Acta Mathematica Vietnamica

, Volume 44, Issue 2, pp 469–491 | Cite as

Elliptic Solutions to Nonsymmetric Monge-Ampère Type Equations I: the d-Concavity and the Comparison Principle

  • Ha Tien NgoanEmail author
  • Thai Thi Kim Chung


We introduce the notion of d-concavity, d ≥ 0, and prove that the nonsymmetric Monge-Ampère type function of matrix variable is concave in an appropriate unbounded and convex set. We prove also the comparison principle for nonsymmetric Monge-Ampère type equations in the case when they are so-called δ-elliptic with respect to compared functions with 0 ≤ δ < 1.


d-concavity δ-elliptic solution The comparison principle 

Mathematics Subject Classification (2010)




  1. 1.
    Aitken, A.C.: Determinants and Matrices. Oliver and Boyd, Edinburgh (1956)zbMATHGoogle Scholar
  2. 2.
    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin (2001)zbMATHGoogle Scholar
  3. 3.
    Ngoan, H.T., Chung, T.T.K.: Elliptic solutions to nonsymmetric Monge-Ampère type equations II. A priori estimates and the Dirichlet problem (in preparation)Google Scholar
  4. 4.
    Jiang, F., Trudinger, N.S., Yang, X.-P.: On the Dirichlet problem for Monge-Ampère type equations. Calc. Var. PDE 49, 1223–1236 (2014)CrossRefzbMATHGoogle Scholar
  5. 5.
    Jiang, F., Trudinger, N.S., Yang, X.-P: On the Dirichlet problem for a class of augmented Hessian equations. J. Diff. Eqns. 258, 1548–1576 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Trudinger, N.S.: Recent developments in elliptic partial differential equations of Monge-Ampère type. Proc. Int. Cong. Math. Madrid 3, 291–302 (2006)zbMATHGoogle Scholar

Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2017

Authors and Affiliations

  1. 1.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam
  2. 2.University of Transport TechnologyHanoiVietnam

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