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The higher order Pompeiu integral operator providing a particular solution to the inhomogeneous polyanalytic equation provides via the higher order Cauchy–Pompeiu representation an attempt for solv...

This paper studies controllability of the Poisson equation on the unit disk in C subject to the third boundary condition when the control is imposed on the boundary. We use complex analytic methods to prove existence and uniqueness of the control when the parameter λ is a nonzero complex number but not a negative integer (not an eigenvalue). Otherwise, due to multiplicity of solutions to the underlying problem, when λ is a negative integer, controllability could only be obtained if proper additional conditions on the boundary are imposed.

The paper provides explicit eigenvalues and eigenfunctions of pluriharmonic functions for the third boundary condition on the unit polydiscs. It is shown that in the case of eigenvalue, for each eigenvalue, there are multiple eigenfunctions. Compatibility and solvability conditions are also studied for the case of inhomogeneous third boundary condition.

The eigenvalue problem of holomorphic functions on the unit disc for the third boundary condition with general coefficient is studied using Fourier analysis. With a general anti-polynomial coefficient a variable number of additional boundary conditions need to be imposed to determine the eigenvalue uniquely. An additional boundary condition is required to obtain a unique eigenvalue when the coefficient includes an essential singularity rather than a pole. In either case explicit solutions are derived.

The Riemann–Hilbert–Poincaré problem with general coefficient for the inhomogeneous Cauchy–Riemann equation on the unit disc is studied using Fourier analysis. It is shown that the problem is well posed only if the coeffcient is holomorphic. If the coefficient has a pole, then the problem is transformed into a system of linear equations and a finite number of boundary conditions are imposed in order to find a unique and explicit solution. In the case when the coefficient has an essential singularity, it is shown that the problem is well posed only for the Robin boundary condition.

The inhomogeneous Robin condition with general coefficient for the overdetermined inhomogeneous Cauchy–Riemann system of equations on the polydisc is studied using Fourier analysis. It is shown that this problem for the case of nonholomorphic general coefficient, is actually a problem with essential singularity in the domain, but still well-posed under certain compatibility conditions. Under proper assumptions, the unique solution is given explicitly.

The Riemann–Hilbert problem is studied for holomorphic functions in higher dimensional poly domains and the explicit constructive solution is given. The connection between the Riemann problem and the Riemann–Hilbert problem for poly domains is presented and proven. Contrary to earlier studies, our results provide explicit solutions and are not attached to any artificial assumptions.

We give results on the boundedness and compactness of wavelet multipliers on L p ( R n ) , 1 ≤ p ≤ ∞ L^p(\mathbb {R}^n),\,1\leq p\leq \infty .

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.

The Cauchy kernel is one of the two significant tools for solving the Riemann boundary value problem for analytic functions. For poly-domains, the Cauchy kernel is modified in such a way that it corresponds to a certain symmetry of the boundary values of holomorphic functions in poly-domains. This symmetry is lost if the classical counterpart of the one-dimensional form of the Cauchy kernel is applied. It is also decisive for the establishment of connection between the Riemann–Hilbert problem and the Riemann problem. Thus, not only the Schwarz problem for holomorphic functions in poly-domains is solved, but also the basis is established for solving some other problems. The boundary values of functions, holomorphic in poly-domains, are classified in the Wiener algebra. The general integral representation formulas for these functions, the solvability conditions and the solutions of the corresponding Schwarz problems are given explicitly. A necessary and sufficient condition for the boundary values of a holo...

The classical Dirichlet problem for the inhomogeneous pluriholomorphic system in a polydisc is studied. To get a unique solution the boundary condition is modified and its solvability conditions as well as the unique solution are given explicitly.

In this paper the Neumann problem for the inhomogeneous Cauchy-Riemann system in a polydisc is considered. The solvability conditions and the explicit solution which is unique up to an arbitrary constant are given.

