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Boundary Properties and Applications of the Differentiated Poisson Integral for Different Domains
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Authors: Sergo Topuria (Georgian Technical University, Georgia) 
Book Description:
This monograph is devoted to the investigation of boundary properties of the differentiated Poisson integral. It is proved that the boundary properties of the differentiated Poisson integral for different types of domains (circle, sphere, half-plane, half-space, bicylinder) differ substantially from each other and depend on in what sense the integral density is differentiable. The theorems proven here are, in a definite sense, improvable. Relying on the obtained results, the Dirichlet problem is solved for a sphere and a half-space (of a any finite dimension) in the case where the boundary function is measurable and finite almost everywhere.

Table of Contents:
Preface

1. Boundary Properties of Derivatives of the Poisson Integral for a
Half-Plane, pp. 1-19
1.1 Notation, Definitions and the Well-Known Statements
1.2 Auxiliary Statements
1.3 The Boundary Properties of Derivatives of the Poisson Integral for a Half-Plane
1.4 The Dirichlet Problem for a Half-Plane

2. Boundary Properties of Derivatives of the Poisson Integral for a
Circle, pp. 21-35
2.1 Notation and Definitions
2.2 Auxiliary Statements
2.3 Boundary Properties of Derivatives of the Poisson Integral for a Circle
2.4 The Dirichlet Problem for a Circle

3. Boundary Properties of Derivatives of the Poisson Integral for a
Ball, pp. 37-85
3.1 Notation, Definitions and Statement of Some Well-Known Facts
3.2 The Poisson Integral for a Ball
3.3 A Generalized Laplace Operator on the Unit Sphere Sk 1
3.4 Boundary Properties of the Integral DkU
3.5 The Boundary Properties of the Integral Drk
3.6 The Dirichlet Problem for a Ball
3.7 Representation by the Laplace Series of an Arbitrary Measurable
Function Defined on the Unit Sphere S2

4 Boundary Properties of Derivatives of the Poisson Integral for a
Space Rk+1 + (k > 1), 87-143
4.1 Generalized Partial First Order Derivatives of a Function of Several Variables
4.2 The Boundary Properties of First Order Partial Derivatives of the
Poisson Integral for a Half-Space Rk+1 + (k > 1)
4.3 Generalized Partial Second Order Derivatives for a Function of Several Variables
4.4 The Boundary Properties of Partial Second Order Derivatives of the Poisson Integral for a Half-Space Rk+1 + (k > 1)
4.5 The Dirichlet Problem for a Half-Space R3+
4.6 Generalized Partial Derivatives of Arbitrary Order
4.7 The Boundary Properties of Arbitrary Order Partial Derivatives of
the Poisson Integral for a Half-Space Rk+1
4.8 Generalized Mixed Derivatives and Differentials of Arbitrary Order
4.9 The Boundary Properties of Mixed Derivatives and Differentials of
Arbitrary Order of The Poisson Integral for the Half-Space Rk+1
+ (k > 1)
4.10 The Generalized Laplace Operator in Rk (k , 2)
4.11 The Boundary Properties of the Integral ­˘rU(f; x; xk+1)

5. Boundary Properties of a Differentiated Poisson Integral for a cylinder,and Representation of a Function of Two Variables by a
Double Trigonometric Series, pp. 145-167
5.1 Notation and Definitions
5.2 Integral for a Bicylinder
5.3 Representation of a Function of Two Variables by a Trigonometric
Series in the Case of Spherical Convergence
5.4 On One Method of Summation of Double Fourier Series
5.5 Representation of a Function of Two Variables by a Double Trigonometric Series in the Case of Ptingsheim’s Convergence

Bibliography, pp. 169-177

Index

   Series:
      Mathematics Research Developments
   Binding: Hardcover
   Pub. Date: 2011 - 2nd quarter
   Pages: 180 pp.
   ISBN: 978-1-60692-704-5
   Status: AV
  
Status Code Description
AN Announcing
FM Formatting
PP Page Proofs
FP Final Production
EP Editorial Production
PR At Prepress
AP At Press
AV Available
  
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Boundary Properties and Applications of the Differentiated Poisson Integral for Different Domains