Schwarz problem for bi-analytic functions in multiply connected domains

The Schwarz problem for bi-analytic functions in unbounded circular multiply connected domains is considered. We combine constructive methods applied to boundary value problems for complex partial differential equations in simply connected domains and for the Riemann–Hilbert type problems in multiply connected domains. A general method is outlined and the case of doubly connected domains is discussed in details. Solution is obtained in the form of a series.     

关于具分形边界连通区域上的谱渐近及弱Weyl-Berry猜想

本文研究了连通分形鼓上的谱渐近，对满足“切口”条件的连通分形鼓以及一类自然连通的分形鼓，分别证明了弱Weyl－Berry猜想是成立的.     

Counting function asymptotics and the weak Weyl-Berry conjecture for connected domains with fractal boundaries

In this paper, we study the spectral asymptotics for connected fractal domains and Weyl-Berry conjecture. We prove, for some special connected fractal domains, the sharp estimate for second term of counting function asymptotics, which implies that the weak form of the Weyl-Berry conjecture holds for the case. Finally, we also study a naturally connected fractal domain, and we prove, in this case, the weak Weyl-Berry conjecture holds as well. Research partially supported by the Natural Science Foundation of China-and the Royal Society of London     

Multiply Connected Quadrature Domains and the Bergman Kernel Function

This paper derives general analytical formulae for the conformal maps from multiply connected circular preimage domains to multiply connected quadrature domains by considering the Bergman kernel functions of the preimage and target domains. The new formulae are expressed in terms of the Schottky–Klein prime function. They generalize, to the case of arbitrary connectivity, a formula relevant to doubly connected domains derived by Y. Avci in 1977. Submitted: September 17, 2007. Accepted: June 5, 2008.     

The Grötzsch and Teichmüller extremal problems on a Riemann surface

The Grötzsch and Teichmüller well-known extremal problems for the moduli of plane doubly connected domains are extended to the case of domains on Riemann surfaces.     

On an exterior boundary-value problem for the time-harmonic maxwell equations with boundary conditions for the normal components of the electric and magnetic field

We treat the time-harmonic Maxwell equations with the boundary condition (ν, E) = (ν, H) = 0 in an exterior multiply connected domain. A uniqueness result by Yee for the case of a simply connected domain is extended to multiply connected domains and existence is obtained by a boundary integral equation approach.     

Conjugate function method for numerical conformal mappings

We present a method for numerical computation of conformal mappings from simply or doubly connected domains onto so-called canonical domains, which in our case are rectangles or annuli. The method is based on conjugate harmonic functions and properties of quadrilaterals. Several numerical examples are given.     

Approximate solution of boundary integral equations for biharmonic problems in non-smooth domains

This paper deals with approximate solutions to integral equations arising in boundary value problems for the biharmonic equation in simply connected piecewise smooth domains. The approximation method considered demonstrates excellent convergence even in the case of boundary conditions discontinuous at corner points. In an application we obtain very accurate approximations for some characteristics of two-dimensional Stokes flow in non-smooth domains. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solution of a singular homogeneous Hilbert boundary-value problem for analytic function in multiply connected circular domain

We offer a new approach for solving the homogeneous Riemann-Hilbert boundary-value problem for analytic function in multiply connected circular domains. The approach is based on determination of analytic function in terms of known boundary values of its argument in a special case.     

Representation of extremal functions in the classes Eq in terms of Green's and Neumann's functions

We consider extremal problems in the classes E_q, q≥1, in finitely connected domains and we establish a representation of extremal functions in terms of Green's and Neumann's functions. The equations which we obtain constitute an extension to the multiply connected case of known representation equations for extremal functions in the classes H_q in the unit disk.     

Some inequalities involving eigenvalues of the Neumann Laplacian

This paper is concerned with the eigenvalues of the Neumann Laplacian on various classes of domains of given measure: simply‐connected Lipschitz planar domains, n‐sided planar polygons and smooth N‐dimensional domains. In each case, we consider some quantities involving low eigenvalues of the Neumann Laplacian for which we obtain new inequalities. Moreover, we sharpen a universal bound derived by M. Ashbaugh and R. Benguria for sum of reciprocal of Neumann eigenvalues. Our investigations make use of some properties of conformal mappings, Bessel functions, symmetric domains or some isoperimetric inequalities for moments of inertia. Copyright © 2013 John Wiley & Sons, Ltd.     

一类椭圆方程的边界唯一延拓性

本文主要讨论了一类二阶椭圆方程Lu=-△u+Vu=0的弱解在连通凸区域边界上的唯一延拓性,并且证明了文[4]中提出的猜测在本文的情形同样成立.     

Inverse boundary-value problems of cauchy type for harmonic functions

We apply two methods for solving the inverse boundary-value problem (the so-called problem (A)) in the Cauchy statement for an analytic function and an unknown curve ??. We obtain criteria for ?? to be the unit circle. We apply the proposed methods for solving a modified Hadamard example and generalize the obtained results for the case of doubly connected domains.     

Carathéodory Convergence and Harmonic Measure

We give several new characterizations of Carathéodory convergence of simply connected domains. We then investigate how different definitions of convergence generalize to the multiply-connected case.

     

Pre-Schwarz导数意义下区域之间的距离

Lehto曾用Schwarz导数定义了边界多于一点的两个单连通区域的Mbius等价类之间的"距离",并猜测它是一个距离.但最近Bozin和Markovic否定了这一猜想.一个自然的问题就是:在Pre-Schwarz导数意义相应情况如何?用Pre-Schwarz导数给出了边界多于一点的两个单连通区域的仿射等价类之间的"距离",并证明了这样定义的"距离"是一个伪距离,即使将其限制在由具有解析边界的单连通区域的仿射等价类空间上也是如此.     

L_p-theory of free boundary problems of magnetohydrodynamics in multi-connected domains

We present the \(L_p\) -theory of solvability of free boundary problems of magnetohydrodynamics of viscous incompressible fluids in multi-connected domains constructed in the paper Solonnikov (Interf Free Bound 14:569–603, 2012) for \(p=2\) . The case of simply connected domains is studied in Padula and Solonnikov (J Math Sci 178:313–344, 2011), Solonnikov ( \(L_p\) -theory of free boundary problems of magnetohydrodynamics in simply connected domains, submitted to AMS Translations).     

Weighted Schur-class functions and functional models

We generalize the class of contractive-valued functions analytic in the unit disk (Schur class) to the case of finitely connected domains with operator-valued weights on the boundary. The two fundamental decompositions (the inner-outer factorization and the decomposition into the orthogonal sum of the pure and unitary parts) are extended to our setting. This generalization allows one to use a uniform approach to functional models developed recently in papers of the author and D. Yakubovich. Moreover, it gives an alternative language to handle multi-valued functions in the multiply connected case. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 5, pp. 221–236, 2006.     

相通连续Domain的若干特征定理

首先引入了相通连续Domain的概念,利用主理想及连通闭集刻画了相通连续Domain; 其次考察了相通完备偏序集的定向完备化,得到了一些好的结果.