{"ID":22964813,"CreatedAt":"2026-09-17T02:29:35.102872001Z","UpdatedAt":"2026-09-17T02:29:35.102872001Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.18889","arxiv_id":"2609.18889","title":"Existence of Admissible Subsolutions to the Dirichlet Problem for Symmetric Augmented $k$-Hessian Type Equations in Bounded Domains","abstract":"We prove the existence of admissible subsolutions to the Dirichlet problem for symmetric augmented $k$-Hessian type equations. An important sufficient condition is the uniform $(k-1)$-$A$-convexity of the domain $Ω,$ where $A(x, z, p)$ is the augmented symmetric matrix appearing in the equation. This condition was originally introduced by F. Jiang, N. S. Trudinger, and X.-P. Yang and we have chosen a special their case. The structural conditions on the matrix $A(x, z, p)$ include its growth with respect to the variables $z$ and $p,$ particularly requiring that some of its first and second derivatives are sufficiently small in a sufficiently small neighborhood of the boundary. Under certain structural conditions on $A(x, z, p),$ the uniform $(k-1)$-$A$-convexity of $Ω$ is also a necessary condition for the existence of admissible subsolutions of the equation in a neighborhood of the boundary. Our results extend the classic result by L. Caffarelli, L. Nirenberg, and J. Spruck from the case $A \\equiv 0$ to the general case $A \\neq 0.$ Our same theorems are valid also for augmented quotient Hessian type equations.","short_abstract":"We prove the existence of admissible subsolutions to the Dirichlet problem for symmetric augmented $k$-Hessian type equations. An important sufficient condition is the uniform $(k-1)$-$A$-convexity of the domain $Ω,$ where $A(x, z, p)$ is the augmented symmetric matrix appearing in the equation. This condition was orig...","url_abs":"https://arxiv.org/abs/2609.18889","url_pdf":"https://arxiv.org/pdf/2609.18889v1","authors":"[\"Quang Hong Dinh\",\"Bang Van Tran\",\"Ngoan Tien Ha\",\"Tho Huu Nguyen\",\"Tien Trong Phan\"]","published":"2026-09-16T16:22:22Z","proceeding":"math.AP","tasks":"[\"math.AP\"]","methods":"[]","has_code":false}