Extending Solutions of Holomorphic Partial Differential Equations Across Real Hypersurfaces

The main result in this paper, Theorem 1.2, generalizes a theoremof Zerner 26] concerning sufficient conditions for the holomorphiccontinuability of a solution of a linear holomorphic partialdifferential equation across a point of a hypersurface, on oneside of which it is holomorphic. The point of the new theoremis, roughly speaking, that it applies also to regular solutionsof partial differential equations whose coefficients may havecertain kinds of singularities. This enables us to deduce somenew results (see 2) on elliptic partial differential equationsin R²:Theorem 2.1 extends a result of Vekua on the size of thedomain of holomorphy of solutions to elliptic equations, inthe case where singularities are permitted in the coefficients;Theorem 2.2 is of an apparently novel type, showing (roughly)that under certain conditions the solution to Cauchy's problemis real-analytic in a domain whose size depends only on theprincipal part of the operator, which is assumed to be the Laplacian,and the Cauchy data on the real axis. (Results of this kindare very delicate, as we shall illustrate in 4 with a simplecounterexample.) Theorem 2.2 is new and non-trivial even forequations with analytic coefficients, in which case though,Theorem 1.2 is not needed for the proof.