Extension theorems for external cusps with minimal regularity

Abstract

Sobolev functions defined on certain simple domains with an isolated singular point (such as power type external cusps) can not be extended in standard, but in appropriate weighted spaces. In this article we show that this result holds for a large class of domains that generalizes external cusps, allowing minimal boundary regularity. The construction of our extension operator is based on a modification of reflection techniques originally developed for dealing with uniform domains. The weight involved in the extension appears as a consequence of the failure of the domain to comply with basic properties of uniform domains, and it turns out to be a quantification of that failure. We show that weighted, rather than standard spaces, can be treated with our approach for weights that are given by a monotonic function either of the distance to the boundary or of the distance to the tip of the cusp.

Gromov hyperbolicity and inner uniformity

“…Proof. Since [76, Lemma 6.34 (1)] implies that the conditions ( 1) and ( 2) in the lemma are quantitatively equivalent, and since the implication from (3) to ( 1) is obvious, we see that we only need to show the implication from (1) to (3). To this end, we let w ∈ X.…”

“…Motivated by these ideas, Bonk, Heinonen and Koskela introduced the concept of a uniform metric space in [14] and established a fundamental two-way correspondence between this class of spaces and proper, geodesic and Gromov hyperbolic spaces. Since its introduction, the concept of the uniformity has played a significant role in the study of geometric function theory in metric spaces, such as the properties of quasiconformal and quasimöbius mappings of metric spaces [22,46,78], the theory of Lipschitz and quasiconformal mappings of Carnot groups (including Heisenberg groups) [26,29], boundary behavior and boundary extensions of Sobolev functions [1,12,30,50], the Martin boundary in potential theory [2], the boundary Harnack principle for second order elliptic partial differential equations [53,54], and even the Brownian motion in probability theory [5,27].…”

Korn's inequalities for generalized external cusps

“…Indeed, first show that a cylindrical set J of the form , with b − a ∼ diam (Ω), is a John domain with constants given by those of ϖ . Then, conclude by observing that Ω j almost agrees with a dilatation of J (actually it is possible to construct a bi‐Lipschitz mapping F such that F (Ω j ) = ℓ i J , see ).…”

“…Finally, before proceeding, let us mention that the extension procedure for external cusps used in can be appropriately combined with the extension operator for uniform domains constructed in . In that way, it is possible to obtain similar results to those given here.…”

“…In that way, it is possible to obtain similar results to those given here. However, by doing that, the notion of ‘locally John’ should be replaced by the notion of ‘locally uniform’ proposed in . This allows to handle a less general class of external cusps than that treated here.…”