Related papers: Sobolev homeomorphic extensions

We study the basic question of characterizing which boundary homeomorphisms of the unit sphere can be extended to a Sobolev homeomorphism of the interior in 3D space. While the planar variants of this problem are well-understood, completely…

We give a full characterization of embeddings of the unit circle that admit a Sobolev homeomorphic extension to the unit disk. As a direct corollary, we establish that for quasiconvex target domains $\mathbb Y$, any homeomorphism $\varphi…

We consider the planar unit disk $\mathbb D$ as the reference configuration and a Jordan domain $\mathbb Y$ as the deformed configuration, and study the problem of extending a given boundary homeomorphism $\varphi \colon \partial \mathbb D…

Given the planar unit disk as the source and a Jordan domain as the target, we study the problem of extending a given boundary homeomorphism as a Sobolev homeomorphism. For general targets, this Sobolev variant of the classical…

Very recently, it was proved that if the hyperbolic metric of a planar Jordan domain is $L^q$-integrable for some $q\in (1,\infty)$, then every homeomorphic parametrization of the boundary Jordan curve via the unit circle can be extended to…

Let X and Y be planar Jordan domains of the same finite connectivity, Y being inner chordarc regular (such are Lipschitz domains). Every homeomorphism h:X->Y in the Sobolev space $W^{1,2}$ extends to a continuous map between closed domains.…

Let $\mathbb{X}$ be a Jordan domain satisfying hyperbolic growth conditions. Assume that $\varphi$ is a homeomorphism from the boundary $\partial \mathbb{X}$ of $\mathbb{X}$ onto the unit circle. Denote by $h$ the harmonic diffeomorphic…

In this paper, we study the relationship between Sobolev extension domains and homogeneous Sobolev extension domains. Precisely, we obtain the following results. 1- Let $1\leq q\leq p\leq \infty$. Then a bounded $(L^{1, p}, L^{1,…

For each $m\ge 1$ and $p>2$ we characterize bounded simply connected Sobolev $L^m_p$-extension domains $\Omega\subset R^2$. Our criterion is expressed in terms of certain intrinsic subhyperbolic metrics in $\Omega$. Its proof is based on a…

We describe a class of Sobolev $W^k_p$-extension domains $\Omega\subset R^n$ determined by a certain inner subhyperbolic metric in $\Omega$. This enables us to characterize finitely connected Sobolev $W^1_p$-extension domains in $R^2$ for…

Each homeomorphic parametrization of a Jordan curve via the unit circle extends to a homeomorphism of the entire plane. It is a natural question to ask if such a homeomorphism can be chosen so as to have some Sobolev regularity. This…

We show that there exists a planar Jordan domains $\Omega$ with boundary of Hausdorff dimension $1$ such that, for any conformal maps $\varphi \colon \mathbb D \to \Omega$, any homeomorphic extension of $\varphi$ or $\varphi^{-1}$ to the…

We give a necessary condition for a domain to have a bounded extension operator from $L^{1,p}(\Omega)$ to $L^{1,p}(\mathbb R^n)$ for the range $1 < p < 2$. The condition is given in terms of a power of the distance to the boundary of…

Let X and Y be bounded multiply connected Lipschitz domains in \R^2. We consider the class H_p (X, Y) of homeomorphisms h : X -> Y in the Sobolev space W^{1,p} (X, \R^2). We prove that the weak and strong closures of H_p (X, Y), 2 \le p<…

We characterize bounded simply-connected planar $W^{1,p}$-extension domains for $1 < p <2$ as those bounded domains $\Omega \subset \mathbb R^2$ for which any two points $z_1,z_2 \in \mathbb R^2 \setminus \Omega$ can be connected with a…

We prove an asymptotically sharp dimension upper-bound for the boundary of bounded simply-connected planar Sobolev $W^{1,p}$-extension domains via the weak mean porosity of the boundary. The sharpness of our estimate is shown by examples.

Let $\Omega \subset \mathbb{R}^n$ be a bounded domain and $1 < p < \infty$. We characterize $(1,p)$-extension domains in terms of inequalities of Bourgain--Brezis--Mironescu type. More precisely, we show that $\Omega$ is a $(1,p)$-extension…

Let $X, Y \subset \mathbb{R}^n$ be Lipschitz domains, and suppose there is a homeomorphism $\varphi \colon \overline{X} \to \overline{Y}$. We consider the class of Sobolev mappings $f \in W^{1,n} (X, \mathbb{R}^n)$ with a strictly positive…

We investigate non-homeomorphic mappings of Riemannian surfaces of Sobolev class. We have obtained some estimates of distortion of moduli of families of curves. We have proved that, under some conditions, these mappings have a continuous…

Given a Moebius homeomorphism $f : \partial X \to \partial Y$ between boundaries of proper, geodesically complete CAT(-1) spaces $X,Y$, and a family of probability measures $\{ \mu_x \}_{x \in X}$ on $\partial X$, we describe a continuous…