Elliptic Solutions to Nonsymmetric Monge-Ampère Type Equations II. A Priori Estimates and the Dirichlet Problem

Abstract

In this paper, we consider the Dirichlet problem for nonsymmetric Monge-Ampère type equations in which a skew-symmetric matrix is introduced. We establish uniform with respect to a class of skew-symmetric matrix bounds for δ-elliptic \(C^{2, \alpha }(\overline {{\Omega }})\)-solutions to the Dirichlet problem. Then, we prove the classical solvability of the Dirichlet problem, provided those skew-symmetric matrices are sufficiently small in some sense.

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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 II. A Priori Estimates and the Dirichlet Problem. Acta Math Vietnam 44, 723–749 (2019). https://doi.org/10.1007/s40306-018-0270-3

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Keywords

  • Monge-Ampère type equations
  • δ-elliptic solutions
  • Second derivative estimates
  • The method of continuity

Mathematics Subject Classification (2010)

  • 35J66