Convergence of minimizers for the \(p\)-Dirichlet integral.

*(English)*Zbl 0798.58022The 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)

##### 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 HĂ¶lder 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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