studies on the relationship of the structure of manifolds and p-harmonic functions

流形结构与p调和函数关系的研究

• 批准号: 16540208
• 负责人: TAKEUCHI Hiroshi
• 金额: $ 2.2万
• 依托单位: Shikoku University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2004
• 资助国家: 日本
• 起止时间: 2004 至 2007
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-16540208/
• 关键词: p-harmonic function; p-harmonic map; Riemannian manifold; graph; 幾何学; P-調和写像; p-ラプラシアン; p-harmonic morphism; スペクトル

P-Laplacian Δ_p(1<P<∞)is defined as operator acting on functions on Riemannian manifolds. The p-harmonic function u is defind by Δ_<p>u=div(|∇u|^<p-2> ∇u)=0. In the case of p=2, it becomes the usual harmonic map. We may consider the p-harmonic function the extension of harmonic function. In fact, the p-harmonic function is a critical point of the p-energy functional. Euler-Lagrange equation of it is that of the p-harmonic function. Because this equation is a nonlinear elliptic partial equation, it is hard to handle. We can define the p-Laplacian on graphs also, and define the p-harmonic function on graphs.We consider the spectrum of the p-Laplacian on graphs, p-harmonic morphisms between two graphs, and estimates for the solutions of p-Laplace equations on graphs. More precisely we prove a Ceeger type inequality and a Brooks type inequality for infinite graphs. We showed p-harmonic morphisms and horizontal conformal maps between two graphs are equivalent. We give some estimates for solutions of p-Laplace equations, which coincide with Green kernels in the case of p=2.Harmonic maps flow and p-harmonics flow are closely related to harmonic maps and p-harmonic maps.The stationary state of harmonic maps flow becomes the harmonic map, and the stationary state of p-harmonic maps flow becomes the p-harmonic map. But they do not necessarily converge, but blow-up of the solutions happen. We report this phenomena as research notes in Bulletin of Shikoku University.

P-Laplacian Δ p（1<P<∞）定义为作用于黎曼流形上函数的算子。p-调和函数u定义为Δ_<p>u=div（|u|（<p-2>^^u）=0.在p=2的情况下，它成为通常的调和映射。p-调和函数可以看作是调和函数的延拓。实际上，p-调和函数是p-能量泛函的临界点。它的欧拉-拉格朗日方程就是p-调和函数的欧拉-拉格朗日方程。由于该方程是一个非线性椭圆型偏微分方程，所以处理起来比较困难。我们也可以定义图上的p-Laplacian，定义图上的p-调和函数，考虑图上的p-Laplacian的谱，两个图之间的p-调和态射，以及图上的p-Laplace方程的解的估计。更精确地证明了无限图的一个Ceeger型不等式和一个布鲁克斯型不等式。我们证明了两个图之间的p-调和态射和水平共形映射是等价的。给出了p-Laplace方程解的估计，在p= 2的情况下，这些估计与绿色核一致.调和映射流和p-调和映射流与调和映射和p-调和映射密切相关，调和映射流的稳态成为调和映射，p-调和映射流的稳态成为p-调和映射.但它们不一定收敛，但解的爆破发生了。我们将这一现象作为研究笔记发表在四国大学学报上。

项目成果

調和写像流について

关于谐波映射流程

• 发表时间: 2008
• 期刊: 四国大学紀要, 自然科学編 26
• 作者: [2] Y.Muroya;E.Ishiwata;N.Guglielmi;N.Fukagai;N.Fukagai;深貝暢良;N.Fukagai;N.Fukagai;N.Fukagai;竹内 博
• 通讯作者: 竹内 博

Cut loci and distance functions

切割轨迹和距离函数

• 发表时间: 2007
• 期刊: Math. J. Okayama Univ. 49
• 作者: Yuji Kobayashi;Tomoko Adachi;小林 ゆう治(編集);小林ゆう治;Hiroaki Taniguchi;Hiroaki Taniguchi;Hiroaki Taniguchi;Hiroaki Taniguchi;Hiroaki Taniguchi;Hiroaki Taniguchi;Hiroaki Taniguchi;Hiroaki Taniguchi;Hiroaki Taniguchi;Hiroaki Taniguchi;HiroakI Taniguchi;Hiroaki Taniguchi;Hiroaki Taniguchi;Hiroaki Taniguchi;谷口浩朗;Hiroaki Taniguchi;Hiroaki Taniguchi;Hiroaki Taniguchi;Hiroaki Taniguchi;Hiroaki Taniguchi;Hiroaki Taniguchi;Hiroaki Taniguchi;Hiroaki Taniguchi;J. Itoh and L. Yuan;H. Oshima;J. Itoh & T. Zamfirescu;J. Itoh & T. Sakai
• 通讯作者: J. Itoh & T. Sakai

Boundary regularity for Ricci equation, geometori Convergence, and Gel'fand's inverse boundary problem

Ricci 方程的边界正则性、几何收敛性和 Gelfand 逆边界问题

• 发表时间: 2004
• 期刊: Invent. Math. 158
• 作者: M. Anderson;A. Katsuda;Y. Kurylev;M. Lassas;M. Taylor
• 通讯作者: M. Taylor

Boundary regularity for Ricci equation,geometric convergence,and Gel'fand's inverse boundary problem

Ricci方程的边界正则性、几何收敛性和Gelfand的逆边界问题

• 发表时间: 2004
• 期刊: Invent.Math. 158
• 作者: M.Anderson;A.Katsuda;et. al.
• 通讯作者: et. al.

Notes on the harmonic map heat flows(In Japanese)

关于调和图热流的注释（日语）

• 发表时间: 2008
• 期刊: Bulletin of Shikoku University Ser. B No 26
• 作者: [2] Y.Muroya;E.Ishiwata;N.Guglielmi;N.Fukagai;N.Fukagai;深貝暢良;N.Fukagai;N.Fukagai;N.Fukagai;竹内 博;Hiroshi Takeuchi
• 通讯作者: Hiroshi Takeuchi