The Integral Equations of the Theory of Elasticity

We will assume that the reader is familiar with the theory of Fredholm integral equations (for example, as presented in one of the books [24, 33, 39, 41]). We also require the reader to know the simplest concepts of Functional Analysis such as Banach Spaces (in particular, the spaces C and L ₂), separable spaces, linear functionals and operators (bounded and unbounded), dual spaces and operators, precompactness and compactness of manifolds (a space is M-precompact, if is compact). The reader will find an adequate introduction to these topics, for example, in [10, 12, 40].

The theory of one-dimensional, singular integral equations, as an independent, scientific discipline, began in the Twentieth Century, almost contemporary with Fredholm’s theory, in the work of D. Hilbert and H. Poincaré. The efforts of F. Noether (1921) and T. Carleman (1922) were of fundamental importance for this theory. Its further developments occurred in the work of F.D. Gakhov and his students, the Tbilissi Mathematical School headed by N.I. Muskhelishvili and I.N. Vekua, and the Kishinev Mathematical School. Of well known significance were also the efforts of the authors of this work on the theory and applications of these equations. In more recent years, important studies were due to S. Prössdorf and his students; in particular, their work on the approximate solution of these equations should be mentioned.

The results of the theory of two-dimensional, singular integral equations extend in an obvious manner to a variety of such equations of any dimensionality, located in Euclidean space of any number of dimensions. We will limit ourselves here to the earlier mentioned simpler cases, having in mind the requirements of relevant applications.

Finite difference methods, widely employed for the solution of differential (or other operator) equations are also used to solve approximately integral equations; they include the methods of Ritz, Bubnov-Galerkin, least squares, collocation as well as a variety of these methods, linked to the application of so called finite elements. One has also been made of grid methods which, in the case of integral equations, assume the form of “methods of mechanical quadrature”. The method of iteration has been applied for the solution of equations of the form u − Au = f, ║A║ < 1.

The two-dimensional problems of the theory of elasticity form a class of problems which lend themselves, in contrast to the majority of three-dimensional problems, to analytical treatment and at the same time do not have such intuitively clear answers as one-dimensional problems. This intermediate position between the one-dimensional and three-dimensional problems determines the special position of the two-dimensional problems of the theory of elasticity.

The classical results of S.G. Mikhlin [45], N.I. Muskhelishvili [54], N.P. Vekua [8], V.D. Kupradze [28], and also the monographs of V.Z. Parton and P.I. Perlin [58], T.V. Burchuladze and T.P. Gegelia [7] and others [69, 9], it might be said, have presented comprehensive answers to questions, linked to applications of integral equations to problems of the theory of elasticity.

An overwhelming majority of practically important engineering problems formulated in terms of the theory of elasticity, are contact problems, and only mathematical difficulties force investigators to replace the action of the second media by transitional boundary conditions. However, at the present time, the requirement of accuracy of computations all too often creates the necessity to give up simplifications and fulfill completely the conditions on the surface of contact.

A general approach to the study of the integral equations for plane as well as space problems for regions with non smooth boundaries was proposed by V.G. Maz’ja [35] and developed by him for the plane problem in cooperation with S.S.Zargaryan [24,25]. An other approach was employed independently by W. Wendland, E. Stephan and M. Costabel [87, 101, 102].