The main theme of this work, to investigate the holomorphic continuation of solutions of holomorphic partial differential equations in domains within the space of several real or complex varibles, will employ techniques from three areas of mathematical analysis. They are partial differential equations, complex analysis and potential theory. The extent to which solutions of elliptic equations extend from hypersurfaces to surrounding domains is believed to be directly related to the maximum extension of the Schwarz potential. It should be mentioned that the question of maximal extent of solutions of equations of hyperbolic type (wave equations) has been thoroughly investigated. In the case of equations of elliptic type, such as Laplace's equation, conjectures of what the maximal extent should be have not even been formulated. Present work will focus on very specific questions regarding homogeneous equations with holomorphic boundary values. The first objective will be to determine whether or not the maximal domain of analyticity of a solution depends only on the differential operator and the boundary, but not the specific Cauchy data. A simple device which has proved valuable in the cases of equations in two real dimensions is the Schwarz function of an analytic curve. The domain of regularity of the Schwarz function plays a crucial role in determining the maximal region of analyticity for solutions of elliptic initial value problems with analytic data. A reasonable extension of the Schwarz function has been formulated which will be examined to determine whether the basic two-dimensional results can be expanded to higher real dimensions and (any) complex ones. Preliminary analysis will be done on problems where the Cauchy data consists of polynomials, with later goals to include entire functions.
