Transformation of the boundary value problems into integral equations

Abstract

A classical method to attain the proofs of existence theorems for the solutions of various boundary value problems consists of reducing the solution of these problems to the solution of an integral equation or of a system of integral equations. This method originated in the works of Fredholm for Δ₂ and the later researches of Hilbert, Poincaré, Picard,Lichten Stein, and others¹, and has been most often applied to the study of particular problems on which we cannot dwell here. Problems under the general conditions in which they will be posed here were treated by this method for the first time by E.E. Levi in the case m = 2. M. Gevrey and G. Giraud successively studied the general case under much less restrictive hypotheses than those considered by Levi. There is not room for doubt that the definitive results along this line are those of Giraud, and on these we shall dwell copiously. Nevertheless, at the end of the chapter we shall not fail to illustrate briefly the research of Levi and of Gevrey also. In § 23 we shall also mention some results of other authors concerning the oblique derivative problem in the non-regular case.