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

Authors:	Jean-Christophe Bourgoin

Institution:	(1) Laboratoire de Mathematiques et Physique Théorique, Université de Tours, Parc Grandmont, 37200 Tours, France

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 $\partial \mathbf{B}^n$ .

Keywords:	Minimizing map p-Harmonic map p-Energy Weighted energy Weakly p-harmonic map
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