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Zurich Open Repository and Archive

Kappeler, T (2002). Perturbations of the harmonic map equation. In: Journées "Équations aux Dérivées Partielles" (Forges-les-Eaux, 2002). Nantes, Ex No. IX, 8. ISBN 2-86939-188-9.

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Abstract

We consider perturbations of the harmonic map equation in the case where the source and target manifolds are closed Riemannian manifolds and the latter is, in addition, of nonpositive sectional curvature. For any semilinear and, under some extra conditions, quasilinear perturbation, the space of classical solutions within a homotopy class is proved to be compact. For generic perturbations the set of solutions is finite and we present a count of this set. An important ingredient for our analysis is a new inequality for maps in a given homotopy class which can be viewed as a version of the Poincaré inequality for such maps.