Suppose that $f$ is a $K$-quasiconformal self-mapping of the unit disk $\mathbb{D}$, which satisfies the following: $(1)$ the biharmonic equation $Δ(Δf)=g$ $(g\in \mathcal{C}(\overline{\mathbb{D}}))$, (2) the boundary condition $Δf=φ$ ($φ\in\mathcal{C}(\mathbb{T})$ and $\mathbb{T}$ denotes the unit circle), and $(3)$ $f(0)=0$. The purpose of this paper is to prove that $f$ is Lipschitz continuos, and, further, it is bi-Lipschitz continuous when $\|g\|_{\infty}$ and $\|φ\|_{\infty}$ are small enough. Moreover, the estimates are asymptotically sharp as $K\to 1$, $\|g\|_{\infty}\to 0$ and $\|φ\|_{\infty}\to 0$, and thus, such a mapping $f$ behaves almost like a rotation for sufficiently small $K$, $\|g\|_{\infty}$ and $\|φ\|_{\infty}$.
The main aim of this paper is to study the Lipschitz continuity of certain $(K, K')$-quasiconformal mappings with respect to the distance ratio metric, and the Lipschitz continuity of the solution of a quasilinear differential equation with respect to the distance ratio metric.
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.
We establish that every $K$-quasiconformal mapping of $w$ of the unit disk $\ID$ onto a $C^2$-Jordan domain $Ω$ is Lipschitz provided that $Δw\in L^p(\ID)$ for some $p>2$. We also prove that if in this situation $K\to 1$ with $\|Δw\|_{L^p(\ID)}\to 0$, and $Ω\to \ID$ in $C^{1,α}$-sense with $α>1/2,$ then the bound for the Lipschitz constant tends to $1$. In addition, we provide a quasiconformal analogue of the Smirnov absolute continuity result over the boundary.
The purpose of this paper is to explore conditions which guarantee Lipschitz-continuity of harmonic maps w.r.t. quasihyperbolic metrics. For instance, we prove that harmonic quasiconformal maps are Lipschitz w.r.t. quasihyperbolic metrics.
In this article, we first discuss the Lipschitz characteristic and the linear measure distortion of $K$-quasiconformal harmonic mappings. Then we give some characterizations of the radial John disks with the help of Pre-Schwarzian of harmonic mappings.
A quasiplane $f(V)$ is the image of an $n$-dimensional Euclidean subspace $V$ of ${\Bbb R}^N$ ($1\leq n\leq N-1$) under a quasiconformal map $f:{\Bbb R}^N\to{\Bbb R}^N$ . We give sufficient conditions in terms of the weak quasisymmetry constant of the underlying map for a quasiplane to be a bi-Lipschitz $n$-manifold and for a quasiplane to have big pieces of bi-Lipschitz images of ${\Bbb R}^n$. One main novelty of these results is that we analyze quasiplanes in arbitrary codimension $N-n$. To establish the big pieces criterion, we prove new extension theorems for "almost affine" maps, which ar…
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.
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.
In this article, we continue our investigations of the boundary behavior of harmonic mappings. We first discuss the classical problem on the growth of radial length and obtain a sharp growth theorem of the radial length of $K$-quasiconformal harmonic mappings. Then we present an alternate characterization of radial John disks. In addition, we investigate the linear measure distortion and the Lipschitz continuity on $K$-quasiconformal harmonic mappings of the unit disk onto a radial John disk. Finally, using Pommerenke interior domains, we characterize certain differential properties of $K$-qua…
Suppose that $f$ is a $K$-quasiconformal ($(K,K')$-quasiconformal resp.) self-mapping of the unit disk $\mathbb{D}$, which satisfies the following: $(1)$ the inhomogeneous polyharmonic equation $Δ^{n}f=Δ(Δ^{n-1} f)=φ_{n}$ $(φ_{n}\in \mathcal{C}(\overline{\mathbb{D}}))$, (2) the boundary conditions $Δ^{n-1}f|_{\mathbb{T}}=φ_{n-1},~\ldots,~Δ^{1}f|_{\mathbb{T}}=φ_{1}$ ($φ_{j}\in\mathcal{C}(\mathbb{T})$ for $j\in\{1,\ldots,n-1\}$ and $\mathbb{T}$ denotes the unit circle), and $(3)$ $f(0)=0$, where $n\geq2$ is an integer and $K\geq1$ ($K'\geq0$ resp.). The main aim of this paper is to prove that $f…
We study the mean radius growth function for quasiconformal mappings. We give a new sub-class of quasiconformal mappings in $\mathbb{R}^n$, for $n\geq 2$, called bounded integrable parameterization mappings, or BIP maps for short. These have the property that the restriction of the Zorich transform to each slice has uniformly bounded derivative in $L^{n/(n-1)}$. For BIP maps, the logarithmic transform of the mean radius function is bi-Lipschitz. We then apply our result to BIP maps with simple infinitesimal spaces to show that the asymptotic representation is indeed quasiconformal by showing t…
In \cite{kamz} the author proved that every quasiconformal harmonic mapping between two Jordan domains with $C^{1,α}$, $0<α\le 1$, boundary is bi-Lipschitz, providing that the domain is convex. In this paper we avoid the restriction of convexity. More precisely we prove: any quasiconformal harmonic mapping between two Jordan domains $Ω_j$, $j=1,2$, with $C^{j,α}$, $j=1,2$ boundary is bi-Lipschitz.
