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Concavity of the Lagrangian phase operator and applications

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Abstract

We study the Dirichlet problem for the Lagrangian phase operator, in both the real and complex setting. Our main result states that if \(\Omega \) is a compact domain in \({\mathbb {R}}^{n}\) or \({\mathbb {C}}^n\), then there exists a solution to the Dirichlet problem with right-hand side h(x) satisfying \(|h(x)| > (n-2)\frac{\pi }{2}\) and boundary data \(\varphi \) if and only if there exists a subsolution.

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Acknowledgements

We would like to thank D.H. Phong for all his guidance and support. We also thank Pei-Ken Hung for many helpful discussions. The authors are grateful to Valentino Tosatti and Mu-Tao Wang for helpful comments.

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Authors and Affiliations

  1. Department of Mathematics, Harvard University, 1 Oxford St., Cambridge, MA, USA

    Tristan C. Collins

  2. Department of Mathematics, Columbia University, 2990 Broadway, New York, NY, USA

    Sebastien Picard & Xuan Wu

Authors
  1. Tristan C. Collins
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  2. Sebastien Picard
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  3. Xuan Wu
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Correspondence to Xuan Wu.

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Communicated by O.Savin.

Supported in part by National Science Foundation Grants DMS-1506652 (T.C.C.) and DMS-12-66033 (S.P.).

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Collins, T.C., Picard, S. & Wu, X. Concavity of the Lagrangian phase operator and applications. Calc. Var. 56, 89 (2017). https://doi.org/10.1007/s00526-017-1191-z

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