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Convergence of minimizers for the $$p$$-Dirichlet integral. (English) Zbl 0798.58022
The introduction: The purpose of this paper is to show that a weakly convergent sequence of minimizing $$p$$-harmonic maps, $$\infty > p > 1$$, between Riemannian manifolds converges to a minimizing $$p$$-harmonic map. For special target manifolds this has been proved by R. Hardt and F. H. Lin [Commun. Pure Appl. Math. 40, 555-588 (1987; Zbl 0646.49007), Theorem 6.4]; and if the domain is a ball, it has been proved by the author [Indiana Univ. Math. J. 37, No. 2, 349-368 (1988; Zbl 0641.58012)], and indeed used crucially in the blow up argument of that paper. The proof given here is a straightforward generalization of the latter method. The result is stated for more general functionals and problems with constraints, as it does not complicate the proof.
Reviewer: A.Ratto (Brest)

##### MSC:
 58E20 Harmonic maps, etc. 49Q20 Variational problems in a geometric measure-theoretic setting
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##### References:
 [1] Hardt, R., Lin, F.H.: Mappings minimizing theL p-norm of the gradient. Commun. Pure Appl. Math.40, 555–588 (1987) · Zbl 0646.49007 [2] Luckhaus, S.: Partial Hölder continuity for minima of certain energies among maps into a Riemannian manifold. Indiana Univ. Math. J.37, 349–367 (1988) · Zbl 0641.58012 [3] White, B.: Homotopy classes in Sobolev spaces and the existence of energy minimizing maps. Acta Math.160, 1–17 (1988) · Zbl 0647.58016
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