The Limitations of the Cauchy-Kovalevsky Theorem

Abstract

The so-called Cauchy-Kovalevsky theorem (C-K theorem) is the fundamental local existence and uniqueness theorem for partial differential equations in the holomor-phic category. In one formulation, one considers a system of m quasilinear partial differential equations of first order for unknown functions u₁, …, u_m of n complex variables z:= (z₁,…, z_n), which moreover are required to coincide on a complex hyperplane (for example, {z_n = 0}) in a neighborhood of some point of C ⁿ, with m specified (germs of) holomorphic functions on that hyperplane. The theorem asserts that, if the system is in “normal form” there is one and only one solution for the holomorphic germs {u_j}.