Convergence of Formal Solutions of Singular Partial Differential Equations

Abstract

We consider partial differential operators of the form

${\rm P}({\rm x},{\rm x}{\partial\over\partial{\rm x}})=\sum\limits_{\vert\ell\vert\leqslant_{d}}{\rm a}_{\ell}({\rm x})({\rm x}{\partial\over\partial{\rm x}})^{\ell}$

and give conditions when the equation Pu = f, f analytic near the origin, has a power series solution which converges in a neighbourhood of the origin. More generally we consider non linear equations of the form Pu = F(x,u) and give some applications. This is a report on the results of [2].