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A Variational Approach to the Eigenvalue Problem for Complex Hessian Operators

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Nonlinear Analysis, Geometry and Applications

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Abstract

Let \(1 \leq m \leq n\) be two integers and \(\Omega \Subset \mathbb {C}^n\) a bounded m-hyperconvex domain in \(\mathbb {C}^n\). Using a variational approach, we prove the existence of the first eigenvalue and an associated eigenfunction which is m-subharmonic with finite energy for general twisted complex Hessian operators of order m. Under some extra assumption on the twist measure we prove Hölder continuity of the corresponding eigenfunction. Moreover we give applications to the solvability of more general degenerate complex Hessian equations with the right hand side depending on the unknown function.

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Acknowledgements

The authors would like to thank the referees for their careful reading of the first version of this article and for their useful comments and suggestions which made it possible to correct certain statements and improve the presentation of the article.

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Authors and Affiliations

  1. Laboratoire de Mathématiques et Applications, Université Assane Seck de Ziguinchor, Ziguinchor, Senegal

    Papa Badiane

  2. Institut de Mathématiques de Toulouse, Université de Toulouse, CNRS, UPS, Toulouse, Cedex 09, France

    Ahmed Zeriahi

Authors
  1. Papa Badiane
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  2. Ahmed Zeriahi
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Corresponding author

Correspondence to Ahmed Zeriahi .

Editor information

Editors and Affiliations

  1. Department of Mathematics of Decision, University Cheikh Anta Diop of Dakar, Dakar, Senegal

    Diaraf Seck

  2. Université Félix H.B. d’Abidjan, Abidjan, Côte d'Ivoire

    Kinvi Kangni

  3. Université Assane Seck de Ziguinchor, Ziguinchor, Senegal

    Marie Salomon Sambou

  4. Laboratoire de Recherche en Mathématiques, École Normale Supérieure, Libreville, Gabon

    Philibert Nang

  5. African Institute for Mathematical Sciences, Mbour, Senegal

    Mouhamed Moustapha Fall

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Badiane, P., Zeriahi, A. (2024). A Variational Approach to the Eigenvalue Problem for Complex Hessian Operators. In: Seck, D., Kangni, K., Sambou, M.S., Nang, P., Fall, M.M. (eds) Nonlinear Analysis, Geometry and Applications. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-031-52681-7_10

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