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 , where p⩾2. We establish the following induction principle: if the map minimizes among maps satisfying u(x)=x on Sn, then the map minimizes among maps 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 .