Formal Gevrey class of formal power series solution for singular first order linear partial differential operators (Q5930120)

The author is interested in the following semilinear complex differential equation NEWLINE\[NEWLINELu:= \sum^n_{k=1} a_k(z) \partial_{z_k} u= F(z,u),NEWLINE\]NEWLINE where the coefficients and right-hand side are holomorphic in their arguments in a neighbourhood of the origin. To characterize the solvability behaviour of \(Lu= F(z,u)\) the following assumptions are posed in a neighbourhood \(U\) of the origin:NEWLINENEWLINENEWLINE\(S= \{z\in U; a_k(z)= 0\) for \(k= 1,\dots, n\}\) is a complex submanifold of codimension \(n_1\);NEWLINENEWLINENEWLINEthe Jacobian \(({\partial a_k\over\partial z_\ell}(0))^n_{k,\ell= 1}\) has a special Jordan normal form;NEWLINENEWLINENEWLINE\(a_1\in M^\delta\) modulo \(I\{a_{i_1},\dots, a_{i_{n_0}}\}\), where \(\delta\geq 2\), \(M\) is related to \(S\), and the ideal \(I\) is related to the Jordan normal form;NEWLINENEWLINENEWLINE\(|\sum^{n_0}_{i= 1}\lambda_i k_i- c|\geq \sigma(|k|+ 1)\) with a positive constant \(\sigma\), for all \(k= (k_1,\dots, k_{n_0})\), \(c= {\partial F\over\partial u} (0,0)\), and \(\lambda_1,\dots, \lambda_{n_0}\) are the nonzero eigenvalues of \(({\partial a_k\over\partial z_\ell}(0))^n_{k, \ell=1}\).NEWLINENEWLINENEWLINEThen it is proved, that \(Lu= F(z,u)\) has a unique formal power series solution, and this solution belongs to a formal Gevrey class basing on suitable local coordinates.