The minimality of the map x/|x| for weighted energy

TL;DR

This paper proves the minimality of the map x/|x| for certain weighted energy functionals on the unit ball, extending known results and exploring cases where minimality does not hold.

Contribution

It establishes minimality of the map x/|x| for specific weighted energies and identifies conditions where minimality fails, broadening understanding of energy minimizers.

Findings

x/|x| minimizes weighted energy for p=1 with non-decreasing f

x/|x| minimizes for p ≤ n-1 with power weights r^α, α ≥ 0

x/|x| does not minimize for certain negative power weights near -n+2 and 4-n when n ≥ 6

Abstract

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 $f= r^{\alpha}$ with $\alpha \geq 0$, 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^{\alpha}$ with $-n+2 < \alpha < 0$ and prove that $\frac{x}{\|x\|}$ does not minimize $E\_{p,f}$ for $\alpha$ close to $-n+2$ and when $n \geq 6$, for $\alpha$ close to $4-n$.