$\mathscr{C}^1$ 光滑 Jordan 域之间的调和拟共形映射

Quasiconformal harmonic mappings between Dini's smooth Jordan domains

Let $D$ and $Ω$ be Jordan domains with Dini's smooth boundaries and and let $f:D\mapsto Ω$ be a harmonic homeomorphism. The object of the paper is to prove the following result: If $f$ is quasiconformal, then $f$ is Lipschitz. This extends some recent results, where stronger assumptions on the boundary are imposed, and somehow is optimal, since it coincides with the best condition for Lipschitz behavior of conformal mappings in the plane and conformal parametrization of minimal surfaces.

Quasiconformal and HQC mappings between Lyapunov Jordan domains

Let $h$ be a quasiconformal (qc) mapping of the unit disk $\mathbb{U}$ onto a Lyapunov domain. We show that $h$ maps subdomains of Lyapunov type of $\mathbb{U}$, which touch the boundary of $\mathbb{U}$, onto domains of similar type. In particular if $h$ is a harmonic qc (hqc) mapping of $\mathbb{U}$ onto a Lyapunov domain, using it, we prove that $h$ is co-Lipschitz (co-Lip) on $\mathbb{U}$. This settles an open intriguing problem.

通过光滑拟共形映射实现Jordan曲线叶状结构的长度控制

Length control for foliations by Jordan curves via smooth quasiconformal maps

针对单连通域上Jordan曲线叶状结构的长度控制问题，构造光滑5-拟共形映射，以ε=2/157肯定回答了Bouchala与Koskela提出的p=2及附近的可积性问题。

Quasiconformal Jordan domains

We extend the classical Carathéodory extension theorem to quasiconformal Jordan domains $( Y, d_{Y} )$. We say that a metric space $( Y, d_{Y} )$ is a quasiconformal Jordan domain if the completion $\overline{Y}$ of $( Y, d_{Y} )$ has finite Hausdorff $2$-measure, the boundary $\partial Y = \overline{Y} \setminus Y$ is homeomorphic to $\mathbb{S}^{1}$, and there exists a homeomorphism $ϕ\colon \mathbb{D} \rightarrow ( Y, d_{Y} )$ that is quasiconformal in the geometric sense. We show that $ϕ$ has a continuous, monotone, and surjective extension $Φ\colon \overline{ \mathbb{D} } \rightarrow \ove…

Quasiconformal mappings and Hölder continuity

We establish that every $K$-quasiconformal mapping $w$ of the unit ball $\IB$ onto a $C^2$-Jordan domain $Ω$ is Hölder continuous with constant $α= 2-\frac{n}{p}$, provided that its weak Laplacean $Δw$ is in $ L^p(\IB)$ for some $n/2<p<n$. In particular it is Hölder continuous for every $0<α<1$ provided that $Δw\in L^n(\IB)$.

Invertible harmonic mappings beyond Kneser theorem and quasiconformal harmonic mappings

In this paper we extend Rado-Choquet-Kneser theorem for the mappings with Lipschitz boundary data and essentially positive Jacobian at the boundary, without restriction on the convexity of image domain. It is an extension of a recent result of Alessandrini and Nesi \cite{ale}. Some applications for the family of quasiconformal harmonic mappings between Jordan domains are given.