An induction principle for the weighted p-energy minimality of x/|x|

In this paper, we investigate minimizing properties of the map x/|x| from the Euclidean unit ball Bn to its boundary Sn−1, for the weighted energy functionals En p,α(u) = Bn |x|α|∇u|p dx, where p 2. We establish the following induction principle: if the map x |x| :Bn+1 →Sn minimizes En+1 p,α among maps u:Bn+1 →Sn satisfying u(x) = x on Sn, then the map y |y| :Bn →Sn−1 minimizes En p,α+1 among maps v :Bn→Sn−1 satisfying v(y) = y on Sn−1. This result enables us to enlarge the range of values of p and α for which x/|x| minimizes En p,α. © 2006 Elsevier Inc. All rights reserved.