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

设u：M → N是黎曼流形与p之间的光滑映射，p是真实的数1 < p < ∞.我们称u为p-调和映射，如果它是p-能量泛函的临界点|杜|^pdx.在p = 2的情况下，它成为通常的调和映射。当N是真实的数时，映射u成为p-调和函数，并且它是Δ_pu =div（|u|（<p-2>^^u）= 0.当M是n维球面S^n，且p等于M的维数（dimM = 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Sacks-Uhlenbeck的结果的推广，即n = p = 2的情形。设N为真实的数。对于p-Laplacian算子Δ p，我们定义p-Laplacian算子的第一特征值为满足方程Δ pu = -λ的最小真实的数λ| u|有<p-2>一个非平凡解u。在此之前，我们对它们在黎曼流形上的一些估计，如Faber-Krahn型不等式、Cheeger型不等式和Cheng型不等式。在这个项目中我们得到了一个离散的类比，即我们定义了图上的p-Laplacian，得到了Cheeger型不等式和布鲁克斯型不等式.设G_1 =（V_1，E_1）和G_2 =（V_2，E_2）是两个图，φ：V_1 → V_2是到上映射。称映射φ是G_1到G_2的p-调和态射，如果对任意的p-调和函数f在y = φ（x）∈ V_2，复合φ^* f = f o φ在x ∈ V_1是p-调和函数。我们证明了p-调和态射与水平共形态射是等价的，然后我们考虑了p = 2时p-Laplace方程满足绿色核的解，并给出了一些估计。

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.