Boundary value problems for generalized analytic functions in functional space

1967 ◽  
Vol 8 (1) ◽  
pp. 86-91
Author(s):  
V. S. Rogozhin
Keyword(s):  
Boundary Value Problems ◽  
Analytic Functions ◽  
Boundary Value ◽  
Functional Space ◽  
Generalized Analytic Functions

scholarly journals Generalized Analytic Functions in Higher Dimensions

2007 ◽  
Vol 14 (3) ◽  
pp. 581-595
Author(s):  
Wolfgang Tutschke
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Boundary Value Problems ◽  
Analytic Functions ◽  
Boundary Value ◽  
Higher Dimensions ◽  
Partial Differential ◽  
Generalized Analytic Functions ◽  
Linear Partial Differential Equations ◽  
Weakly Singular

Abstract Originally I. N. Vekua's theory of generalized analytic functions dealt only with linear systems of partial differential equations in the plane. The present paper shows why I. N. Vekua's ideas are also fruitful for the solution of linear and non-linear partial differential equations in higher dimensions. One of the highlights of the theory of generalized analytic functions in the plane is the reduction of boundary value problems for general (linear or nonlinear) equations to boundary value problems for holomorphic functions using the well-known weakly singular and strongly singular 𝑇- and П-operators, respectively. The present paper is mainly aimed at reducing boundary value problems in higher dimensions to boundary value problems for monogenic functions.

Logarithmic potential and generalized analytic functions

2021 ◽  
Vol 18 (1) ◽  
pp. 12-36
Author(s):  
Vladimir Gutlyanskii ◽  
Olga Nesmelova ◽  
Vladimir Ryazanov ◽  
Artyem Yefimushkin
Keyword(s):  
Boundary Value Problems ◽  
Analytic Functions ◽  
Harmonic Functions ◽  
Boundary Data ◽  
Boundary Value ◽  
Logarithmic Capacity ◽  
Functional Interpretation ◽  
Arbitrary Boundary ◽  
Generalized Analytic Functions ◽  
Complex Valued

The study of the Dirichlet problem in the unit disk $\mathbb D$ with arbitrary measurable data for harmonic functions is due to the famous dissertation of Luzin [31]. Later on, the known monograph of Vekua \cite{Ve} has been devoted to boundary-value problems (only with H\"older continuous data) for the generalized analytic functions, i.e., continuous complex valued functions $h(z)$ of the complex variable $z=x+iy$ with generalized first partial derivatives by Sobolev satisfying equations of the form $\partial_{\bar z}h\, +\, ah\, +\ b{\overline h}\, =\, c\, ,$ where it was assumed that the complex valued functions $a,b$ and $c$ belong to the class $L^{p}$ with some $p>2$ in smooth enough domains $D$ in $\mathbb C$. The present paper is a natural continuation of our previous articles on the Riemann, Hilbert, Dirichlet, Poincar\'{e} and, in particular, Neumann boundary-value problems for quasiconformal, analytic, harmonic, and the so-called $A-$harmonic functions with boundary data that are measurable with respect to logarithmic capacity. Here, we extend the corresponding results to the generalized analytic functions $h:D\to\mathbb C$ with the sources $g$ : $\partial_{\bar z}h\ =\ g\in L^p$, $p>2\,$, and to generalized harmonic functions $U$ with sources $G$ : $\triangle\, U=G\in L^p$, $p>2\,$. This paper contains various theorems on the existence of nonclassical solutions of the Riemann and Hilbert boundary-value problems with arbitrary measurable (with respect to logarithmic capacity) data for generalized analytic functions with sources. Our approach is based on the geometric (theoretic-functional) interpretation of boundary-values in comparison with the classical operator approach in PDE. On this basis, it is established the corresponding existence theorems for the Poincar\'{e} problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations $\triangle\, U=G$ with arbitrary boundary data that are measurable with respect to logarithmic capacity. These results can be also applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media.

scholarly journals Boundary value problems for the generalized analytic and harmonic functions

2019 ◽  
Vol 33 ◽  
pp. 66-82
Author(s):  
Vladimir Gutlyanskii ◽  
Olga Nesmelova ◽  
Vladimir Ryazanov ◽  
Artem Yefimushkin
Keyword(s):  
Boundary Value Problems ◽  
Analytic Functions ◽  
Harmonic Functions ◽  
Boundary Data ◽  
Boundary Value ◽  
Logarithmic Capacity ◽  
Functional Interpretation ◽  
Arbitrary Boundary ◽  
Generalized Analytic Functions ◽  
Complex Valued

The study of the Dirichlet problem with arbitrary measurable data for harmonic functions is due to the famous dissertation of Luzin. Later on, the known monograph of Vekua has been devoted to boundary value problems (only with H\"older continuous data) for the generalized analytic functions, i.e., continuous complex valued functions $h(z)$ of the complex variable $z=x+iy$ with generalized first partial derivatives by Sobolev satisfying equations of the form $\partial_{\bar z}h\, +\, ah\, +\ bh\, =\, c\, ,$ where $\partial_{\bar z}\ :=\ \frac{1}{2}\left(\ \frac{\partial}{\partial x}\ +\ i\cdot\frac{\partial}{\partial y}\ \right),$ and it was assumed that the complex valued functions $a,b$ and $c$ belong to the class $L^{p}$ with some $p>2$ in the corresponding domains $D\subset \mathbb C$. The present paper is a natural continuation of our articles on the Riemann, Hilbert, Dirichlet, Poincare and, in particular, Neumann boundary value problems for quasiconformal, analytic, harmonic and the so-called $A-$harmonic functions with boundary data that are measurable with respect to logarithmic capacity. Here we extend the correspon\-ding results to the generalized analytic functions $h:D\to\mathbb C$ with the sources $g$ : $\partial_{\bar z}h\ =\ g\in L^p$, $p>2\,$, and to generalized harmonic functions $U$ with sources $G$ : $\triangle\, U=G\in L^p$, $p>2\,$. It was also given relevant definitions and necessary references to the mentioned articles and comments on previous results. This paper contains various theorems on the existence of nonclassical solutions of the Riemann and Hilbert boundary value problems with arbitrary measurable (with respect to logarithmic capacity) data for generalized analytic functions with sources. Our approach is based on the geometric (theoretic-functional) interpretation of boundary values in comparison with the classical operator approach in PDE. On this basis, it is established the corresponding existence theorems for the Poincare problem on directional derivatives and, in particular, for the Neumann problem to the Poisson equations $\triangle\, U=G$ with arbitrary boundary data that are measurable with respect to logarithmic capacity. These results can be also applied to semi-linear equations of mathematical physics in anisotropic and inhomogeneous media.

scholarly journals Riemann-Hilbert boundary value problems in BMO classes for generalized analytic functions

2011 ◽  
Author(s):  
С.Б. Климентов
Keyword(s):  
Boundary Value Problems ◽  
Analytic Functions ◽  
Boundary Value ◽  
Generalized Analytic Functions

В работе рассматривается разрешимость краевой задачи Римана - Гильберта в классе BMO для обобщенных аналитических функций в предположении, что коэффициент краевого условия принадлежит пространству мультипликаторов класса BMO. Ранее автором построены примеры, когда задача с неотрицательным индексом в такой наиболее стественной постановке неразрешима в классе голоморфных функций BMOA [2] и были даны достаточные условия на коэффициент, при которых имеет место обычная картина разрешимости. В этой работе результаты для голоморфных функций из [2] переносятся на обобщенные аналитические функции.

Sign in / Sign up

Export Citation Format

Share Document