## 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:

 5.8e+31 Variational principles in infinite-dimensional spaces

Zbl 0652.58023
Full Text: