measure and the Structure of the Free Boundary of the Lower Dimensional Obstacle Problem

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A Correction to this article was published on 04 July 2018

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Abstract

We provide a thorough description of the free boundary for the lower dimensional obstacle problem in \({\mathbb{R}^{n+1}}\) up to sets of null \({\mathcal{H}^{n-1}}\) measure. In particular, we prove

  1. (i)

    local finiteness of the (n−1)-dimensional Hausdorff measure of the free boundary,

  2. (ii)

    \({\mathcal{H}^{n-1}}\)-rectifiability of the free boundary,

  3. (iii)

    classification of the frequencies up to a set of Hausdorff dimension at most (n−2) and classification of the blow-ups at \({\mathcal{H}^{n-1}}\) almost every free boundary point.

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Change history

  • 04 July 2018

    One of the propositions of the paper on the classification of homogeneous solutions was found to be incorrect.

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Authors and Affiliations

  1. DiMaI, Università degli Studi di Firenze, Viale Morgagni 67/A, 50134, Florence, Italy

    Matteo Focardi

  2. Universität Leipzig, Augustusplatz 10, 04109, Leipzig, Germany

    Emanuele Spadaro

  3. Università di Roma La Sapienza, P.le Aldo Moro 5, 00185, Rome, Italy

    Emanuele Spadaro

Authors
  1. Matteo Focardi
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  2. Emanuele Spadaro
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Correspondence to Matteo Focardi.

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Communicated by A. Figalli

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Focardi, M., Spadaro, E. On the Measure and the Structure of the Free Boundary of the Lower Dimensional Obstacle Problem. Arch Rational Mech Anal 230, 125–184 (2018). https://doi.org/10.1007/s00205-018-1242-4

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