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

Jean-Christophe Bourgoin (1)

1 LMPT - Laboratoire de Mathématiques et Physique Théorique

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 $4-n$.

Mots clés: minimizing map; $p$-harmonic map; $p$-energy; weighted energy.; weighted energy

Domaines: Géométrie différentielle [math.DG]; Géométrie métrique [math.MG]; Equations aux dérivées partielles [math.AP]

Jean-Christophe Bourgoin. "The minimality of the map x/|x| for weighted energy".. Calculus of Variation and P.D.E's, 2006, 25, pp.469-489. ⟨hal-00022138⟩