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On the Asymptotic Dirichlet Problem for the Minimal Hypersurface Equation in a Hadamard Manifold

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Abstract

We study the Dirichlet problem at infinity on a Cartan-Hadamard manifold M of dimension n ≥ 2 for a large class of operators containing, in particular, the p-Laplacian and the minimal graph operator. We extend several existence results obtained for the p-Laplacian to our class of operators. As an application of our main result, we prove the solvability of the asymptotic Dirichlet problem for the minimal graph equation for any continuous boundary data on a (possibly non rotationally symmetric) manifold whose sectional curvatures are allowed to decay to 0 quadratically.

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Authors and Affiliations

  1. Département de Mathématique, Université Libre de Bruxelles, CP 214, Boulevard du Triomphe, B-1050, Bruxelles, Belgium

    Jean-Baptiste Casteras

  2. Department of Mathematics and Statistics, University of Helsinki, P.O. Box 68, 00014, Helsinki, Finland

    Ilkka Holopainen

  3. UFRGS, Instituto de Matemática, Av. Bento Goncalves 9500, 91540-000, Porto Alegre-RS, Brasil

    Jaime B. Ripoll

Authors
  1. Jean-Baptiste Casteras
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  2. Ilkka Holopainen
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  3. Jaime B. Ripoll
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Corresponding author

Correspondence to Ilkka Holopainen.

Additional information

J.-B.C. supported by the CNPq (Brazil) project 501559/2012-4; I.H. supported by the Academy of Finland, project 252293; J.R. supported by the CNPq (Brazil) project 302955/2011-9

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Casteras, JB., Holopainen, I. & Ripoll, J.B. On the Asymptotic Dirichlet Problem for the Minimal Hypersurface Equation in a Hadamard Manifold. Potential Anal 47, 485–501 (2017). https://doi.org/10.1007/s11118-017-9624-z

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  • DOI: https://doi.org/10.1007/s11118-017-9624-z

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