Deepening and application of Sobolev inequality studies using reproducing kernel theory

• Project/Area Number: 17K05374
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Nihon University
• Principal Investigator: TAKEMURA Kazuo 日本大学, 理工学部, 准教授 (60367216)
• Co-Investigator(Kenkyū-buntansha): 亀高 惟倫 大阪大学, その他部局等, 名誉教授 (00047218) ; 山岸 弘幸 東京都立産業技術高等専門学校, ものづくり工学科, 准教授 (10448053)
• Project Period (FY): 2017-04-01 – 2022-03-31
• Project Status: Completed (Fiscal Year 2021)
• Budget Amount: ¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000); Fiscal Year 2021: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000); Fiscal Year 2020: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000); Fiscal Year 2019: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2018: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000); Fiscal Year 2017: ¥650,000 (Direct Cost: ¥500,000、Indirect Cost: ¥150,000)
• Keywords: グリーン関数 / ソボレフ不等式 / 最良定数 / グリーン行列 / 関数方程式論 / 解析・評価

Outline of Final Research Achievements

A representative result in Sobolev inequalities is the successful extension of the results for one-dimensional L2 Sobolev-type inequalities to Lp Sobolev-type inequalities. In the discrete Sobolev inequality, the best constants were obtained for 1812 isomers of C60 fullerene. The best estimate of the discrete Sobolev inequality was found to be related to the physical properties of materials with crystal structures. In particular, the best　constant is considered to be an indicator of the stiffness of the mechanical model under consideration; the smaller the best-constant, the stiffer the material under consideration. The results obtained provide evidence that buckyball is the stiffest of the 1812 isomers.

Academic Significance and Societal Importance of the Research Achievements

本研究成果の中でも特に，離散ソボレフ不等式をC60フラーレンに対する異性体1812個に対して適用して得られた結果は数理的な問題を現実の物質に適用することで，物質の特徴付けを行うことができたという点において，学術的に意義がある。こうして得られた知見は，工学的にも応用が可能であることから，社会的意義も同時に兼ね備えている。本研究成果は結晶構造を持つ物質を対象として，それらの剛性を調べることが可能であることから，今後，工学分野へ波及効果をもたらすものと考えられる。

Research Products

• [Journal Article] SOBOLEV INEQUALITY FOR DEFLECTION PROBLEM OF A STRING WITH NONLOCAL PERIODIC BOUNDARY CONDITIONS (2022)
  • Author(s): Kazuo Takemura, Atsushi Nagai and Yoshinori Kametaka
  • Journal Title: Far East Journal of Mathematical Sciences (FJMS) Volume: 134 Pages: 47-63
  • DOI: 10.17654/0972087122004
  • Peer Reviewed / Open Access
• [Journal Article] The best constant of discrete Sobolev inequality on 1812 C60 fullerene isomers (2020)
  • Author(s): Kametaka Yoshinori、Watanabe Kohtaro、Nagai Atsushi、Takemura Kazuo、Yamagishi Hiroyuki、Sekido Hiroto
  • Journal Title: JSIAM Letters Volume: 12 Issue: 0 Pages: 49-52
  • DOI: 10.14495/jsiaml.12.49
  • NAID: 130007881802
  • ISSN: 1883-0609, 1883-0617
  • Peer Reviewed / Open Access
• [Journal Article] Evaluation of the One-Dimensional Lp Sobolev Type Inequality (2020)
  • Author(s): Kazuo Takemura and Yoshinori Kametaka
  • Journal Title: Mathematics Volume: 8 Issue: 2 Pages: 296-296
  • DOI: 10.3390/math8020296
  • Peer Reviewed / Open Access
• [Journal Article] Sharp Constants of Discrete Sobolev Inequalities on a Cyclic Rectangular Lattice (2019)
  • Author(s): Kazuo Takemura and Yoshinori Kametaka
  • Journal Title: Far East Journal of Mathematical Sciences (FJMS) Volume: 112 Issue: 2 Pages: 175-194
  • DOI: 10.17654/ms112020175
  • Peer Reviewed / Open Access
• [Journal Article] A Hierarchical Structure for the Sharp Constants of Discrete Sobolev Inequalities on a Weighted Complete Graph (2017)
  • Author(s): Kazuo Takemura, Yoshinori Kametaka and Atsushi Nagai
  • Journal Title: Symmetry Volume: 2017 Issue: 1 Pages: 1-1
  • DOI: 10.3390/sym10010001
  • Peer Reviewed / Open Access
• [Presentation] 弾性基盤上の張力をかけた半無限長の棒のたわみのグリーン関数の階層 構造とソボレフ不等式の最良定数 (2022)
  • Author(s): 亀高惟倫，渡辺宏太郎，永井 敦，武村一雄，山岸弘幸
  • Organizer: 日本数学会