The critical Hardy inequality and its application to partial differential equations with logarithmic singularity

• Project/Area Number: 15K17575
• Research Category: Grant-in-Aid for Young Scientists (B)
• Allocation Type: Multi-year Fund
• Research Field: Mathematical analysis
• Research Institution: Ehime University
• Principal Investigator: ioku norisuke 愛媛大学, 理工学研究科(理学系), 准教授 (50624607)
• Project Period (FY): 2015-04-01 – 2018-03-31
• Project Status: Completed (Fiscal Year 2017)
• Budget Amount: ¥4,160,000 (Direct Cost: ¥3,200,000、Indirect Cost: ¥960,000); Fiscal Year 2017: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2016: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000); Fiscal Year 2015: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
• Keywords: スケール不変性 / 熱方程式 / Hardyの不等式 / 最良定数 / 対数型特異性 / 偏微分方程式 / 半線形熱方程式 / 存在・一意性の分類

Outline of Final Research Achievements

We clarified a scale invariance structure of the critical Hardy inequality. Namely, we proved that the higher order remainder term is automatically determined by the scale invariance property. Moreover, we investigated classification results of existence and nonexistence of the heat equation with general nonlinearity, and proved its threshold integrability of the initial data. In addition, if the nonlinear term is an exponential type, we obtained the classification of uniqueness and non uniqueness results.

Research Products

• [Int'l Joint Research] Firenze University/Milano University(Italy)
• [Int'l Joint Research] Firenze大学(Italy)
• [Int'l Joint Research] ミラノ大学(イタリア)
• [Journal Article] Canceling effects in higher-order Hardy-Sobolev inequalities (2017)
  • Author(s): N. Ioku and A. Cianchi
  • Journal Title: Calculus of Variations and Partial Differential Equations Volume: 56 Issue: 2
  • DOI: 10.1007/s00526-017-1112-1
  • Peer Reviewed / Int'l Joint Research
• [Journal Article] Hardy type inequalities in $L_p$ with sharp remainders (2017)
  • Author(s): N. Ioku, M. Ishiwata, T. Ozawa
  • Journal Title: Journal of Inequalities and Applications Volume: - Issue: 1 Pages: 1-7
  • DOI: 10.1186/s13660-016-1271-1
  • Peer Reviewed / Open Access
• [Journal Article] A Note on the Scale Invariant Structure of Critical Hardy Inequalities (2016)
  • Author(s): N. Ioku and M. Ishiwata
  • Journal Title: Geometric Properties for Parabolic and Elliptic PDE's, Springer Proceedings in Mathematics & Statistics Volume: 176 Pages: 97-120
  • DOI: 10.1007/978-3-319-41538-3_7
  • ISBN: 9783319415369, 9783319415383
  • Peer Reviewed
• [Journal Article] On a variational problem associated with a Hardy type inequality involving a mean oscillation (2015)
  • Author(s): N. Ioku, M. Ishiwata
  • Journal Title: Calculus of Variations and Partial Differential Equations Volume: 54
  • Peer Reviewed
• [Journal Article] A scale invariant form of a critical Hardy inequality (2015)
  • Author(s): N. Ioku, M. Ishiwata
  • Journal Title: International Mathematics Research Notices Volume: 18
  • Peer Reviewed
• [Journal Article] Existence, Non-existence, and Uniqueness for a Heat Equation with Exponential Nonlinearity in $\mathbb{R}^2$ (2015)
  • Author(s): N. Ioku, B. Ruf, E. Terraneo
  • Journal Title: Mathematical Physics Analysis and Geometry Volume: 18
  • Peer Reviewed / Int'l Joint Research
• [Presentation] Existence and nonexistence of solutions for the heat equation with a superlinear source term (2017)
  • Author(s): Norisuke Ioku
  • Organizer: Pacific Rim Mathematical Association PRIMA 2017
  • Int'l Joint Research
• [Presentation] 高階Hardy-Sobolevの不等式が持つ相殺効果 (2017)
  • Author(s): 猪奥倫左
  • Organizer: 第11回実解析と関数解析による偏微分方程式論
• [Presentation] 半線形熱方程式の可解性の分類 (2017)
  • Author(s): 猪奥倫左
  • Organizer: 愛媛大学における微分方程式セミナー
• [Presentation] 半線形熱方程式の可解性の分類 (2017)
  • Author(s): 猪奥倫左
  • Organizer: 九州関数方程式セミナー
• [Presentation] 半線形熱方程式の可解性の分類 (2017)
  • Author(s): 猪奥倫左
  • Organizer: 金沢解析セミナー
• [Presentation] Existence and nonexistence of solutions for the heat equation with a superlinear source term (2017)
  • Author(s): 猪奥倫左
  • Organizer: 南大阪解析セミナー