Boundary Value Problems For Second Order Elliptic Equations

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