权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

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.对几类齐次型低项Turrittin方程进行了第二次微局部分析。在这种情况下，类似于Mizohata方程的情况，很明显，基本解可以使用第二微局部傅里叶变换理论来构造。在这里，我们使用的结果的增长阶的一些常微分方程的整体解的空间中的傅立叶变换。在更特殊的情况下，逆傅里叶变换满足一些Fuchsian方程。我们得到了Fuchsian方程的解、奇性、指数以及解的一些性质。2.另一方面，我们可以考虑在一个真实的区域和一个复区域的乘积空间上具有全纯参数的微函数的概念。它们被表示为全纯函数的边值，而它们的边值定义了第二微函数的概念，等等，文[1]中的方程在很小的区域内具有微局部椭圆性，但我们可以通过它们的逆函数对具有全纯参数的微函数的作用来构造它们的第二微局部解。这一结果与Kataoka的伪微分算子边值理论之间的关系将是进一步的研究课题。3.线性偏微分方程的基本解可以看作是积分变换的核函数。我们在一个期望下研究了核，即一个具有核的积分变换应该用类似于连续性的概念来刻画，并通过引入半连续性的概念，得到了解析范畴中的核定理。在这里，我们使用了曲线Radon变换的复杂核的构造。