Archive for Rational Mechanics and Analysis

, Volume 230, Issue 1, pp 125–184 | Cite as

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

  • Matteo Focardi
  • Emanuele Spadaro


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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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.DiMaIUniversità degli Studi di FirenzeFlorenceItaly
  2. 2.Universität LeipzigLeipzigGermany
  3. 3.Università di Roma La SapienzaRomeItaly

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