加权能量映射x/ |x|的极小性
The minimality of the map x/|x| for weighted energy
摘要：
本文研究了$E\_{p,f}= \int\_{\mathbf{B}^n}f(r)\|\nabla u\|^p dx$型加权能量泛函从欧氏单位球$\mathbf{B}^n$到其边界$\mathbb{S}^{n-1}$的映射$\frac{x}{\|x\|}$的极小性，其中$f$是非负函数.我们证明了在以下两种情况下：i）$p=1$和$f$是非减的，i））$p$是一个整数，$p \leq n-1$和$f= r^α$与$α\geq 0$，映射$\frac{x}{\|x\|}$在$W^{1,p}(\mathbf{B}^n, \mathbb{S}^{n-1})$中与$\frac{x}{\|x\|}$在$\partial \mathbf{B}^n$上重合的映射中最小化$E\_{p,f}$.我们还研究了$ f(r)= r^α$与$-n+2 < α< 0$的情况，并证明了$\frac{x}{\|x\|}$不会使$α$接近$-n+2$的$E\_{p,f}$最小化，当$n \geq 6$时，$α$接近$4-n$。
英文摘要：
In this paper, we investigate the minimality of the map $\frac{x}{\|x\|}$ from the euclidean unit ball $\mathbf{B}^n$ to its boundary $\mathbb{S}^{n-1}$ for weighted energy functionals of the type $E\_{p,f}= \int\_{\mathbf{B}^n}f(r)\|\nabla u\|^p dx$, where $f$ is a non-negative function. We prove that in each of the two following cases: i) $p=1$ and $f$ is non-decreasing, i)) $p$ is an integer, $p \leq n-1$ and $f= r^α$ with $α\geq 0$, the map $\frac{x}{\|x\|}$ minimizes $E\_{p,f}$ among the maps in $W^{1,p}(\mathbf{B}^n, \mathbb{S}^{n-1})$ which coincide with $\frac{x}{\|x\|}$ on $\partial \mathbf{B}^n$. We also study the case where $ f(r)= r^α$ with $-n+2 < α< 0$ and prove that $\frac{x}{\|x\|}$ does not minimize $E\_{p,f}$ for $α$ close to $-n+2$ and when $n \geq 6$, for $α$ close to $4-n$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
