# Boundary Value Problems For Second Order Elliptic Equations

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Applied Mathematics and Mechanics, Volume 5: Boundary Value Problems: For Second Order Elliptic Equations is a revised and augmented version of a lecture course on non-Fredholm elliptic boundary value problems, delivered at the Novosibirsk State University in the academic year 1964-1965. This seven-chapter text is devoted to a study of the basic linear boundary value problems for linear second order partial differential equations, which satisfy the condition of uniform ellipticity. The opening chapter deals with the fundamental aspects of the linear equations theory in normed linear spaces. This topic is followed by discussions on solutions of elliptic equations and the formulation of Dirichlet problem for a second order elliptic equation. A chapter focuses on the solution equation for the directional derivative problem. Another chapter surveys the formulation of the Poincaré problem for second order elliptic systems in two independent variables. This chapter also examines the theory of one-dimensional singular integral equations that allow the investigation of highly important classes of boundary value problems. The final chapter looks into other classes of multidimensional singular integral equations and related boundary value problems.

## Table of Contents

Preface

Ch. I. Introductory Remarks

1. Some definitions and notations

2. General information on second order elliptic equations and boundary value problems

3. Fundamental aspects of the theory of linear equations in normed linear spaces

4. Fredholm integral equations of the second kind

5. Singular integral equations

6. Fredholm integral equations of the first kind

7. Conventional classification of boundary value problems

Ch. II. Certain Qualitative and Constructive Properties of the Solutions of Elliptic Equations

1. The extremum principle

2. The Hopf principle

3. The Zaremba-Giraud principle

4. The extremum principle for a class of elliptic systems

5. Adjoint operators. Green's formula

6. Existence of solutions

7. Elementary solutions

8. The principle elementary solution

9. Generalised potentials and their properties

10. General representation of the solutions of a class of elliptic systems

11. Harmonic potentials of a simple and double layer and integrals of Cauchy type

Ch. III. The Dirichlet Problem for a Second Order Elliptic Equation

1. Formulation of the problem. Uniqueness of the solution

2. Existence of a solution of the problem (2.1), (3.1)

3. The Dirichlet problem for the Laplace equation in two independent variables. Green's function

4. The Riemann-Hilbert problem and integral representations of holomorphic functions

Ch. IV. The Dirichlet Problem for Elliptic Systems

1. Preliminary remarks

2. Uniqueness of the solution of the Dirichlet problem

3. Elliptic systems (4.1) with the principal part in the form of the Laplace operator

4. The Dirichlet problem for the elliptic system (4.11) with analytic coefficients

5. The Dirichlet problem for system (4.1 )

6. General representation of the solutions of system (4.71)

7. The Dirichlet problem for a weakly connected system (4.71)

8. Some remarks concerning strongly connected systems

9. The Dirichlet problem for system (4.96)

10. Influencing effect of coefficients of the smaller derivatives

Ch. V. The Directional Derivative Problem for Equation (2.1 ) When the Direction of Inclination is Not Tangential to the Boundary

1. Formulation of the problem

2. Investigation of the Neumann problem

3. The adjoint problem

4. Investigation of the directional derivative problem (5.1), (5.2) when condition (5.4) is satisfied

Ch. VI. The Poincaré Problem for Second Order Elliptic Systems in Two Independent Variables

1. General remarks

2. The Poincaré problem for system (4.18) with analytic coefficients

3. Certain special cases of the problem (4.18), (6.1)

4. The Poincaré problem for the uniformly elliptic system (4.1)

5. The Poincaré problem for elliptic systems (4.71) with constant coefficients

Ch. VII. Certain Classes of Multidimensional Singular Integral Equations and Related Boundary Value Problems

1. Preliminary remarks

2. Concept of a holomorphic vector

3. Integral of Cauchy type and singular integral in the sense of the Cauchy principal value

4. Limiting values of an integral of Cauchy type and an interchange formula for singular integrals

5. Inversion of a system of singular integral equations

6. An integral representation of a holomorphic vector in a halfspace

7. The directional derivative problem for harmonic functions with two independent variables

8. The directional derivative problem for harmonic functions with three independent variables

9. The directional derivative problem with polynomial coefficients for harmonic functions

10. A class of multidimensional singular integral equations

References

Subject Index

## Product details

- No. of pages: 211
- Language: English
- Copyright: © North Holland 1968
- Published: January 1, 1968
- Imprint: North Holland
- eBook ISBN: 9780323162265

## About the Author

### A.V. Bitsadze

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