The torus related Riemann problem

The Riemann jump problem is solved for analytic functions of several complex variables with the unit torus as the jump manifold. A well-posed formulation is given which does not demand any solvability conditions. The higher dimensional Plemelj-Sokhotzki formula for analytic functions in torus domains is established. The canonical functions of the Riemann problem for torus domains are represented and applied in order to construct solutions for both of the homogeneous and inhomogeneous problems. Thus contrary to earlier research the results are similar to the respective ones for just one variable. A connection between the Riemann and the Riemann-Hilbert boundary value problem for the unit polydisc is explained.     

Hyperanalytic Riemann boundary value problem on d-summable closed curves

We are interested in finding solvability conditions for the Riemann boundary value problems for hyperanalytic functions in a simply connected bounded open subset of the complex plane whose boundary is merely required to be a d-summable closed curve.     

The Riemann Hilbert Problem for General Elliptic Complex Equations of Fourth Order

§１．　ＣｏｎｄｉｔｉｏｎｓｆｏｒＧｅｎｅｒａｌＥｌｌｉｐｔｉｃＣｏｍｐｌｅｘＥｑｕａｔｉｏｎｓｏｆＦｏｕｒｔｈＯｒｄｅｒ　　ＬｅｔＤｂｅａｂｏｕｎｄｅｄｄｏｍａｉｎ,ｗｅｃｏｎｓｉｄｅｒｔｈｅｇｅｎｅｒａｌｅｌｌｉｐｔｉｃｃｏｍｐｌｅｘｅｑｕａｔｉｏｎｏｆｆｏｕｒｔｈｏｒｄｅｒｉｎｔｈｅｆｏｌｌｏｗｉｎｇｆｏｒｍｗｚ２ｚ２＝Ｆ（ｚ,ｗ,ｗｚ,ｗｚ,…,ｗｚ３,ｗｚ４,ｗｚ３ｚ,ｗｚ３ｚ,ｗｚ４）＋Ｇ（ｚ,ｗ,ｗｚ,ｗｚ,…,ｗｚ３）,Ｆ＝∑ｊ＋ｋ＝４（ｊ,ｋ）≠（２,２）Ｑｊ…     

Integration over non-rectifiable curves and Riemann boundary value problems

In this paper we introduce an alternative way of defining the curvilinear Cauchy integral over non-rectifiable curves in the case of complex functions of one complex variable. Especially the jump behavior on the boundary is considered. As an application, solvability conditions of the Riemann boundary value problem are derived on very weak conditions to the boundary. Besides the complex case the consideration can be extended to the theory of Douglis algebra valued functions.     

二阶复椭圆Riemann—Hilbert边值问题

讨论了一般二阶非线性椭圆复方程的Riemann-Hilbert边值问题,首先给出Riemann-Hilbert问题及其适应性的概念,其次给出改进后的边值问题解的表述并证明了它的可解性,最后导出原Rremann-Hilbert边值问题的可解条件。     

一阶椭圆型复方程组Riemann－Hilbert边值问题

本文以解析函数的边值问题Ｂ的解的存在性为基础，根据它们的先验估计式及利用参数开拓法，导出了满足条件Ｃ的多个复变量的一阶拟线性椭圆型复方程组的Ｒｉｅｍａｎｎ－Ｈｉｌｂｅｒｔ边值问题的可解条件，并给出了解的积分表达式．     

Koch曲线上的齐次Riemann边值问题

当L为典型的分形曲线一Koch曲线时,提出了Riemann边值问题,但在一般情况下,在Koch曲线上所做的Cauchy型积分无意义.当对已知函数G(z),g(z)增加一定的解析条件,同时利用一列Cauchy型积分的极限函数,对定义在Koch曲线上的齐次Riemann边值问题进行了讨论,并得到与经典解析函数边值问题相类似的结果.     

On the spectrum of an integro-differential equation arising in viscoelasticity theory

We study the spectral properties of a boundary value problem for an integro-differential equation arising in viscoelasticity theory. We prove that the spectrum of the boundary value problem is either completely real or contains, together with the real part, only finitely many complex eigenvalues. Bibliography: 8 titles.     

ON RIEMANN BOUNDARY VALUE PROBLEM FOR POLYANALYTIC FUNCTIONS ON THE REAL AXIS

In this article, Riemann boundary value problem with different factors for polyanalytic functions on the real axis is studied. The expression of solution and sufficient and necessary condition for solvability of the non-homogeneous Riemann boundary value problem are obtained.     

The Discontinuous Riemann–Hilbert Problem for Elliptic Complex Equations of First Order

In this article, we first transform the general uniformly elliptic systems of first order equations with certain conditions into the complex equations, and propose the discontinuous Riemann- Hilbert problem and its modified well-posedness for the complex equations. Then we give a priori estimates of solutions of the modified discontinuous Riemann-Hilbert problem for the complex equations and verify its solvability. Finally the solvability results of the original discontinuous Riemann-Hilbert boundary value problem can be derived. The discontinuous boundary value problem possesses many applications in mechanics and physics etc.