Dimension estimates for the boundary of planar Sobolev extension domains

TL;DR

This paper establishes a nearly optimal upper bound on the boundary dimension of planar Sobolev extension domains, linking geometric boundary properties to functional extension capabilities.

Contribution

It introduces a sharp dimension estimate for the boundary of Sobolev extension domains based on weak mean porosity, with examples demonstrating the estimate's optimality.

Findings

Boundary dimension upper-bound is asymptotically sharp.

Weak mean porosity characterizes boundary regularity for extension domains.

Examples confirm the sharpness of the dimension estimate.

Abstract

We prove an asymptotically sharp dimension upper-bound for the boundary of bounded simply-connected planar Sobolev $W^{1,p}$-extension domains via the weak mean porosity of the boundary. The sharpness of our estimate is shown by examples.