ABOUT PLANAR (α, β)–ACCESSIBLE DOMAINS

Quasiconformally Homogeneous Planar Domains

In this paper we explore the ambient quasiconformal homogeneity of planar domains and their boundaries. We show that the quasiconformal homogeneity of a domain D and its boundary E implies that the pair (D,E) is in fact quasiconformally bi-homogeneous. We also give a geometric and topological characterization of the quasiconformal homogeneity of D or E under the assumption that E is a Cantor se...

Solutions of the Divergence and Analysis of the Stokes Equations in Planar Hölder-α Domains

If Ω ⊂ R is a bounded domain, the existence of solutions u ∈ H 0 (Ω) of divu = f for f ∈ L(Ω) with vanishing mean value, is a basic result in the analysis of the Stokes equations. In particular it allows to show the existence of a solution (u, p) ∈ H 0 (Ω) n × L(Ω), where u is the velocity and p the pressure. It is known that the above mentioned result holds when Ω is a Lipschitz domain and tha...

A Right Inverse of the Divergence for Planar Hölder-α Domains

Abstract. If Ω ⊂ R is a bounded domain, the existence of solutions u ∈ H 0 (Ω) n of divu = f for f ∈ L(Ω) with vanishing mean value, is a basic result in the analysis of the Stokes equations. In particular it allows to show the existence of a solution (u, p) ∈ H 0 (Ω) n × L(Ω), where u is the velocity and p the pressure. It is known that the above mentioned result holds when Ω is a Lipschitz do...

Remarks about Euclidean Domains

We will call (1) the d-inequality. Sometimes it is expressed in a different way: for nonzero a and b, if a|b then d(a) ≤ d(b). This is equivalent to the d-inequality, taking into account of course the different roles of a and b in the two descriptions. Examples of Euclidean domains are Z (with d(n) = |n|), F [T ] for any field F (with d(f) = deg f ; this example is the reason that one doesn’t a...

Reasoning about Uncertain Domains