Homotopy classes in Sobolev spaces and the existence of energy minimizing maps. (English) Zbl 0647.58016

White, Brian

Acta Math. 160, No. 1-2, 1-17 (1988).

Let M and N be compact Riemannian manifolds, and \(p\geq 1\). It is not always possible to minimize \(\int_{M}| Df|\) p in a given homotopy class, since such a class need not be closed in the weak topology of an appropriate Sobolev space. The central theme of this paper is that it is however possible to minimize among maps f whose restrictions to a lower dimensional skeleton of a triangulation of M belong to a given homotopy class. More specifically, one of the main results tells that if d is the greatest integer strictly less than p, then nearby (in a Sobolev sense) Lipschitz maps from M to N have restrictions to the d-skeleton M d of M in the same homotopy class. Consequently each f in a Sobolev space \(H^{1,p}(M,N)\) has a well- defined d-homotopy type, which is preserved under weak convergence with uniformly bounded derivatives. Thus \(\int_{M}| Df|\) p can be minimized in every such homotopy class.
The author also shows that a Lipschitz map from \(\partial M\) to N is a trace of \(L^{1,p}(M,N)\), the strong \(\| f\|_ p+\| Df\|_ p\)-closure of Lipschitz maps, if and only if it has an extension to a continuous map from \(\partial M\cup M^{[p]}\) to N.

Reviewer: P.Mattila

MSC:

58E30	Variational principles in infinite-dimensional spaces
58B05	Homotopy and topological questions for infinite-dimensional manifolds

Keywords:

p-energy; homotopy class; Sobolev space

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