An Induction Principle for the Weighted p-Energy Minimality of x/|x|

Abstract

In this paper, we investigate minimizing properties of the map $x/\Vert x\Vert$ from the Euclidean unit ball ${\mathbf{B}}^n$ to its boundary ${\mathbb{S}}^{n-1}$, for the weighted energy functionals $E^n\_p,\alpha (u)=\int \_{\mathbf{B}}^n\Vert x{\Vert }^{\alpha }\Vert \nabla u{\Vert }^{p}dx$. We establish the following induction principle: if the map $\frac{x}{\Vert x\Vert }:{\mathbf{B}}^{n+1}\to {\mathbb{S}}^n$ minimizes $E^{n+1}\_p,\alpha$ among the maps $u:{\mathbf{B}}^{n+1}\to {\mathbb{S}}^n$ satisfying $u(x)=x$ on ${\mathbb{S}}^n$, then the map $\frac{y}{\Vert y\Vert }:{\mathbf{B}}^n\to {\mathbb{S}}^{n-1}$ minimizes $E^n\_p,\alpha +1$ among the maps $v:{\mathbf{B}}^n\to {\mathbb{S}}^{n-1}$ satisfying $v(y)=y$ on ${\mathbb{S}}^{n-1}$. This result enables us to enlarge the range of values of $p$ and $\alpha$ for which $x/\Vert x\Vert$ minimizes $E^n\_p,\alpha$.

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