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 extrinsic $k$-energy functional is defined on the Sobolev space $W^{k,2}\left (B^n,\mathbb{S} ^n \right )$ as follows: $E_{k}^{{\textrm{ext}}}(u)=\int _{B^n}|\Delta ^s u|^2\ dx$ when $k=2s$, and $E_{k}^{{\textrm{ext}}}(u)=\int _{B^n}|\nabla \Delta ^s u|^2\ dx$ 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 $E_{k}^{{\textrm{ext}}}(u)$ 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.

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关于高阶能量泛函赤道图的稳定性

令 $B^n\subset \mathbb{R} ^{n}$ 和 $\mathbb{S} ^n\subset \mathbb{R} ^{n+1}$ 表示欧几里得 $n$ 维单位球和球体，分别。外在 $k$-energy 泛函在 Sobolev 空间 $W^{k,2}\left (B^n,\mathbb{S} ^n \right )$ 上定义如下： $E_{k}^{ {\textrm{ext}}}(u)=\int _{B^n}|\Delta ^su|^2\dx$ 当$k=2s$ 和$E_{k}^{{\textrm{ ext}}}(u)=\int _{B^n}|\nabla \Delta ^su|^2\dx$ 当$k=2s+1$。这些能量泛函是经典外在双能的自然高阶版本，也称为 Hessian 能量。赤道图$u^*: B^n \to \mathbb{S} ^n$，定义为$u^*(x)=(x/|x|,0)$，是$E_的临界点{k}^{{\textrm{ext}}}(u)$ 假设 $n \geq 2k+1$。本文的主要目的是建立关于$k$ 和$n$ 的充要条件，其中$u^*：