Refinement of classical inequalities and its application to elliptic variational problems

2015 Fiscal Year Final Research Report

• Project/Area Number: 24540157
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Basic analysis
• Research Institution: Ibaraki University
• Principal Investigator: Horiuchi Toshio 茨城大学, 理学部, 教授 (80157057)
• Co-Investigator(Kenkyū-buntansha): NAKAI EIICHI 茨城大学, 理学部, 教授 (60259900) ; SHIMOMURA KATUNORI 茨城大学, 理学部, 教授 (00201559) ; ANDO HIROSHI 茨城大学, 理学部, 講師 (60292471)
• Co-Investigator(Renkei-kenkyūsha): HOSHIRO TOSHIHIKO 兵庫県立大学, 物質理学研究科, 教授 (40211544) ; SATO TOKUSHI 東北大学, 理学研究科, 助教 (00261545)
• Project Period (FY): 2012-04-01 – 2016-03-31
• Keywords: ＣＫＮ型不等式 / ハーディ不等式 / ソボレフ不等式 / 楕円型変分問題 / ミッシング・ターム / p－ラプラシアン / 加藤の不等式 / 強最大値原理

Outline of Final Research Achievements

(1) Existence of the solutions for the best constants of the Caffarelli-Kohn-Nirenberg type inequalities, continuity with respect to parameters of the best constant, symmetry breaking were studied systematically. For p=1, the Caffarelli-Kohn-Nirenberg type inequalities were established by effective use of isoperimetric inequalities, and symmetry breaking in p=1 was proved. (2) A theory of super logarithm based on logarithmic infinite product was introduced, and existence of infinitely many missing term was shown for the weighted Hardy type inequalities. (3) Differential equations characterizing super logarithm was studied to deal with elliptic equations with nonlinear term of very fast growth order.

Free Research Field

偏微分方程式論