A Sobolev Extension Domain That is Not Uniform

Abstract

Abstract. In this paper we construct a Sobolev extension domain which, together with its complement, is topologically as nice as possible and yet not uniform. This shows that the well known implication that Uniform ⇒ Sobolev extension can not be reversed under strongest possible topological conditions.

Elliptic and parabolic regularity for second‐ order divergence operators with mixed boundary conditions

“…4.2], compare also for further information. Although the uniformness property is not necessary for a domain to be a Sobolev extension domain , it seems presently to be the broadest class of domains for which this extension property holds – at least if one aims at all p ∈]1, ∞ [. For example, it contains Koch's snowflake, compare with .…”

“…Although the uniformness property is not necessary for a domain to be a Sobolev extension domain (see [64]) it seems presently to be the broadest class of domains for which this extension property holds -at least if one aims at all p ∈ ]1∞[. E.g.…”

Hardy’s Inequality for Functions Vanishing on a Part of the Boundary

“…5.11] for a sketch of proof. (ii) Although the uniformity property is not necessary for a domain to be a Sobolev extension domain [49] it seems presently to be the broadest known class of domains for which this extension property holds -at least if one aims at all p ∈ ]1, ∞[. For example Koch's snowflake is an (ε, δ)-domain [21].…”