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[ratio ]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.