Analysis IV

Linear and Boundary Integral Equations

• V. G. Maz’ya
• S. M. Nikol’skiĭ
Book

Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 27)

1. Front Matter
Pages i-vii
2. S. Prössdorf
Pages 1-125
3. V. G. Maz’ya
Pages 127-222
4. Back Matter
Pages 223-236

Introduction

A linear integral equation is an equation of the form XEX. (1) 2a(x)cp(x) - Ix k(x, y)cp(y)dv(y) = f(x), Here (X, v) is a measure space with a-finite measure v, 2 is a complex parameter, and a, k, f are given (complex-valued) functions, which are referred to as the coefficient, the kernel, and the free term (or the right-hand side) of equation (1), respectively. The problem consists in determining the parameter 2 and the unknown function cp such that equation (1) is satisfied for almost all x E X (or even for all x E X if, for instance, the integral is understood in the sense of Riemann). In the case f = 0, the equation (1) is called homogeneous, otherwise it is called inhomogeneous. If a and k are matrix functions and, accordingly, cp and f are vector-valued functions, then (1) is referred to as a system of integral equations. Integral equations of the form (1) arise in connection with many boundary value and eigenvalue problems of mathematical physics. Three types of linear integral equations are distinguished: If 2 = 0, then (1) is called an equation of the first kind; if 2a(x) i= 0 for all x E X, then (1) is termed an equation of the second kind; and finally, if a vanishes on some subset of X but 2 i= 0, then (1) is said to be of the third kind.

Keywords

Integralgleichungen Nichtstetige Randintegrale Potential theory Potentialtheorie Randintegraltheorie Singuläre Integralgleichungen boundary integral equations eigenvalue problem integral equation integral equations kernel measure non-smooth boundary integrals singular integral equations system

Editors and affiliations

• V. G. Maz’ya
• 1
• S. M. Nikol’skiĭ
• 2