发表机构
Hanoi Pedagogical University 2; University of Economics - Technology for Industries; Hanoi University of Science and Technology; Quang Binh University（河内师范大學二校; 經濟與工業技術大學; 河內科技大學; 廣平大學）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明有界域中对称增广$k$-Hessian型方程Dirichlet问题可容许下解的存在性，以一致$(k-1)$-$A$-凸性为充要条件，推广了Caffarelli-Nirenberg-Spruck的经典结果。
AI 中文摘要
我们证明了有界域中对称增广$k$-Hessian型方程Dirichlet问题可容许下解的存在性。一个重要的充分条件是区域$\Omega$的一致$(k-1)$-$A$-凸性，其中$A(x, z, p)$是方程中出现的增广对称矩阵。该条件最初由F. Jiang、N. S. Trudinger和X.-P. Yang引入，我们选取了其特殊情况。矩阵$A(x, z, p)$的结构条件包括其关于变量$z$和$p$的增长性，特别要求其某些一阶和二阶导数在边界的一个充分小邻域内充分小。在$A(x, z, p)$的某些结构条件下，$\Omega$的一致$(k-1)$-$A$-凸性也是方程在边界邻域内存在可容许下解的必要条件。我们的结果将L. Caffarelli、L. Nirenberg和J. Spruck的经典结果从$A \equiv 0$的情形推广到一般$A \neq 0$的情形。我们的相同定理也适用于增广商Hessian型方程。
英文摘要
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.
