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

（2005）我们在各种边界条件下构造了Green函数，并证明了Green函数再现了合适的Hilbert空间的核。在此基础上，通过对Green函数对角值的详细考察，成功地计算出Sobolev不等式的最佳常数和最佳函数。针对弦弯曲问题的Diriclet型、Neumann型和周期型条件等边值问题，计算了Sobolev不等式的最佳常数。如果相应的特征值问题具有非正特征值，我们通过对边值问题施加可解性和正交性条件，用所谓的对称正交方法构造一个广义格林函数。（2006）我们具体计算了2m阶微分算子边值问题对应的Sobolev不等式的最佳常数，该方程包含聚集型、Diriclet型、Neumann型、自由端、周期型条件。特别地，找到了Dirichlet、Neumann和周期边界条件的最佳常数，并用伯努利多项式和Riemann zeta函数表示出来。这个结果给出了黎曼ζ函数的变分意义。在另外两种情况下，我们通过检查Green函数的对角线值来计算Sobolev不等式的最佳常数。

(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.