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

Abstract

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.

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References

  1. 1.

    Aitken, A.C.: Determinants and Matrices. Oliver and Boyd, Edinburgh (1956)

    Google Scholar 

  2. 2.

    Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin (2001)

    Google 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)

  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)

    Article  MATH  Google 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)

    MathSciNet  Article  MATH  Google 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)

    MATH  Google Scholar 

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Correspondence to Ha Tien Ngoan.

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Ngoan, H.T., Chung, T.T.K. Elliptic Solutions to Nonsymmetric Monge-Ampère Type Equations I: the d-Concavity and the Comparison Principle. Acta Math Vietnam 44, 469–491 (2019). https://doi.org/10.1007/s40306-017-0231-2

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Keywords

  • d-concavity
  • δ-elliptic solution
  • The comparison principle

Mathematics Subject Classification (2010)

  • 35J66