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

设u: M→N为黎曼流形与p（实数1 < p <∞）之间的光滑映射。如果u是p能泛函∫_M | du |^pdx的临界点，我们称它为p调和映射。在p = 2的情况下，它变成了通常的调和映射。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是n维球面S^n且p等于M的维数（dim M = n = p）时，我们可以得到S^n到n的n调和映射的存在性。这是Sacks-Uhlenbeck在n = p = 2情况下的推广。设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方程Δ_pu = -λ|u|^<p-2>u具有非平凡解u的最小实数λ。之前，我们在黎曼流形上对它们有几个估计，如Faber-Krahn型不等式，Cheeger型不等式和Cheng型不等式。在本课题中，我们得到了一个离散的类比，即我们定义了图上的p-拉普拉斯不等式，得到了Cheeger型不等式和Brooks型不等式。设G_1 = (V_1, E_1)和G_2 = (V_2, E_2)为两个图，φ: V_1→V_2为映上映射。如果对于任意p-调和函数f (y = φ(x)∈V_2)，复合函数φ^* f = f ο φ是x∈V_1的p-调和函数，则表示映射φ是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，我们考虑在p = 2的情况下与格林核重合的p-拉普拉斯方程的解，并给出一些估计。

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.