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On the Dirichlet problem for \(p\)-harmonic maps I: compact targets

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Abstract

In this paper we solve the relative homotopy Dirichlet problem for \(p\)-harmonic maps from compact manifolds with boundary to compact manifolds of non-positive sectional curvature. The proof, which is based on the direct calculus of variations, uses some ideas of B. White to define the relative d-homotopy type of Sobolev maps. One of the main points of the proof consists in showing that the regularity theory by Hardt and Lin can be applied. A comprehensive uniqueness result for general complete targets with non-positive curvature is also given.

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Correspondence to Stefano Pigola.

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The second author has been partially supported by INdAM Fellowships in Mathematics and/or Applications for Experienced Researchers cofunded by Marie Curie.

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Pigola, S., Veronelli, G. On the Dirichlet problem for \(p\)-harmonic maps I: compact targets. Geom Dedicata 177, 307–322 (2015). https://doi.org/10.1007/s10711-014-9991-1

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