This article contains several results for λ-Robertson functions, i.e., analytic functions $f$ defined on the unit disk $D$ satisfying $f(0) = f'(0)-1=0$ and $Re e^{-iλ} {1+zf"(z)/f'(z)} > 0$ in $D$, where $λε(-π/2,π/2)$. We will discuss about conditions for boundedness and quasiconformal extension of Robertson functions. In the last section we provide another proof of univalence for Robertson functions by using the theory of Löwner chains.
Let $\overline{\mathbb{D}}$ be the closure of the unit disk $\mathbb{D}$ in the complex plane $\mathbb{C}$ and $g$ be a continuous function in $\overline{\mathbb{D}}$. In this paper, we discuss some characterizations of elliptic mappings $f$ satisfying the Poisson's equation $Δf=g$ in $\mathbb{D}$, and then establish some sharp distortion theorems on elliptic mappings with the finite perimeter and the finite radial length, respectively. The obtained results are the extension of the corresponding classical results.
We study $C^2$ extremal quasiconformal mappings in space and establish necessary and sufficient conditions for a `localized' form of extremality in the spirit of the work of G. Aronsson on absolutely minimizing Lipschitz extensions. We also prove short time existence for smooth solutions of a gradient flow of QC diffeomorphisms associated to the extremal problem.
It is known for some time that there exists an energy-minimal diffeomorphism between two doubly-connected domains $Ω$ and $D$ provided that $\mathrm{Mod}(Ω)\le \mathrm{Mod}{D}$ and that diffeomorphism is harmonic \cite{tedi}. In this note we give a short proof of the fact that for given annuli $Ω$ and $D$ satisfying the condition $\mathrm{Mod}(Ω)\le \mathrm{Mod}(D)$ there exist a $K-$quasiconformal harmonic diffeomorphism $f:Ω\onto D$, where $K=K(τ)$, $τ=\mathrm{Mod}(Ω)/ \mathrm{Mod}(D)$ and $\lim_{τ\to 1}K(τ)=1$.
If $Ω$ is a simply connected domain in $\overline{\mathbb C}$ then, according to the Ahlfors-Gehring theorem, $Ω$ is a quasidisk if and only if there exists a sufficient condition for the univalence of holomorphic functions in $Ω$ in relation to the growth of their Schwarzian derivative. We extend this theorem to harmonic mappings by proving a univalence criterion on quasidisks. We also show that the mappings satisfying this criterion admit a homeomorphic extension to $\overline{\mathbb C}$ and, under the additional assumption of quasiconformality in $Ω$, they admit a quasiconformal extension …
We first investigate the Lipschitz continuity of $(K, K')$-quasiregular $C^2$ mappings between two Jordan domains with smooth boundaries, satisfying certain partial differential inequalities concerning Laplacian. Then two applications of the obtained result are given: As a direct consequence, we get the Lipschitz continuity of $ρ$-harmonic $(K, K')$-quasiregular mappings, and as the other application, we study the Lipschitz continuity of $(K,K')$-quasiconformal self-mappings of the unit disk, which are the solutions of the Poisson equation $Δw=g$. These results generalize and extend several re…
In this paper, we establish a three circles type theorem, involving the harmonic area function, for harmonic mappings. Also, we give bounds for length and area distortion for harmonic quasiconformal mappings. Finally, we will study certain Lipschitz-type spaces on harmonic mappings.
