Discretization of Sobolev inequalities and its engineering applications

2018 Fiscal Year Final Research Report

• Project/Area Number: 25400146
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Tsuda University (2017-2018); Nihon University (2013-2016)
• Principal Investigator: Nagai Atsushi 津田塾大学, 学芸学部, 教授 (90304039)
• Co-Investigator(Kenkyū-buntansha): 亀高 惟倫 大阪大学, その他部局等, 名誉教授 (00047218)
• Project Period (FY): 2013-04-01 – 2019-03-31
• Keywords: グリーン関数 / ソボレフ不等式 / 離散化 / 最良定数 / C60 / グラフ

Outline of Final Research Achievements

We first formulated boundary value problems for differential or difference equations which appear in the field of engineerings and found their Green functions or Green matrices. The Green functions or matrices are reproducing kernels for a suitable Hilbert space. From reproducing relations, Sobolev inequalities and their discrete version are derived. The equality conditions for the inequalities are found, that is to say, the best constant and the best function which attains = in the inequality are found by investigating the Green functions or matrices. In particular, discrete Sobolev inequalities for C60 fullerene and fundamental graphs are derived, together with the best constants and the best functions.

Free Research Field

微分方程式と差分方程式

Academic Significance and Societal Importance of the Research Achievements

工学上重要な微分方程式や差分方程式の境界値問題に対してグリーン関数やグリーン行列を厳密に求めることは、工学の問題の数学的基礎付けを与えることに相当する。また対応するソボレフ不等式や離散ソボレフ不等式はC60フラーレンを例にとると、C60を構成する各分子の変位の最大値をC60のエネルギーの定数倍で評価する不等式である。また最良定数はC60の硬さを表す１つの指標であり、工学上の意味は大きいと確信している。