On singular solutions for some first order partial differential equations with degenerate principal symbols (Q1097414)

This paper is exceptionally well written, well organized and a pleasure to read. It begins with the study of the PDE \[ (1)\quad L(u)=\sum^{n}_{i=1}(\lambda_ iy_ i+a_ i)\partial u/\partial_ ix=u, \] where \(\lambda_ i\in {\mathbb{C}}\) \((i=1,2,...,n)\) and \(a_ i\) \((i=1,2,...,n)\) are all holomorphic in a neighborhood of the origin of \({\mathbb{C}}^ n.\) Each \(a_ i\) vanishes at the origin with an order larger than two. It investigates solutions, \(u(x_ 1,...,x_ n)\) having the form \[ (2)\quad u(x_ 1,...,x_ n)=u_ 0(x_ 1,...,x_ n)H(x_ 1,...,x_ n) \] where \(u_ 0(x_ 1,x_ 2,...,x_ n)\) is a solution of the equation \(L_ 0(\gamma)\equiv \sum^{n}_{i=0}\lambda_ ix_ i\partial \gamma /\partial x_ i=\gamma\), and \(H(x_ 1,...,x_ n)\) is a holomorphic function at the origin. A brief review of the ordinary differential equation problem is included in the paper giving it a very easy to read atmosphere. It then develops the problem (1) and considers solutions (2) in the following three cases: Case \(1: u_ 0(x_ 1,...,x_ n)=x_ 1^{\alpha_ 1}...x_ n^{\alpha_ n}\), \(\sum^{n}_{i=1}\alpha_ i\lambda_ i=1\) Case \(2: u_ 0(x_ 1,...,x_ n)=(\sum^{n}_{i=1}c_ ix_ i^{1/\lambda_ ir}))\) r, \(r\neq 0\) Case \(3: u_ 0(x_ 1,...,x_ n)=(\sum^{n}_{i=1}c_ ix_ iq/\lambda_ i)\) p(\(\sum^{n}_{i=1}d_ ix_ i^{r/\lambda_ i})\) s, \(pq+rs=1.\) The solution to (1) is of type (2) in Case 1 if and only if \(a_ k\equiv 0(mod x_ k)\) for k such that \(a_ k\neq 0\). The other cases are developed into several subcases. It opens Case 2 with the assumption (A) \(1/\lambda_ ir\not\in Z\) for all i. This then resolves itself with the result that under assumption (A) we have the requisite solution iff \(a_ i=\lambda_ ix_ ib\) where b is holomorphic for all i such that \(c_ i\neq 0.\) Case 3 is also refined with an additional assumption. The paper continues in this manner replacing, relaxing and strengthening the assumptions within each case.