V.G. Mazya – författare

The first part of this volume is a survey of the current state of the theory of linear integral equations. Material from abstract operator theory is presented first, before moving on to a detailed discussion of Fredholm operators and estimates for the eigenvalues and singular numbers of integral operators. Other sections of this part are devoted to the theories of one-, and higher-dimensional singular operators. The second part of the volume concentrates on theoretical aspects of the boundary integral equation method, with particular emphasis on the results for equations on non-smooth surfaces obtained during the last few years. A survey of the classical theory of harmonic potentials and the boundary integral equations on smooth surfaces generated by such potentials is given and the theory is then extended to elastic and hydrodynamic potentials and to other problems of mathematical physics, including the oblique derivative problem, the biharmonic equation, the heat and wave equations and viscoelasticity.

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 integralequations 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.