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Smooth solutions to a class of mixed type Monge–Ampère equations

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Abstract

We prove the existence of C ∞ local solutions to a class of mixed type Monge–Ampère equations in the plane. More precisely, the equation changes type to finite order across two smooth curves intersecting transversely at a point. Existence of C ∞ global solutions to a corresponding class of linear mixed type equations is also established. These results are motivated by and may be applied to the problem of prescribed Gaussian curvature for graphs, the isometric embedding problem for 2-dimensional Riemannian manifolds into Euclidean 3-space, and also transonic fluid flow.

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Authors and Affiliations

  1. Department of Mathematics, University of Notre Dame, Notre Dame, IN, 46556, USA

    Qing Han

  2. Beijing International Center for Mathematical Research, Peking University, Beijing, 100871, China

    Qing Han

  3. Department of Mathematics, Stony Brook University, Stony Brook, NY, 11794, USA

    Marcus Khuri

Authors
  1. Qing Han
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  2. Marcus Khuri
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Correspondence to Qing Han.

Additional information

Communicated by C.S. Lin.

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Han, Q., Khuri, M. Smooth solutions to a class of mixed type Monge–Ampère equations. Calc. Var. 47, 825–867 (2013). https://doi.org/10.1007/s00526-012-0547-7

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  • DOI: https://doi.org/10.1007/s00526-012-0547-7

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