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…
We demonstrate that the complex plane and a class of generalized Grushin planes $G_r$, where $r$ is a function satisfying specific requirements, are quasisymmetrically equivalent. Then using conjugation we are able to develop an analytic definition of quasisymmetry for homeomorphisms on $G_r$ spaces. In the last section we show our analytic definition of quasisymmetry is consistent with earlier notions of conformal mappings on the Grushin plane. This leads to several characterizations of conformal mappings on the generalized Grushin planes.
We will prove a global estimate for the gradient of the solution to the {\it Poisson differential inequality} $|Δu(x)|\le a|\nabla u(x)|^2+b$, $x\in B^{n}$, where $a,b<\infty$ and $u|_{S^{n-1}}\in C^{1,α}(S^{n-1}, \Bbb R^m)$. If $m=1$ and $a\le (n+1)/(|u|_\infty4n\sqrt n)$, then $|\nabla u| $ is a priori bounded. This generalizes some similar results due to E. Heinz (\cite{EH}) and Bernstein (\cite{BS}) for the plane. An application of these results yields the theorem, which is the main result of the paper: A quasiconformal mapping of the unit ball onto a domain with $C^2$ smooth boundary, sat…
We prove that every quasiconformal mapping from the harmonic $β$-Bloch space between the unit ball and a spatial domain with $C^1$ boundary is globally $α$-Hölder continuous for $α<1-β$, with the Hölder coefficient that does not depend neither on the mapping nor on $β$. An analogous result also holds for Lipschitz continuous, quasiconformal harmonic mappings for $α<1$. This extends some results from the complex plane obtained by Warschawski in \cite{Warschawski} for conformal mappings and Kalaj in \cite{Kalaj6} for quasiconformal harmonic mappings.
Using normal family arguments, we show that the degree of the first nonzero homogenous polynomial in the expansion of $n$ dimensional Euclidean harmonic $K$-quasiconformal mapping around an internal point is odd, and that such a map from the unit ball onto a bounded convex domain, with $K< 3^{n-1}$, is co-Lipschitz. Also some generalizations of this result are given, as well as a generalization of Heinz's lemma for harmonic quasiconformal maps in $\mathbb R^n$ and related results.
Given a bounded domain $D \subset {\mathbb R}^n$ strictly starlike with respect to $0 \in D\,,$ we define a quasi-inversion w.r.t. the boundary $\partial D \,.$ We show that the quasi-inversion is bi-Lipschitz w.r.t. the chordal metric if and only if every "tangent line" of $\partial D$ is far away from the origin. Moreover, the bi-Lipschitz constant tends to $1,$ when $\partial D$ approaches the unit sphere in a suitable way. For the formulation of our results we use the concept of the $α$-tangent condition due to F. W. Gehring and J. Väisälä (Acta Math. 1965). This condition is shown to be e…
A quantitative version of an inequality obtained in \cite[Theorem~2.1]{mathz} is given. More precisely, for normalized $K$ quasiconformal harmonic mappings of the unit disk onto a Jordan domain $Ω\in C^{1,μ} $ ($0<μ\le 1$) we give an explicit Lipschitz constant depending on the structure of $Ω$ and on $K$. In addition we give a characterization of q.c. harmonic mappings of the unit disk onto an arbitrary Jordan domain with $C^{2,α}$ boundary in terms of boundary function using the Hilbert transformations. Moreover it is given a sharp explicit quasiconformal constant in terms of the boundary fu…
We prove the following theorem: every quasiconformal harmonic mapping between two plane domains with $C^{1,α}$ ($α<1$), respectively $C^{1,1}$ compact boundary is bi-Lipschitz. The distance function with respect to the boundary of the image domain is used. This in turn extends a similar result of the author in \cite{kalajan} for Jordan domains, where stronger boundary conditions for the image domain were needed.
We obtain a complete description of closed ideals in weighted Lipschitz algebras $Λ_ω$ of analytic functions on the unit disk satisfying the following condition $$\frac{|f(z)-f(w)|}{ω(|z-w|)}=o(1)\qquad(as |z-w| \longrightarrow 0),$$ where $ω$ is a modulus of continuity satisfying some regularity conditions.
