Boundary Value Problems For Second Order Elliptic Equations

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.

Contenu

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
8. The extremum principle
9. The Hopf principle
10. The Zaremba-Giraud principle
11. The extremum principle for a class of elliptic systems
12. Adjoint operators. Green's formula
13. Existence of solutions
14. Elementary solutions
15. The principle elementary solution
16. Generalised potentials and their properties
17. General representation of the solutions of a class of elliptic systems
18. Harmonic potentials of a simple and double layer and integrals of Cauchy type
    Ch. III. The Dirichlet Problem for a Second Order Elliptic Equation
19. Formulation of the problem. Uniqueness of the solution
20. Existence of a solution of the problem (2.1), (3.1)
21. The Dirichlet problem for the Laplace equation in two independent variables. Green's function
22. The Riemann-Hilbert problem and integral representations of holomorphic functions
    Ch. IV. The Dirichlet Problem for Elliptic Systems
23. Preliminary remarks
24. Uniqueness of the solution of the Dirichlet problem
25. Elliptic systems (4.1) with the principal part in the form of the Laplace operator
26. The Dirichlet problem for the elliptic system (4.11) with analytic coefficients
27. The Dirichlet problem for system (4.1 )
28. General representation of the solutions of system (4.71)
29. The Dirichlet problem for a weakly connected system (4.71)
30. Some remarks concerning strongly connected systems
31. The Dirichlet problem for system (4.96)
32. 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
33. Formulation of the problem
34. Investigation of the Neumann problem
35. The adjoint problem
36. 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
37. General remarks
38. The Poincaré problem for system (4.18) with analytic coefficients
39. Certain special cases of the problem (4.18), (6.1)
40. The Poincaré problem for the uniformly elliptic system (4.1)
41. 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
42. Preliminary remarks
43. Concept of a holomorphic vector
44. Integral of Cauchy type and singular integral in the sense of the Cauchy principal value
45. Limiting values of an integral of Cauchy type and an interchange formula for singular integrals
46. Inversion of a system of singular integral equations
47. An integral representation of a holomorphic vector in a halfspace
48. The directional derivative problem for harmonic functions with two independent variables
49. The directional derivative problem for harmonic functions with three independent variables
50. The directional derivative problem with polynomial coefficients for harmonic functions
51. A class of multidimensional singular integral equations
    References
    Subject Index