Local analytic continuation of holomorphic solutions of partial differential equations

Abstract

We ask. When is it possible to continue analytically holomorphic solutions of partial differential equations. Using objects called cones of analytic continuation we get sufficient conditions generalizing results by J.-M. Bony and P. Schapira and by Y. Tsuno. The results are counterparts of earlier results by the author on local uniqueness in the Cauchy problem. We also give a necessary condition by constructing solutions with singularities. We think that the technique used here should also have other applications. Anyhow this result is a generalization of a result by Y. Tsuno related to simple characteristic hypersurfaces. It corresponds to the existence of local null solution when the initial hypersurface has constant multiplicity in the Cauchy problem.