Holomorphic extension of solutions to homogeneous analytic partial differential equations

In this dissertation we derive sufficient conditions on a pseudoconvex domain O and a linear, analytic differential operator P for the existence of solutions to Pu = 0 which are holomorphic O in near p ∈ ∂O, but cannot be prolonged across p.;We first define the notion of strong P-convexity of O, given by the existence of a supporting everywhere characteristic analytic hypersurface, and show how under the assumption of strong P-convexity a theorem of Tsuno [Tsu74] can be used to construct our desired solutions.;The rest of the work consists of finding necessary conditions for strong P-convexity. In Chapter 2 we consider the case in which is strictly pseudoconvex at p and show that strong P-convexity follows from a positivity condition given by Kawai and Takei [KT90].;In Chapter 3 we discuss bicharacteristic convexity, and show that strong P-convexity follows from the combination of bicharacteristic convexity and the convexifiability of a local projection along bicharacteristics.;In the final two chapters we extend the results of Chapter 2 to the case in which O is weakly pseudoconvex. Under some simplifying assumptions, we find new invariants of the pair (P,M), and use them to give sufficient conditions for strong P-convexity in terms of a defining equation for and the coefficients of P.