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年）我们在各种边界条件下构建了绿色功能，并表明绿色功能正在繁殖合适的希尔伯特空间的内核。基于这一事实，我们通过以详细的方式检查了绿色功能的对角线值，成功地计算了Sobolev不平等的最佳常数和最佳功能。我们计算了与索博莱夫不平等的最佳常数相对于几个边界价值问题，包括透明型型透明类型，neumann类型，neumann类型和字符串类型条件的定期条件。 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类型，Neumann类型，自由端，周期性类型条件。特别是，通过Bernoulli多项式和Riemann Zeta功能，发现并表达了Dirichlet，Neumann和周期性边界条件的最佳常数。该结果给出了riemann zeta函数的各种含义。在其他两种情况下，我们通过检查绿色功能的对角线值来计算Sobolev不等式的最佳常数。