On the minimality of the p-harmonic map $x/\|x\|$ for weighted energy

Abstract

In this paper, we study the minimality of the map $\frac{x}{\|x\|}$ for the weighted energy functional $E_{f,p}= \int_{\mathbf{B}^n}f(r)\|\nabla u\|^p dx$, where $f : [0,1] \rightarrow \mathbb{R}^{+}$ is a continuous function. We prove that for any integer $p \in \{2, \ldots, n-1\}$ and any non-negative, non-decreasing continuous function f, the map $\frac{x}{\|x\|}$ minimizes E _f,p among the maps in $W^{1,p}(\mathbf{B}^n, \mathbb{S}^{n-1})$ which coincide with $\frac{x}{\|x\|}$ on $\partial \mathbf{B}^n$. The case p = 1 has been already studied in [Bourgoin J.-C. Calc. Var. (to appear)]. Then, we extend results of Hong (see Ann. Inst. Poincaré Anal. Non-linéaire 17: 35–46 (2000)). Indeed, under the same assumptions for the function f, we prove that in dimension n ≥  7 for any real $p \in [2,n)$ with $p \in (n-2\sqrt{n-1},n)$, the map $\frac{x}{\|x\|}$ minimizes E _f,p among the maps in $W^{1,p}(\mathbf{B}^n, \mathbb{S}^{n-1})$ which coincide with $\frac{x}{\|x\|}$ on $\partial \mathbf{B}^n$.