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The properties of P-harmonic maps and the application to Geometry

P调和映射的性质及其在几何中的应用

基本信息

批准号：
11640221
负责人：
TAKEUCHI Hiroshi
金额：
$ 2.11万
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：
1999
资助国家：
日本
起止时间：
1999 至 2001
项目状态：
已结题

项目摘要

Let u : M → N be a smooth map between Riemannian manifolds and p a real number 1 < p < ∞. We call u a p-harmonic map if it is a critical point of the p-energy functional ∫_M | du |^pdx. In the case of p = 2, it becomes the usual harmonic map. When N is a real number, the map u becomes the p-harmonic function and it is the solution of Δ_pu =div(|∇u|^<p-2>∇u) = 0. When M is the n-dimensional sphere S^n and p is equal to the dimension of M (dim M = n = p), we can get the existence of n-harmonic maps from S^n to N. This is the generalization of the results of Sacks-Uhlenbeck, which is the case of n = p = 2.Let N be a real number. For the p-Laplacian Δ_p, we define the first eigenvalue of the p-Laplacian as the least real number λ for which the equation Δ_pu = -λ|u|^<p-2>u has a nontrivial solution u. Before, we had several estimates for them on Riemannian manifolds, such as the Faber-Krahn type inequality, the Cheeger type inqulity, and the Cheng type inequality. We get a discrete analogue in this project term, that is, we define the p-Laplacian on graphs and get the Cheeger type inequality and the Brooks type inequality. Let G_1 = (V_1, E_1) and G_2 = (V_2, E_2) be two graphs and φ : V_1 → V_2 an onto mapping. The map φ is said to be a p-harmonic morphism of G_1 to G_2 if for any p-harmonic function f at y = φ(x) ∈ V_2, the composition φ^* f = f ο φ is p-harmonic function at x ∈ V_1. We show the p-harmonic morphism is equivalent to the horizontally conformal.Next we consider the solution of p-Laplace equations which coincide with Green kernels in the case of p = 2 and give some estimates.

令 u : M → N 为黎曼流形之间的平滑映射，p 为实数 1 < p < ∞。如果 u 是 p 能量泛函 ∫_M | 的临界点，我们称 u 为 p 调和映射。杜|^pdx。当p=2时，就变成了通常的调和图。当 N 为实数时，映射 u 成为 p 调和函数，它是 Δ_pu =div(|∇u|^<p-2>∇u) = 0 的解。当 M 为 n 维球体 S^n 且 p 等于 M 的维数（dim M = n = p）时，我们可以得到从 S^n 到 N 的 n 调和映射的存在性。这是结果的推广 Sacks-Uhlenbeck，即 n = p = 2 的情况。令 N 为实数。对于 p-拉普拉斯算子 Δ_p，我们将 p-拉普拉斯算子的第一个特征值定义为最小实数 λ，其中方程 Δ_pu = -λ|u|^<p-2>u 具有非平凡解 u。之前，我们对黎曼流形进行了一些估计，例如 Faber-Krahn 型不等式、Cheeger 型不等式和 Cheng 型不等式。在这个项目项中我们得到了一个离散类比，即我们在图上定义了p-拉普拉斯算子，并得到了Cheeger型不等式和Brooks型不等式。令 G_1 = (V_1, E_1) 和 G_2 = (V_2, E_2) 为两个图，并且 φ : V_1 → V_2 为一个到映射。如果对于任何在 y = φ(x) ε V_2 处的 p 调和函数 f，组合 φ^* f = f ο φ 是在 x ∈ V_1 处的 p 调和函数，则映射 φ 被称为 G_1 到 G_2 的 p 调和态射。我们证明了p调和态射等价于水平共形。接下来我们考虑p-拉普拉斯方程的解，它在p = 2的情况下与格林核一致，并给出一些估计。