Extending Solutions of Holomorphic Partial Differential Equations Across Real Hypersurfaces

Abstract

The main result in this paper, Theorem 1.2, generalizes a theorem of Zerner [26] concerning sufficient conditions for the holomorphic continuability of a solution of a linear holomorphic partial differential equation across a point of a hypersurface, on one side of which it is holomorphic. The point of the new theorem is, roughly speaking, that it applies also to regular solutions of partial differential equations whose coefficients may have certain kinds of singularities. This enables us to deduce some new results (see §2) on elliptic partial differential equations in ℝ2: Theorem 2.1 extends a result of Vekua on the size of the domain of holomorphy of solutions to elliptic equations, in the 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 problem is real-analytic in a domain whose size depends only on the principal part of the operator, which is assumed to be the Laplacian, and the Cauchy data on the real axis. (Results of this kind are very delicate, as we shall illustrate in §4 with a simple counterexample.) Theorem 2.2 is new and non-trivial even for equations with analytic coefficients, in which case though, Theorem 1.2 is not needed for the proof.