A necessary condition for Sobolev extension domains in higher dimensions

Abstract

We give a necessary condition for a domain to have a bounded extension operator from 𝐿1,𝑝(𝛺) to 𝐿1,𝑝(R𝑛) for the range 1 < 𝑝 < 2. The condition is given in terms of a power of the distance to the boundary of 𝛺 integrated along the measure theoretic boundary of a set of locally finite perimeter and its extension. This generalizes a characterizing curve condition for planar simply connected domains, and a condition for 𝑊1,1-extensions. We use the necessary condition to give a quantitative version of the curve condition. We also construct an example of an extension domain in R3 that is homeomorphic to a ball and has 3-dimensional boundary.