A removability theorem for Sobolev functions and detour sets

Abstract

We study the removability of compact sets for continuous Sobolev functions. In particular, we focus on sets with infinitely many complementary components, called “detour sets”, which resemble the Sierpiński gasket. The main theorem is that if \(K \subset \mathbb {R}^n\) is a detour set and its complementary components are sufficiently regular, then K is \(W^{1,p}\)-removable for \(p>n\). Several examples and constructions of sets where the theorem applies are given, including the Sierpiński gasket, Apollonian gaskets, and Julia sets.

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Acknowledgements

The author would like to thank Mario Bonk for introducing him to the problem of removability, and for several fruitful discussions and explanations on the background of the problem and the proofs of previous results. Additional thanks go to Huy Tran for pointing out the connection of the problem to SLE, to Ville Tengvall for pointing out the reference [14], and to Pekka Koskela for a motivating discussion. The author is also grateful to Vasiliki Evdoridou, Malik Younsi, and the anonymous referee for their comments and corrections. This paper was written while the author was visiting University of Helsinki. He thanks the faculty and staff of the Department of Mathematics at the University of Helsinki for their hospitality.

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Correspondence to Dimitrios Ntalampekos.

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The author was partially supported by NSF grant DMS-1506099.

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Ntalampekos, D. A removability theorem for Sobolev functions and detour sets. Math. Z. 296, 41–72 (2020). https://doi.org/10.1007/s00209-019-02405-7

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Keywords

  • Removability
  • Sobolev functions
  • Hölder domains
  • Detour sets
  • Sierpiński gasket

Mathematics Subject Classification

  • Primary 46E35
  • Secondary 30C65