The Treatment of Window Problems by Transform Methods

Our paper is devoted to the so-called window problems of elliptic differential equations over domains in IR 2 and R. The kind of problems appear as a special class of mixed boundary value problems, where the window is a finite part of the boundary. It means that the connection with the outside only through a. window can happen. By unconventionally chosen unknown functions we will deduce canonical representations of these problems in form of perturbed singular integral equations with singular integrals over the window. We use methods of transform analysis which are different from Green's method and potential methods. These ideas go back to D.S. Jones 18J. In a similar way S.R. Bland (lJ, F. Erdogan 121 and E. Venturino 1131 transformed different problems of mechanics to singular integral equations with generalized kernels. For more complicated domains and a certain class systems of partial differential equations the previous methods to not generalize and we will use methods of quaternionic analysis. Basic ideas in this field are given in the book of K. Gürlebeck and W. Spróig 161. Finally a. numerical consideration on the singular integral equations deduced above conclude the paper.