Study of global solutions to Fuchsian equations and local solutions to linear PDE

Fuchsian方程全局解和线性偏微分方程局部解的研究

基本信息

批准号：
15540156
负责人：
OKADA Yasunori
金额：
$ 1.86万
依托单位：
Chiba University
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2003
资助国家：
日本
起止时间：
2003 至 2005
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540156/
关键词：
Fuchsian equations linear PDE microlocal analysis second microlocalization

项目摘要

1.We tried second microlocal analysis for some class of Turrittin equations with lower terms of homogeneous type. In this case, similarly to the case of Mizohata equations, it became clear that fundamental solutions can be constructed using the theory of second microlocal Fourier transformations. Here we used the result about the growth order for the global solutions to some ordinary differential equations in the space where Fourier transforms live. In more special cases, the inverse Fourier transform satisfies some Fuchsian equations. We got the information of the Fuchsian equations, singularities, exponents, and some behaviors of their solutions.2.On the other hand, we can consider the notion of microfunctions with holomorphic parameters on a product space of a real domain and a complex domain. They are represented as boundary values of holomorphic functions, and their boundary values define the notion of second microfunctions, and so on. The equations in 1 have microlocal ellipticity in rather small region, but we can construct their second microlocal solutions by the actions of their inverses to microfunctions with holomorphic parameters. The relation between this result and the theory of boundary values of pseudo-differential operators due to Kataoka will be a further subject.3.A fundamental solution to linear PDE can be regarded as a kernel function of an integral transformation. We studied the kernels under an expectation that an integral transformation with kernel should be characterized by some notion similar to continuity, and got the kernel theorems in analytic category, by introducing notion of semi-continuity. Here we used the construction of complex kernels with curvilinear Radon transformations.

1.我们尝试了某些具有较低均匀类型术语的Turrittin方程的第二个微局部分析。在这种情况下，与Mizohate方程相似，很明显，可以使用第二个微局部傅立叶变换理论来构建基本解决方案。在这里，我们使用了有关在傅立叶变换生存的空间中某些普通微分方程的全球解决方案的增长顺序的结果。在更多特殊情况下，反傅立叶变换满足一些紫红色方程。我们得到了紫红色方程，奇异性，指数及其解决方案的某些行为的信息。2.在另一方面，我们可以考虑在真实域和复杂域的产品空间上使用全体形态参数的微函数的概念。它们被表示为全态函数的边界值，其边界值定义了第二个微函数的概念，依此类推。 1中的方程在相当小的区域中具有微局部椭圆度，但是我们可以通过其对具有全体形态参数的微函数的作用来构建其第二个微局部溶液。由于Kataoka引起的伪分化算子的边界值之间的关系将是一个进一步的主题。3.线性PDE的基本解决方案可以被视为积分转换的核心函数。我们在期望与内核的积分转换应以某些类似于连续性的概念为特征，并通过引入半持续点的概念来获得内核定理，并获得了内核定理。在这里，我们使用了具有曲线ra rantransations的复杂核的构造。