We mainly investigate some properties of quasiconformal mappings between smooth 2-dimensional surfaces with boundary in the Euclidean space, satisfying certain partial differential equations (inequalities) concerning Laplacian, and in particular satisfying Laplace equation and show that that these mappings are Lipschitz. Conformal parametrization of such surfaces and the method developed in our paper \cite{km} have important role in this paper.dan curves and is extended to the case of $C^{2,α}$ surfaces with smooth and compact boundary.
Suppose that $f$ satisfies the following: $(1)$ the polyharmonic equation $Δ^{m}f=Δ(Δ^{m-1} f)$$=φ_{m}$ $(φ_{m}\in \mathcal{C}(\overline{\mathbb{B}^{n}},\mathbb{R}^{n}))$, (2) the boundary conditions $Δ^{0}f=φ_{0},Δ^{1}f=φ_{1},~\ldots,~Δ^{m-1}f=φ_{m-1}$ on $\mathbb{S}^{n-1}$ ($φ_{j}\in \mathcal{C}(\mathbb{S}^{n-1},\mathbb{R}^{n})$ for $j\in\{0,1,\ldots,m-1\}$ and $\mathbb{S}^{n-1}$ denotes the boundary of the unit ball $\mathbb{B}^{n}$), and $(3)$ $f(0)=0$, where $n\geq3$ and $m\geq1$ are integers. Initially, we prove a Schwarz type lemma and use it to obtain a Heinz type inequality of mapping…
We provide sufficient conditions so that a homeomorphism of the real line or of the circle admits an extension to a mapping of finite distortion in the upper half-plane or the disk, respectively. Moreover, we can ensure that the quasiconformal dilatation of the extension satisfies certain integrability conditions, such as $p$-integrability or exponential integrability. Mappings satisfying the latter integrability condition are also known as David homeomorphisms. Our extension operator is the same as the one used by Beurling and Ahlfors in their celebrated work. We prove an optimal bound for th…
Let $Γ$ be a Lipschitz curve on the complex plane $\mathbb{C}$ and $Ω_+$ is the domain above $Γ$, we define Hardy space $H^p(Ω_+)$ as the set of holomorphic functions $F$ satisfying $\sup_{τ>0}(\int_Γ |F(ζ+\mathrm{i}τ)|^p |\,\mathrm{d}ζ|)^{\frac1p}< \infty$. We mainly focus on the case of $1<p<\infty$ in this paper, and prove that if $F(w)\in H^p(Ω_+)$, then $F(w)$ has non-tangential boundary limit $F(ζ)$ a.e. on $Γ$, and $F(w)$ is the Cauchy integral of $F(ζ)$. We denote the conformal mapping from $\mathbb{C}_+$ onto $Ω_+$ as $Φ$, and then prove that, $ H^p(Ω_+)$ is isomorphic to $H^p(\mathbb…
Let $0<p<\infty$, $Γ$ be a Lipschitz curve on the complex plane~$\mathbb{C}$ and $Ω_+$ is the domain above $Γ$, we define Hardy space $H^p(Ω_+)$ as the set of analytic functions $F$ satisfying $\sup_{τ>0}(\int_Γ |F(ζ+\mathrm{i}τ)|^p |\,\mathrm{d}ζ|)^{\frac1p}< \infty$. We denote the conformal mapping from $\mathbb{C}_+$ onto $Ω_+$ as $Φ$, and prove that, $H^p(Ω_+)$ is isomorphic to $H^p(\mathbb{C}_+)$, the classical Hardy space on the upper half plane~$\mathbb{C}_+$, under the mapping $T\colon F\to F(Φ)\cdot (Φ')^{\frac1p}$. Besides, $T$ and $T^{-1}$ are both bounded. We also prove that if $F(…
It is established a series of criteria for continuous and homeomorphic extension to the boundary of the so-called lower $Q$-homeomorphisms $f$ between domains in $\overline{\Rn}=\Rn\cup\{\infty\}$, $n\geqslant2$, under integral constraints of the type $\intΦ(Q^{n-1}(x))\,dm(x)<\infty$ with a convex non-decreasing function $Φ:[0,\infty]\to[0,\infty]$. It is shown that integral conditions on the function $Φ$ found by us are not only sufficient but also necessary for a continuous extension of $f$ to the boundary. It is given also applications of the obtained results to the mappings with finite ar…
