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Pré-Publication, Document De Travail Année : 2006

AN INDUCTION PRINCIPLE FOR THE WEIGHTED p-ENERGY MINIMALITY OF x/llxll.

Résumé

In this paper, we investigate minimizing properties of the map $x/\|x\|$ from the Euclidean unit ball $\mathbf{B}^{n}$ to its boundary $\mathbb{S}^{n-1}$, for the weighted energy functionals $E^n_{p,\alpha}(u)=\int_{\mathbf{B}^{n}} \|x\|^{\alpha}\|\nabla u\|^p dx$. We establish the following induction principle: if the map $\frac{x}{\|x\|}:\mathbf{B}^{n+1}\to \mathbb{S}^n$ 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 $\frac{y}{\|y\|}:\mathbf{B}^n\to\mathbb{S}^{n-1}$ 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 $x/\|x\|$ minimizes $E^n_{p,\alpha}$.
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Dates et versions

hal-00018444 , version 1 (02-02-2006)

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Jean-Christophe Bourgoin. AN INDUCTION PRINCIPLE FOR THE WEIGHTED p-ENERGY MINIMALITY OF x/llxll.. 2006. ⟨hal-00018444⟩
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