Study on the structure of nonnegative solutions for parabolic equations and the perturbation theory of elliptic operators

抛物型方程非负解结构及椭圆算子摄动理论研究

• 批准号: 21540164
• 负责人: MURATA Minoru
• 金额: $ 2.58万
• 依托单位: Tokyo Institute of Technology
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2009
• 资助国家: 日本
• 起止时间: 2009 至 2011
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-21540164/
• 关键词: 解析学; 関数方程式; 関数解析学; 確率論; 放物型方程式; 熱核; 積分表示; 楕円型方程式; グリーン関数; 半小摂動; マルチン境界; 非負値解

We investigated the structure of nonnegative solutions to parabolic equations in cylinders on Riemannian manifolds, and gave explicit integral representation formulas for any solutions under the general and optimal condition that the constant function 1 is a semismall perturbation of the associated elliptic operator ; whose geometric characterization was also given in the case of the heat equation on rotationally symmetric Riemannian manifolds. Furthermore, by using the characterization and giving a sharp sufficient condition for the uniqueness of nonnegative solutions to the Cauchy problem, we determined the structure of nonnegative solutions to the heat equation on rotationally symmetric Riemannian manifolds.

研究了黎曼流形上圆柱抛物型方程非负解的结构，给出了在常数函数1是相关椭圆算子的半小扰动的一般最优条件下任意解的显式积分表示公式；在旋转对称黎曼流形上的热方程的情况下也给出了其几何特征。此外，通过利用该表征并给出柯西问题非负解唯一性的尖锐充分条件，我们确定了旋转对称黎曼流形上热方程非负解的结构。

项目成果

Riemann mappings of invariant components of Kleinian groups

• DOI: 10.1112/jlms/jdp052
• 发表时间: 2009-12
• 期刊: Journal of the London Mathematical Society
• 作者: H. Shiga

Nonnegative solutions of the heat equation on rotationally symmetric Riemannian manifolds and semismall perturbations, Rev

旋转对称黎曼流形和半小扰动上热方程的非负解，Rev

• DOI: 10.4171/rmi/656
• 发表时间: 2011
• 期刊: Mat. Iberoam
• 作者: B.Ding;S.Grove;藤井斉亮;T. Mabuchi;M. Murata
• 通讯作者: M. Murata

Teichmuller spaces and holomorphic maps, Teichmuller theory

Teichmuller 空间和全纯映射，Teichmuller 理论

• 发表时间: 2010
• 作者: Motoko Kotani;Masanori Hino and Takashi Kumagai;Kazuhiro Ishige;大塚孝治;澤野達哉;H. Shiga
• 通讯作者: H. Shiga

歪積型楕円型方程式の正値解の構造

应变积型椭圆方程正解的结构

• 发表时间: 2011
• 作者: Heittokangas;J.;Thoge;K.;羽鳥理;村田實
• 通讯作者: 村田實

The hitting distributions of a line for two dimensional Random Walks

二维随机游走的一条线的命中分布

• 发表时间: 2010
• 期刊: Trans.American Mathematical Soc.
• 作者: Ikehata;M;M.Ohya;西山陽一;Kohei Uchiyama
• 通讯作者: Kohei Uchiyama