The property of compactness of the quasi-linearly perturbed harmonic-map equation
G Yu Kokarev · Sbornik Mathematics · 2003
For maps u:M{yields}M' of closed Riemannian manifolds a study is made of the quasi-linearly perturbed harmonic-map equation {tau}(u)(x)=G(x,u(x)).du(x)+g(x,u(x), x element of M. In the case of a non-positively curved manifold M' and a small linear part of the perturbation G it is proved that the space of classical solutions in a fixed homotopy class is compact. The proof is based on a uniform estimate for the norm of the differential of a solution of the perturbed equation in terms of its energy and the C{sup 1}-norms of G and g. The crux of this analysis is an inequality called the monotonicity property.