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Dimensional lower bounds for contact surfaces of Cheeger sets

Profile image of Marco CarocciaMarco Caroccia

2021, Journal de Mathématiques Pures et Appliquées

https://doi.org/10.1016/J.MATPUR.2021.11.010
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35 pages

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Abstract

We carry out an analysis of the size of the contact surface between a Cheeger set E and its ambient space Ω ⊂ R d . By providing bounds on the Hausdorff dimension of the contact surface ∂E ∩ ∂Ω, we show a fruitful interplay between this size itself and the regularity of the boundaries. Eventually, we obtain sufficient conditions to infer that the contact surface has positive (d -1) dimensional Hausdorff measure. Finally we prove by explicit examples in two dimensions that such bounds are optimal.

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Related topics

  • Mathematics
  • Applied Mathematics
  • Combinatorics
  • Pure Mathematics
  • Hausdorff Dimension
  • Hausdorff space
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