##
**Minimizing p-harmonic maps into spheres.**
*(English)*
Zbl 0677.58021

The authors show that certain discontinuous mappings from the Euclidean unit ball \(B^ m\) into the unit sphere \(S^ n\) minimize the p-energy
\[
E_ p(u)=\int_{B^ m}| \nabla u|^ p dx
\]
among all mappings v: \(B^ m\to S^ n\) having the same boundary values as u, where p is a given integer \(\geq 1\). One specific example is the mapping u defined by choosing a totally geodesic \(S^ n\subset S^{m-1}=\partial B^ m\) and assigning to each \(x\in B^ m\) the nearest point \(u(x)\in S^ n\). This mapping is shown to minimize \(E_ p\) provided that \(p\leq n\leq m-1\). A second, and simpler, proof for this example has been given by M. Avellaneda and F.-H. Lin [C. R. Acad. Sci., Paris, Sér. I 306, No.8, 355-358 (1988; Zbl 0652.58023)]. Another example is the Hopf mapping u: \(B^{2n}\to S^ n\) for \(n=2,4\) or 8, defined by
\[
u(z,w)=\frac{(| z|^ 2-| w|^ 22z\bar w)}{| z|^ 2+| w|^ 2}\in {\mathbb{R}}\times {\mathbb{R}}^ n
\]
where \(z\bar w\) denotes multiplication on \({\mathbb{R}}^ n\) of complex numbers, quaternions or Cayley numbers, respectively, which is shown to minimize \(E_ 2\) and \(E_ n\). In general, the authors require: (i) that u be Lipschitz-continuous on the complement of a set of codimension \(p+1\); (ii) that Du map the orthogonal complement of its kernel homothetically a.e. (horizontal conformality); and (iii) that the inverse image \(u^{- 1}(S^{n-p})\) of almost every totally geodesic (n-p)-sphere in \(S^ n\) have minimum (m-p)-dimensional mass for its boundary (smooth mappings with these properties are called harmonic morphisms). A key part of the proof is the strong density in \(W^{1,p}(B^ m,S^ p)\) of mappings which are smooth except on sets of codimension \(p+1\), with prescribed discontinuous values near these sets. The proof also exploits the coarea formula, along with an averaging lemma for \(p\times p\) Jacobians of u: \(B^ m\to S^ n\), in which u is composed with the nearest-point projection \(S^ n\to S^ p\) and an average is formed over all totally geodesic \(S^ p\subset S^ n\). The authors also show that for any mapping u: \(B^{n+1}\to S^ n\), which satisfies \(u(-x)=-u(x)\) for \(x\in \partial B^{n+1}\), the n-energy
\[
E_ n(u)\geq n^{n/2}Vol(S^ n)=E_ n(x/| x|).
\]

Reviewer: J.-M.Coron

### MSC:

58E30 | Variational principles in infinite-dimensional spaces |