Best evaluation of Sobolev inequality based on the perspective of special function theory

在2M阶（-1）^M (d/dx)^2M微分算子的自伴随边值问题中，得到了与无箝位边界条件相对应的Sobolev不等式的最佳评价（最佳常数，最佳函数）。我们还得到了对应于n阶Hurwitz微分算子的Sobolev型不等式的最佳评价。在与连续版Sobolev不等式的最佳评价并行进行的离散版Sobolev不等式中，我们能够计算M=4、6、8、12、20的正则M-面体上离散版Sobolev不等式的最佳常数。在此基础上，我们得到了离散Sobolev不等式对应于弦的弯曲问题的最佳评价。这些结果将成为今后研究离散Sobolev不等式的重要线索。

In the self-adjoint boundary value problem of 2M-th order (-1)^M (d/dx)^2M differential operator, best evaluation (best constant, best function) of a Sobolev inequality corresponding to clamped-free boundary condition were obtained. We also obtained the best evaluation of a Sobolev type inequality corresponding to the n-th order Hurwitz differential operators. In the Sobolev inequality of the discrete version that has proceeded in parallel with the best evaluation of the Sobolev inequality of the continuous version, we were able to compute the best constant of discrete Sobolev inequality on regular M-hedron for M=4, 6, 8, 12, 20. In addition to this result, we obtained the best evaluation of the discrete Sobolev inequality corresponding to a bending problem of a string. These are important results to become the clue in studying the future discrete Sobolev inequality.