On continuation of regular solutions of linear partial differential equations

Abstract

Our research mainly concerns problems of type (2b). As a survey, however, we shall start with the general story. As is well known, the origin of this study lies in the problem of removable singularities in the theory of functions of a single complex variable (category (2a) in this classification). Already as early as 1851, Riemann [1] had pointed out that an isolated singularity of a holomorphic function is removable if the function is bounded around the singularity. Later, in 1906, Hartogs [1] proved that in the case of holomorphic functions of several variables any convex compact set is removable, irrespective of the boundedness of the function. After that, studies extending these results were made by researchers in complex analysis. Among them we mention that Hartogs’ theorem was generalized by these people to any pair K ⊂ U , where K is the part of a convex compact set lying in a half space and U is a neighborhood (although this form of the assertion also bears Hartogs’ name nowadays). See Struppa [1] for a detailed history of these. In 1952 Bochner [1] gave a new proof of Hartogs’ theorem employing the notion of an overdetermined system of differential equations. This initiated an abundant study of the problem of continuation of solutions in the theory of partial