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.

33 pages, 8 figures; added references and figures, corrected typos, revised arguments in Section 7, results unchanged

References in corpus (3)

• On removable sets for Sobolev spaces in the plane
• Non-removability of the Sierpinski Gasket
• Non-removability of Sierpinski spaces

Cited by in corpus (2)

• On Dynamical Gaskets Generated by Rational Maps, Kleinian Groups, and Schwarz Reflections
• CNED sets: countably negligible for extremal distances