C1,α and W2,1+ε regularity for the $2$-Hessian equation
Abstract
In this paper, for every ε>0, we establish interior and global a priori C1,α estimates for some α∈(0,1) from W2,1+ε bounds for bounded smooth $2$-admissible solutions of the $2$-Hessian equation, with the interior estimate requiring only an upper bound on the density. Under positive upper and lower density bounds, we prove global W2,1+ε estimates for some ε>0 on bounded strictly mean convex C2,β domains with C2,β boundary values. For densities bounded above and below by positive constants, the interior C1,α estimate follows by combining the first result with an interior W2,1+ε estimate obtained by a slight modification of an existing argument. Combining the first result with the global W2,1+ε estimate yields the global C1,α estimate. Known examples show that the C2,β regularity assumptions on the boundary and boundary values are optimal in the Hölder scale for both the global W2,1+ε estimate and the global C1,α estimate.
Related
- Interior C2,α Regularity for the Quadratic Hessian Equation
- Boundary regularity of k-Hessian eigenfunctions
- C1,1 Regularity Global Estimates for Homogeneous Complex Hessian Equations on Punctured Domains
- Interior Hessian estimates for the quadratic Hessian equation
- A Liouville type rigidity result for entire solutions of the k-Hessian equation
BibTeX
@misc{lyu2026c1α,
title = {{$C^1,α$ and $W^2,1+\varepsilon$ regularity for the $2$-Hessian equation}},
author = {Yasheng Lyu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35839},
eprint = {2610.35839},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35839},
}
RIS
TY - UNPB
TI - C^1,α and W^2,1+ε regularity for the $2$-Hessian equation
AU - Yasheng Lyu
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35839
UR - https://arxiv.org/abs/2610.35839
AB - In this paper, for every ε>0, we establish interior and global a priori C^1,α estimates for some α∈(0,1) from W^2,1+ε bounds for bounded smooth $2$-admissible solutions of the $2$-Hessian equation, with the interior estimate requiring only an upper bound on the density. Under positive upper and lower density bounds, we prove global W^2,1+ε estimates for some ε>0 on bounded strictly mean convex C^2,β domains with C^2,β boundary values. For densities bounded above and below by positive constants, the interior C^1,α estimate follows by combining the first result with an interior W^2,1+ε estimate obtained by a slight modification of an existing argument. Combining the first result with the global W^2,1+ε estimate yields the global C^1,α estimate. Known examples show that the C^2,β regularity assumptions on the boundary and boundary values are optimal in the Hölder scale for both the global W^2,1+ε estimate and the global C^1,α estimate.
ER -