Planar Sobolev extension domains

Abstract

This doctoral thesis deals with geometric characterizations of bounded planar simply connected Sobolev extension domains. It consists of three papers. In the first and third papers we give full geometric characterizations of W 1, p-extension domains for 1 < p < 2 and p = 1, respectively. The second paper establishes a density result for Sobolev functions on planar domains, necessary for the solution for the case p = 1. Combining with the known results, we obtain a full geometric characterization of W 1, p-extension domains for every 1 ≤ p ≤ ∞.