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 ≡ 0 to the general case A ≠ 0. Our same theorems are valid also for augmented quotient Hessian type equations.
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BibTeX
@misc{dinh2026existence,
title = {{Existence of Admissible Subsolutions to the Dirichlet Problem for Symmetric Augmented $k$-Hessian Type Equations in Bounded Domains}},
author = {Quang Hong Dinh and Bang Van Tran and Ngoan Tien Ha and Tho Huu Nguyen and Tien Trong Phan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18889},
eprint = {2609.18889},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18889},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Existence of Admissible Subsolutions to the Dirichlet Problem for Symmetric Augmented k-Hessian Type Equations in Bounded Domains
AU - Quang Hong Dinh
AU - Bang Van Tran
AU - Ngoan Tien Ha
AU - Tho Huu Nguyen
AU - Tien Trong Phan
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18889
UR - https://arxiv.org/abs/2609.18889
AB - 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 ≡ 0 to the general case A ≠ 0. Our same theorems are valid also for augmented quotient Hessian type equations.
ER -