Study of the singularities of solutions of partial differential equations

1.在n维复域上考虑偏微分算子，当算子在特征曲面上的定位为Fuchsian型时，证明了特征曲面上的奇异点是可去除的。将所得结果应用于二阶非对合二重特征方程的Cauchy问题中奇异点的传播。在二维空间中，研究了一类偏微分算子，并将Holmgren唯一性定理推广到双特征点。考虑带有多个Fuchsian变量的偏微分算子，证明了其支持包含在某象限且包含原点的分布解（零解）的存在性。研究了一类退化抛物型方程的柯西问题的适定性，给出了次主符号及其迹的一个必要条件。通过构造渐近解来证明。作为相关研究，利用电荷模拟方法研究了数值保角映射。研究了具有多个点状磁场的薛定谔算子散射振幅的渐近特性。

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.