Boundary Value Problems for Systems of Linear Partial Differential Equations with Regular Singularities
Boundary Value Problems for Systems of Linear Partial Differential Equations with Regular Singularities
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具有正则奇点的线性偏微分方程组的边值问题
DOI：
10.2969/aspm/00410391
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发表时间：
1984
期刊：
影响因子：
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通讯作者：
T. Oshima
中科院分区：
文献类型：
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作者：
T. Oshima
[K-O]引入了具有正则奇点的线性偏微分方程组及其边值问题的概念。一个典型的例子是复平面C中单位圆盘上的拉普拉斯函数Li=(L-lzlzyaZjazaz)，它沿着圆盘的边界具有规则的奇性。在这种情况下，S.Helason证明了李灿的任一本征函数都可以由边界上超函数的泊松积分得到。逆对应由取[K-O]中定义的解的边值映射给出。一般说来，非紧型黎曼对称空间上不变微分算子的任何同时特征函数都可以由对称空间边界上的超函数的泊松积分给出。[K-O]的主要目的是证明这一说法，实际上它在[K-K-]中得到了解决。当我们考虑黎曼(或半单)对称空间在良好紧致流形中的实现时(参看。[0 2]和[O-S])，不变微分算子沿边界具有正则奇性。因此，为了更深入地分析对称空间，我们需要对具有正则奇点的微分方程组进行更深入的研究。这是写这篇论文的主要动机，本文对这一主题的几个应用将在随后的论文中出现。其中之一将在[毛]中找到。我们将提到[K-O]和本文之间的一些区别。在这篇文章中，我们讨论了一个微分方程组，它不一定有一个未知函数，而是有限个。这使我们能够研究定义在对称空间上的向量丛中的微分方程组。此外，在[K-O]中，我们只考虑其个数恰好等于边界余维的微分方程组。但在这里，我们取消了这个限制，我们可以考虑更多的方程，以满足溶胶)。则解的边值可以满足某些方程。这些推导出的方程将是
A concept of systems of linear partial differential equations with regular singularities and their boundary value problems were introduced by [K-O]. A typical example is the Laplacian LI=(l-lzlzyaZjazaz on the unit disc in the complex plane C, which has regular singularity along the boundary of the disc. In this case S. Helgason proved that any eigenfunction of LI can be obtained by the Poisson integral of a hyperfunction on the boundary. The inverse correspondence is given by the map of taking the boundary value of the solution, which was defined in [K-O]. In general any simultaneous eigenfunction of the invariant differential operators on a Riemannian symmetric space of the non-compact type can be given by the Poisson integral of a hyperfunction on a boundary of the symmetric space. The main purpose of [K-O] was to prove this statement and in fact it was solved in [K-K-]. When we consider a realization of a Riemannian (or semisimple) symmetric space in a nice compact manifold (cf. [0 2] and [O-S]), the invariant differential operator has regular singularities along the boundaries. Hence for a deeper analysis on a symmetric space, we need a deeper study on systems of differential equations with regular singularities. This is a main motivation to write this paper and several applications of this paper to this subject will appear in subsequent papers. One of them will be found in [MaO]. We will mention some differences between [K-O] and this paper. In this paper we discuss a system of differential equations which has not necessarily one unknown function but finitely many. This enables us to study a system of differential equations defined in a vector bundle over a symmetric space. Moreover in [K-O] we only consider a system of differential equations whose number equals just the codimension of the boundary. But here we remove this restriction and we can consider more equations that the sol).ltion satisfies. Then the boundary value of the solution may satisfy some equations. These induced equations will be
DOI：
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发表时间：
2006
期刊：
影响因子：
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作者：
M.Mimura;T.Miyaji and I.Ohnishi;S.KAWANO and S.KONISIiI;T. Oshima
通讯作者：
T. Oshima