Project/Area Number  09440049 
Research Category 
GrantinAid for Scientific Research (B)

Section  一般 
Research Field 
解析学

Research Institution  TOKYO INSTITUTE OF TECHNOLOGY 
Principal Investigator 
MURATA Minoru Graduate School of Science and Engineering, tokyo Institute ofTechnology, Professor, 大学院・理工学研究科, 教授 (50087079)

CoInvestigator(Kenkyūbuntansha) 
ISHIGE Kazuhiro Graduate School of Polymathmatics, Nagoya University, Assistant Professor, 大学院・多元数理科学研究科, 助教授 (90272020)
KURATA Kazuhiro Graduate School of Science, Tokyo Metropolitan University, Assistant Professor, 大学院・理学研究科, 助教授 (10186489)
AIKAWA Hiroaki Interdiscriplinary Faculity of Science and Engineering, Simane University, Profe, 総合理工学部, 教授 (20137889)
UCHIYAMA Kouhei Graduate School of Science and Engineering, Tokyo Institute of Technology, Profe, 大学院・理工学研究科, 教授 (00117566)
宮岡 礼子 東京工業大学, 理学部, 助教授 (70108182)
SHIGA Hiroshige Graduate School of Science and Engineering, Tokyo Institute of Technology, Assis, 大学院・理工学研究科, 助教授 (10154189)

Project Fiscal Year 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥5,700,000 (Direct Cost : ¥5,700,000)
Fiscal Year 1998 : ¥2,700,000 (Direct Cost : ¥2,700,000)
Fiscal Year 1997 : ¥3,000,000 (Direct Cost : ¥3,000,000)

Keywords  Parabolic PDE / Elliptic PDE / Positive solution / Initial value problem / Unigueness / Intrinsi metric / Harnack inequality / Martin boundary / 放物型偏微分方程式 / 楕円型偏微分方程式 / 正値解 / 初期値問題 / 一意性 / 内在的距離 / Harnactの不等式 / Martin境界 / 内圧的距離 / Harnackの不等式 / Martin 境界 / 放物型偏徴分方程式 
Research Abstract 
M.Murata and K.Ishige studied uniqueness of nonnegative solutions of the Cauchy problem to second order parabolic equations, and gave optimal growth rates at infinity of the coefficients of equations via intrinsic metrics for equations. They further investigated the uniqueness problem deeply, and established a sharp and general uniqueness theorem which unifies all previous results on the uniqueness. This result was reviewed by such experts as A.Ancona, L.SalloffCoste, K.Th. Sturm, E.B.Davies, A.Grigor'yan, and has been submitted. M.Murata introduced a notion of semismall perturbation in the Martin theory for positive solutions of second order elliptic equations, established stably of the structure of positive solutions under semismall perturbations, and gave sufficient conditions for perturbations to be semismall. This result is related to nonuniqueness of nonnegative solutions of the Cauchy problem to second order parabolic equations, lifetime estimates of diffusion processes in probability theory, and the structure of positive solutions to elliptic equations. It is highly estimated by such experts as Y.Pinchover and R.O.Pinsky, and results related to it have been given recently by H.Aikawa and Y.Pinchover. M.Murata also published a survey on the structure of positive solutions to stationary Schrdinger equations. M.Murata and K.Kurata published a book on the theory of elliptic and parabolic partial differential equations which is fundamental for the above and other investigations on partial differential equations . K.Kurata also gave several results on elliptic equations.
