Study of singularities of solutions of partial differential equations in the complex domain

1.研究了复域上偏微分方程解的奇性，得到了以下结果：(1)完全确定了非线性Fuchsian型偏微分方程奇异解的结构，不需要对特征指数作任何假设；(2)得到了正规型一阶非线性偏微分方程解的存在性和不存在性的一个很好的结果；2.研究了偏微分方程解的形式幂级数解及其解析意义，得到了如下结果：某类线性偏微分方程解的每个形式幂级数解都是多可和的研究了关于p-椭圆型方程的非线性偏微分方程解的存在性。利用Fuchsian偏微分方程解的一个结果，证明了解的每个分数型奇点都是解析奇点。

1.Singularities of solutions of partial differential equations in the complex domain were studied and the followingresults were obtained:(1)Structure of singular solutions of nonlinear Fuchsian type partial differential equations were determined completely without any assumptions on characteristic exponents.(2)A sharp result was obtained on the existence and non-existence of singular solutions of non-linear first order partial differential equations of normal type.2.A formal power series solutions of partial differential equations and its analytic meanings were studied and the following result was obtained: every formal power series solutions of certain class of linear partial differential equations are multisummable.3.Analytic singularities of solutions of non-linear partial differential equations concerning p-elliptic equation were studied. By using a result on Fuchsian partial differential equations, it was proved that every fractional type singularity of solutions is an analytic singularity.