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

Study of Green function of higher order / fractional order differential equations from a viewpoint of a reproducing kernel theory

从再生核理论的角度研究高阶/分数阶微分方程的格林函数

基本信息

批准号：
17540175
负责人：
TAKEMURA Kazuo
金额：
$ 1.54万
依托单位：
Tokyo University of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2005
资助国家：
日本
起止时间：
2005 至 2006
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-17540175/
关键词：
Green function Reproducing kernel Sobolev inequality Best constant Ordinary differential equation

项目摘要

(2005) We constructed Green functions under various boundary conditions and showed that the Green functions are reproducing kernels of suitable Hilbert spaces. Based on this fact, we succeeded in calculation of the best constant and the best function for Sobolev inequality by examining a diagonal value of Green function in a detailed manner.We calculated the best constant of a Sobolev inequality corresponding to several boundary value problems including Diriclet type, Neumann type and the periodic type conditions for a string bending problem. If the corresponding eigenvalue problem has a nonpositive eigenvalue, we constitute a generalied Green function by the so-called symmetric orthogonalization method by imposing the solvability and orthogonality condition to the boundary value problem.(2006) We calculated concretely the best constant of a Sobolev inequality corresponding to boundary value problems for 2M-th order differential operator, which contains clumped type, Diriclet type, Neumann type, a free end, a periodic type condition. In particular, the best constant of Dirichlet, Neumann and periodic boundary condition is found and expressed by means of Bernoulli polynomials and Riemann zeta function. This result give a variational meaning of Riemann zeta function. In the other 2 cases, we calculated the best constant of a Sobolev inequality by examining a diagonal value of Green function.

(2005)我们构造了不同边界条件下的格林函数，证明了格林函数是合适的Hilbert空间的再生核。基于这一事实，我们通过详细研究格林函数的对角值，成功地计算了Sobolev不等式的最佳常数和最佳函数，并计算了Sobolev不等式的最佳常数对应于几个边值问题，包括Diriclet型、Neumann型和弦弯曲问题的周期型条件。如果相应的特征值问题有一个非正的特征值，我们通过对边值问题施加可解性和正交性条件，用所谓的对称正交化方法构造一个广义格林函数。(2006)我们具体地计算了2m阶微分算子边值问题对应的Sobolev不等式的最佳常数，它包含块状型、Diriclet型、Neumann型、自由端、周期型条件。特别地，求出了Dirichlet、Neumann和周期边界条件的最佳常数，并用Bernoulli多项式和Riemann Zeta函数表示。这一结果给出了Riemann Zeta函数的变分意义。在另外两种情况下，我们通过检查格林函数的对角值来计算Soblev不等式的最佳常数。