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

Abstract

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