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<p>We deal with the Dirichlet problem for nonsymmetric augmented Hessian quotient type equations. First, we look for an admissible solution to the problem for corresponding symmetric augmented Hessian quotient type equations. Then we apply the Banach fixed point theorem to prove the existence of a <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\delta\)</EquationSource> </InlineEquation>-admissible solution in <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(C^{2,\alpha }\)</EquationSource> </InlineEquation> to the problem by assuming that the augmented skew-symmetric matrix is sufficiently small in a certain sense. We also give a necessary condition for the existence and sufficient conditions for uniqueness of this kind of <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\delta\)</EquationSource> </InlineEquation>-admissible solutions.</p>

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The Dirichlet Problem for Nonsymmetric Augmented Hessian Quotient Type Equations

  • 期刊论文

  • 发表日期：

    2025年12月1日

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  • Van-Bang Tran,
  • Tien-Ngoan Ha,
  • Huu-Tho Nguyen,
  • Trong-Tien Phan,
  • Hong-Quang Dinh

摘要

We deal with the Dirichlet problem for nonsymmetric augmented Hessian quotient type equations. First, we look for an admissible solution to the problem for corresponding symmetric augmented Hessian quotient type equations. Then we apply the Banach fixed point theorem to prove the existence of a \(\delta\) -admissible solution in \(C^{2,\alpha }\) to the problem by assuming that the augmented skew-symmetric matrix is sufficiently small in a certain sense. We also give a necessary condition for the existence and sufficient conditions for uniqueness of this kind of \(\delta\) -admissible solutions.