The Overdetermined Cauchy Problem for $omega$-ultradifferentiable Functions

In this paper we study the Cauchy problem for overdetermined systems of linear partial differential operators with constant coefficients in some spaces of $omega$-ultradifferentiable functions in the sense of Braun, Meise and Taylor, for non-quasianalytic weight functions $omega$. We show that existence of solutions of the Cauchy problem is equivalent to the validity of a Phragmen-Lindelof principle for entire and plurisubharmonic functions on some irreducible affine algebraic varieties.