The Overdetermined Cauchy Problem for $\omega$-ultradifferentiable Functions

TL;DR

This paper investigates the existence of solutions to overdetermined linear PDE systems within spaces of $$-ultradifferentiable functions, linking solvability to a Phragme9n-Lindelf6f principle on algebraic varieties.

Contribution

It establishes a novel equivalence between the solvability of the Cauchy problem and a Phragme9n-Lindelf6f principle for entire functions in the context of $$-ultradifferentiable functions.

Findings

Existence of solutions is equivalent to a Phragme9n-Lindelf6f principle.

The study extends the theory to non-quasianalytic weight functions.

Provides a characterization of solvability for overdetermined systems.

Abstract

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 Phragm\'en-Lindel\"of principle for entire and plurisubharmonic functions on some irreducible affine algebraic varieties.