The minimality of the map x/|x| for weighted energy
| dc.creator | Bourgoin, Jean-Christophe | |
| dc.date | 2006-04-03 | |
| dc.date.accessioned | 2026-07-07T07:39:40Z | |
| dc.date.available | 2026-07-07T07:39:40Z | |
| dc.description | 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 $4-n$. | |
| dc.identifier | https://arxiv.org/abs/math/0604038 | |
| dc.identifier | http://arxiv.org/abs/math/0604038 | |
| dc.identifier | Calculus of Variation and P.D.E's 25 (2006) 469-489 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/121537 | |
| dc.subject | Differential Geometry | |
| dc.subject | Analysis of PDEs | |
| dc.subject | 58E20 53C43 | |
| dc.title | The minimality of the map x/|x| for weighted energy | |
| dc.type | text |