On planar Sobolev Lpm-extension domains

Estimation of Sobolev-type embedding constant on domains with minimally smooth boundary using extension operator

In this paper, we propose a method for estimating the Sobolev-type embedding constant fromW1,q( ) to Lp( ) on a domain ⊂Rn (n = 2, 3, . . . ) with minimally smooth boundary (also known as a Lipschitz domain), where p ∈ (n/(n – 1),∞) and q = np/(n + p). We estimate the embedding constant by constructing an extension operator fromW1,q( ) toW1,q(Rn) and computing its operator norm. We also present...

Sobolev Extension Property for Tree-shaped Domains with Self-contacting Fractal Boundary

In this paper, we investigate the existence of extension operators fromW (Ω) toW (R) (1 < p < ∞) for a class of tree-shaped domains Ω with a self-similar fractal boundary previously studied by Mandelbrot and Frame. When the fractal boundary has no self-contact, the results of Jones imply that there exist such extension operators for all p ∈ [1,∞]. In the case when the fractal boundary self-inte...

On the Bi-sobolev Planar Homeomorphisms and Their Approximation

The first goal of this paper is to give a short description of the planar bi-Sobolev homeomorphisms, providing simple and self-contained proofs for some already known properties. In particular, for any such homeomorphism u : Ω → ∆, one has Du(x) = 0 for almost every point x for which Ju(x) = 0. As a consequence, one can prove that ∫

Planar Sobolev Homeomorphisms and Hausdorff Dimension Distortion

We investigate how planar Sobolev-Orlicz homeomorphisms map sets of Hausdorff dimension less than two. With the correct gauge functions the generalized Hausdorff measures of the image sets are shown to be zero.

Elliptic Equations in Weighted Sobolev Spaces on Unbounded Domains