Formal Gevrey solutions -- in analytic germs -- for higher order holomorphic PDEs

TL;DR

This paper investigates formal solutions to higher order holomorphic PDEs near singularities, establishing conditions for existence, uniqueness, and divergence rates, and introduces a Gevrey class framework relative to the analytic map $P$.

Contribution

It introduces a new Gevrey class analysis for PDE solutions relative to the singular locus, extending classical results and providing optimal examples.

Findings

Solutions are Gevrey in $P$, revealing divergence behavior.

Poincaré conditions ensure convergence when $P$ is non-singular.

Examples demonstrate the optimality of the Gevrey class in $P$.

Abstract

We consider a family of holomorphic PDEs whose singular locus is given by the zero set of an analytic map $P$ with $P(0)=0$. Our goal is to establish conditions for the existence and uniqueness of formal power series solutions and to determine their divergence rate. In fact, we prove that the solution is Gevrey in $P$, giving new information on divergency while compared to the classical Gevrey classes. If $P$ is not singular at $0$, we also provide Poincar\'e conditions to recover convergent solutions. Our strategy is to extend the dimension and lift the given PDE to a problem where results of singular PDEs can be applied. Finally, examples where the Gevrey class in $P$ is optimal are included.