Logarithmic Holder continuous mappings and Beltrami equation

Hölder and Lipschitz continuity in Orlicz-Sobolev classes, distortion and harmonic mappings

In this article, we consider the Hölder continuity of injective maps in Orlicz-Sobolev classes defined on the unit ball. Under certain conditions on the growth of dilatations, we obtain the Hölder continuity of the indicated class of mappings. In particular, under certain special restrictions, we show that Lipschitz continuity of mappings holds. We also consider Hölder and Lipschitz continuity of harmonic mappings and in particular of harmonic mappings in Orlicz-Sobolev classes. In addition in planar case, we show in some situations that the map is bi-Lipschitzian if Beltrami coefficient is Hö…

On the Riemann-Hilbert problem for the Beltrami equation

It is developed the theory of the Dirichlet problem for harmonic functions. On this basis, for the nondegenerate Beltrami equations in the quasidisks and, in particular, in the smooth domains, it is proved the existence of regular solutions of the Riemann-Hilbert problem with coefficients of bounded variation and boundary data that are measurable with respect to the absolute harmonic measure (logarithmic capacity). Moreover, it is shown that the dimension of the spaces of the given solutions is infinite. One more section was added for the case of coefficients of countably bounded variation.

A lower bound of the hyperbolic dimension for meromorphic functions having a logarithmic Hölder tract

We improve existing lower bounds of the hyperbolic dimension for meromophic functions that have a logarithmic tract Ω which is a Hölder domain. These bounds are given in terms of the fractal behavior, measured with integral means, of the boundary of Ω at infinity.

On Hilbert problem for Beltrami equations in quasihyperbolic domains

It is studied the Hilbert boundary value problem for the nondegenerate Beltrami equations in domains $D$ of the complex plane $\mathbb C$ with the so--called quasihyperbolic boundary condition. It is proved the existence of solutions of this problem with coefficients of countably bounded variation and with arbitrary boundary data that are measurable with respect to logarithmic capacity. It is shown that the dimension of the spaces of such solutions is infinite. It is also given applications to the Dirichlet, Neumann and Poincare boundary value problems for analogs of the Laplace equation in an…

To the theory of semi-linear Beltrami equations

The present paper is devoted to the study of semi-linear Beltrami equations which are closely relevant to the corresponding semi-linear Poisson type equations of mathematical physics on the plane in anisotropic and inhomogeneous media. In its first part, applying completely continuous ope\-ra\-tors by Ahlfors-Bers and Leray--Schauder approach, we prove existence of regular solutions of the semi-linear Beltrami equations with no boundary conditions. Moreover, here we derive their representation through solutions of the Vekua type equations and generalized analytic functions with sources. As con…

Variation of quasiconformal mappings on lines

We obtain improved regularity of homeomorphic solutions of the reduced Beltrami equation, as compared to the standard Beltrami equation. Such an improvement is not possible in terms of Holder or Sobolev regularity; instead, our results concern the generalized variation of restrictions to lines. Specifically, we prove that the restriction to any line segment has finite p-variation for all p>1 but not necessarily for p=1.

Hölder regularity of generic manifold

In this paper we study Hölder continuity of the pluricomplex Green function with logarithmic growth at infinity of a smooth generic submanifold of $\C^n$. In particular we prove that the pluricomplex Green function of any $C^2$-smooth generic compact submanifold of $\C^n$ (without boundary) is Lipschitz continuous in $\C^n$.

无源带标架Beltrami-Vekua正规形的模空间

Moduli of the Sourceless Framed Beltrami-Vekua Normal Form

本研究分析无源带标架Beltrami-Vekua方程在变换下的模空间，给出无零点情形的双曲质量密度与相位曲率流分类及不变量范围，并将分类扩展至带零点场的相位可积扇区，构造了等质量等电荷的不等价方程实例。

非齐次Cauchy-Riemann方程在开放Riemann表面族上的研究

The nonhomogeneous Cauchy-Riemann equation on families of open Riemann surfaces

本文通过非齐次Beltrami方程解决非齐次Cauchy-Riemann方程问题，并应用该结果得到复线丛在开放Riemann表面族上的Oka-Grauert原理。

On boundary value problems for inhomogeneous Beltrami equations

We give here results on the existence of nonclassical solutions of the Hilbert boundary value problem in terms of the so-called angular limits (along nontangent curves to the boundary) for Beltrami equations with sources in Jordan domains satisfying the quasihyperbolic boundary condition by Gehring-Martio, generally speaking, without $(A)-$condition by Ladyzhenskaya-Ural'tseva and, in particular, without the known outer cone condition that were standard for boundary value problems in the PDE theory. Assuming that the coefficients of the problem are functions of countable bounded variation and …

关于一类映射到具有局部拟共形边界区域上的行为

On behavior of one class of mappings acting onto domains with a locally quasiconformal boundary

本文研究满足逆 Poletsky 不等式的映射，在多类具有局部拟共形或正则边界的域之间建立了边界点邻域内的 Hölder 对数连续性。

论广义拟共形映射理论

On the theory of generalized quasiconformal mappings

本文在逆 Poletsky 不等式框架下研究广义拟共形映射，分析其局部与边界行为，并证明该映射类具有对数 Hölder 连续性。

扩张 Thurston 映射的素轨道定理：强不可积条件的通有性

Prime orbit theorems for expanding Thurston maps: Genericity of strong non-integrability condition

本文证明在扩张 Thurston 映射的素轨道定理中，α-强不可积条件在 α-Hölder 连续函数类中具有通有性，从而完善了该系列关于素数定理类比的研究。