CHAPTER I - INTRODUCTION

Abstract

This chapter discusses the work that has been done in the field of multi-dimensional singular equation. The first important work on multi-dimensional singular equations is due to Tricomi, who investigated double singular integrals. The next important work on multi-dimensional singular integrals was that of Giraud. This author investigated integrals taken over a closed Liapounov manifold of any dimension m; these manifolds are not necessarily connected but do not have one-sided parts. Later, Mikhlin provided work on double integrals taken over a two-dimensional plane. Moreover, Trjitzinsky presented work with singular integrals taken over a two-dimensional surface with a boundary. In this work, there is an interesting attempt to construct for a three-dimensional space a theory of boundary problems of the type of Riemann's problem for the plane. In addition, Ebanoidze provided work on double singular integrals with a stationary singularity; analogous equations with a single independent variable have been studied by Khvoles.