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Holomorphic extension of solutions to homogeneous analytic partial dierential equations

  • Author(s): Armel, Jonathan J.
  • et al.
Abstract

In this dissertation we derive sufficient conditions on a pseudoconvex domain \[Omega\] and a linear, analytic differential operator P for the existence of solutions to Pu = 0 which are holomorphic in \[Omega\] near p \[epsilon\]\[theta\]\[Omega\], but cannot be prolonged across p. We first define the notion of strong P-convexity of \[Omega\], 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 chapter we extend the results of Chapter 2 to the case in which \[Omega\] 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 \[Omega\] and the coefficients of P.

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