Study of the singularities of solutions of partial differential equations

1.研究了n维复域上的偏微分算子，证明了当算子在其特征曲面上的局部化为Fuchsian型时，其特征曲面上的奇点是可去的.将所得结果应用于具有非对合重特征的二阶方程Cauchy问题的奇性传播。2.在二维空间中，研究了一类偏微分算子，并将Holmgren的唯一性定理推广到双特征点. 3.研究了多Fuchsian变元的偏微分算子，证明了其分布解的存在性4.研究了一类退化抛物型方程Cauchy问题的适定性，给出了关于次主符号和迹的一个必要条件.通过构造渐近解的方法证明了这一点。5.作为相关研究，研究了模拟电荷法的数值保角变换。研究了具有多个点状磁场的Shrodinger算子的散射振幅的渐近行为。

1.Partial differential operators were considered in an n-dimensional complex domain, and, when the localization of the operator on its characteristic surface is Fuchsian type, it was proved that the singularities on the characteristic surface are removable. The result was applied to the propagation of singularities in the Cauchy problem for 2nd order equation with non-involutive double characteristics.2.In a 2-dimensional space, the same kind of partial differential operators were studied and Holmgren's uniqueness theorem was extended to doubly characteristic points.3.Partial differential operators with several Fuchsian variables were considered and the existence of distribution solution ( null solution) whose support is contained in a certain quadrant and contains the origin was proved.4.The wellposedness of the Cauchy problem was studied for some degenerate parabolic equations and a necessary condition concerning the subprincipal symbol and the trace was given. It was proved by constructing asymptotic solutions.5.As relevant studies, the numerical conformal mapping by using the charge simulation method were studied. The asymptotic behavior of scattering amplitude was also investigated for the Shrodinger operator with several point-like magnetic fields.