Formal Gevrey solutions: in analytic germs—for higher order holomorphic PDEs

Abstract

AbstractWe consider a family of holomorphic PDEs whose singular locus is given by the zero set of an analytic map P with $$P(0)=0$$ 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 Poincaré 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.