Convergence of formal solutions of first order singular partial differential equations of nilpotent type
Volume 97 / 2012
Banach Center Publications 97 (2012), 91-99
MSC: Primary 35A20; Secondary 34M03, 34M25.
DOI: 10.4064/bc97-0-6
Abstract
Let $(x,y,z)\in{\mathbb C}^3$. In this paper we shall study the solvability of singular first order partial differential equations of nilpotent type by the following typical example: \[ Pu(x,y,z):=(y\partial_x-z\partial_y)u(x,y,z)=f(x,y,z)\in{\cal O}_{x,y,z}, \] where \[ P=y\partial_x-z\partial_y:{\cal O}_{x,y,z}\to{\cal O}_{x,y,z}. \] For this equation, our aim is to characterize the solvability on ${\cal O}_{x,y,z}$ by using the $\mathop{\rm Im} P$, $\mathop{\rm Coker} P$ and $\mathop{\rm Ker} P$, and we give the exact forms of these sets.
