The Dirichlet problem for mixed Hessian quotient equations

This paper is concerned with the solvability of the Dirichlet problem of mixed Hessian quotient equations, which precisely are sigma(m)(D(2)u)=sigma(m)(D(2)u)[& sum;(k=1)(m)& sum;(k-1)(j=0)l(kj)(k-j+1)(x)sigma j/sigma k(D(2)u)],for all 1 <= m <= d, which include k-Hessian equation, Hessian quotient equation and mixed k-Hessian equation. We establish the existence of global C (1,1) solution under degenerate case, and furthermore the existence of global C- 2 ,C-alpha solution under nondegenerate case. The uniqueness is also obtained. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.