In this paper, a corona ring design optimization for a 400kV non-ceramic insulator was investigated. Two parameters were changed during the study which are: the ring diameter (R) and the diameter of the ring tube (r). In this study, with the aid of a developed mathematical model the maximum electric field curves were recreated and an estimated electric field equation in terms of r and R while fixing H was found. Finally, using the estimated equation, a non-linear optimization technique was applied. The solution for this optimization problem gives the optimal dimensions of r and R as well as the ratio of R/r which will reduce the electric field value most effectively for a given constant cost.

This paper presents results on some boundary value problems for holomorphic functions of several complex variables in polydomains. The Cauchy kernel is one of the significant tools for solving the Riemann and the Riemann-Hilbert boundary value problems for holomorphic functions as well as for establishment of the connection between them. For polydomains, the Cauchy kernel is modified in such a way that it corresponds to a certain symmetry of the boundary values of holomorphic functions in polydomains. This symmetry is lost if the classical counterpart of the one-dimensional form of the Cauchy kernel is applied. The general integral representation formulas for the functions, holomorphic in polydomains, the solvability conditions and the solutions of the corresponding Schwarz problems are given explicitly. A necessary and sufficient condition for the boundary values of a holomorphic function for arbitrary polydomains is given and an exact, yet compact way of notation for holomorphic functions in arbitrary polydomains is introduced and applied. The Riemann jump problem and the Riemann-Hilbert problem are solved for holomorphic functions of several complex variables with the unit torus as the jump manifold. The higher-dimensional Plemelj-Sokhotzki formula for holomorphic functions in polydomains is established. The canonical functions of the Riemann problem for polydomains are represented and applied in order to construct solutions for both of the homogeneous and inhomogeneous problems. For all three boundary value problems, well-posed formulations are given which does not demand more solvability conditions than in the one variable case. The connection between the Riemann and the Riemann-Hilbert problem for polydomains is proven. Thus contrary to earlier research the results are similar to the respective ones for just one variable.

Discrete formulas for pseudo-differential operators based on the Shannon-Whittaker sampling formula and the Poisson summation formula are given.

Cover and Contents 1 1 The Dirichlet Problem 9 2 The Neumann Problem 33 3 The Schwarz Problem 65 4 The Riemann-Hilbert-Poincare Problem 93 Bibliography 113, Until now there has been still a great deal of activity in the study of integral representations for holomorphic functions of one and several complex variables in terms of their boundary values. These problems are of interest both because of their theoretical importance and the implications for applications, such as quantum field theory. In contrast with the one variable case, studies on holomorphic functions of several complex variables encounter mainly two kinds of difficulties: the geometrical complexity of general higher dimensional domains and the analytic complexity of complex integrability conditions from overdetermination. Many new tools are developed and many new insights are achieved. The aim of the PhD project is finding integral representations for elliptic equations and overdetermined elliptic systems as well as well-posed formulation of the classical problems in higher dimensional space regarding the torus., Integraldarstellungen für holomorphe Funktionen einer und mehrerer komplexen Veränderlichen mit Hilfe ihrer Randwerte stehen seit vielen Jahren im Mittelpunkt von Untersuchungen. Sie sind wegen ihrer theoretischen Bedeutung und für Anwendungen wie etwa die Quantenfeldtheorie von Interesse. Im Gegensatz zur Theorie einer Veränderlichen entstehen bei holomorphen Funktionen mehrerer komplexer Veränderlichen im Wesentlichen zwei Schwierigkeiten: die geometrische Komplexität allgemeiner Gebiete in höheren Dimensionen und die analytische Komplexität von komplexen Integrabilitätsbedingungen durch Überbestimmtheit. Viele neue Hilfsmittel werden entwickelt und viele neue Einsichten gewonnen. Ziel dieser Dissertation ist sowohl das Auffinden neuer Integraldarstellungen für elliptische Gleichungen und überbestimmte elliptische Systeme als auch die Formulierung wohlgestellter klassischer Probleme für Polyzylinder (Tori) in höherdimensionalen Räumen.