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
Aitken, A.C.: Determinants and Matrices. Oliver and Boyd, Edinburgh (1956)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin (2001)
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)
Jiang, F., Trudinger, N.S., Yang, X.-P.: On the Dirichlet problem for Monge-Ampère type equations. Calc. Var. PDE 49, 1223–1236 (2014)
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)
Trudinger, N.S.: Recent developments in elliptic partial differential equations of Monge-Ampère type. Proc. Int. Cong. Math. Madrid 3, 291–302 (2006)
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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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DOI: https://doi.org/10.1007/s40306-017-0231-2