论文标题
关于高阶能量功能的赤道图的稳定性
On the stability of the equator map for higher order energy functionals
论文作者
论文摘要
令$ b^n \ subset {\ mathbb r}^{n} $和$ {\ mathbb s}^n \ subset {\ mathbb r}^{n+1} $表示Euclidean $ n $ n $ - dimementional-dimensional单位球和球体分别分别分别。 \ textIt {外部$ k $ -energy函数}在sobolev空间$ w^{k，2} \ left（b^n，{\ mathbb s}^n \ right）$如下：当$ k = 2s $，而$ e_ {k}^{\ rm ext}（u）= \ int_ {b^n} | \nablaΔ^s u |^2 \，dx $，$ k = 2s+1 $。这些能量功能是经典外部生物体的自然高阶版本，也称为Hessian Energy。赤道映射$ u^*：b^n \ to {\ mathbb s}^n $，由$ u^**（x）=（x/| x |，0）$定义，是$ e_ {k}^{\ rm ext} {\ rm ext}（u rm ext}（u）的关键点。本文的主要目的是在$ k $和$ n $上建立必要的条件，根据$ u^*：b^n \ to {\ mathbb s}^n $，对于外部$ k $ - 富集而言是最小化或不稳定的。
Let $B^n\subset {\mathbb R}^{n}$ and ${\mathbb S}^n\subset {\mathbb R}^{n+1}$ denote the Euclidean $n$-dimensional unit ball and sphere respectively. The \textit{extrinsic $k$-energy functional} is defined on the Sobolev space $W^{k,2}\left (B^n,{\mathbb S}^n \right )$ as follows: $E_{k}^{\rm ext}(u)=\int_{B^n}|Δ^s u|^2\,dx$ when $k=2s$, and $E_{k}^{\rm ext}(u)=\int_{B^n}|\nabla Δ^s u|^2\,dx$ when $k=2s+1$. These energy functionals are a natural higher order version of the classical extrinsic bienergy, also called Hessian energy. The equator map $u^*: B^n \to {\mathbb S}^n$, defined by $u^*(x)=(x/|x|,0)$, is a critical point of $E_{k}^{\rm ext}(u)$ provided that $n \geq 2k+1$. The main aim of this paper is to establish necessary and sufficient conditions on $k$ and $n$ under which $u^*: B^n \to {\mathbb S}^n$ is minimizing or unstable for the extrinsic $k$-energy.
