Geodesic
The property of compactness of the quasi-linearly perturbed
Sbornik. Mathematics, Tome 194 (2003) no. 7, pp. 1055-1068 Cet article a éte moissonné depuis la source Math-Net.Ru

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For maps $u\colon M\to M'$ of closed Riemannian manifolds a study is made of the quasi-linearly perturbed harmonic-map equation $$ \tau(u)(x)=\mathsf G(x,u(x))\cdot du(x)+\mathsf g(x,u(x)), \qquad x\in M. $$ In the case of a non-positively curved manifold $M'$ and a small linear part of the perturbation $\mathsf G$ 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 $C^1$-norms of $\mathsf G$ and $\mathsf g$. The crux of this analysis is an inequality called the monotonicity property. 
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G. Yu. Kokarev. The property of compactness of the quasi-linearly perturbed. Sbornik. Mathematics, Tome 194 (2003) no. 7, pp. 1055-1068. http://geodesic.mathdoc.fr/item/SM_2003_194_7_a5/

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