The minimality of the map frac{x}{|{x}|} for weighted energy

In this paper, we investigate the minimality of the map from the Euclidean unit ball Bⁿ to its boundary 핊ⁿ⁻¹ for weighted energy functionals of the type E_p,f = ∫_Bⁿ f(r)‖∇ 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,
ii)	p is integer, p ≤ n−1 and f = rα with α ≥ 0, the map minimizes Ep,f among the maps in W1,p(Bn, 핊n−1) which coincide with on ∂ Bn. We also study the case where f(r) = rα with −n+2 < α < 0 and prove that does not minimize Ep,f for α close to −n+2 and when n ≥ 6, for α close to 4−n.

Mathematics Subject Classification (2000) 58E20; 53C43