PreprintCC BY 4.0arXiv
A Liouville Theorem for the Real k-Hessian Equation with a One-Sided Growth Condition
Abstract
We establish a scale-invariant interior C2 estimate for smooth solutions of the real k-Hessian equation. As an application, we prove that every smooth entire solution of σk(D2u)=1 satisfying u(x)≥ b|x|2−B for some constants b,B with b>0 is a quadratic polynomial. For smooth solutions, admissibility is forced by the growth condition. The results also hold for viscosity solutions and for weak solutions in Trudinger's sense. Admissibility is required in the latter case.
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BibTeX
@misc{hu2026liouville,
title = {{A Liouville Theorem for the Real k-Hessian Equation with a One-Sided Growth Condition}},
author = {Jingchen Hu and Heming Jiao},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.35565},
eprint = {2610.35565},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.35565},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Liouville Theorem for the Real k-Hessian Equation with a One-Sided Growth Condition
AU - Jingchen Hu
AU - Heming Jiao
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.35565
UR - https://arxiv.org/abs/2610.35565
AB - We establish a scale-invariant interior C^2 estimate for smooth solutions of the real k-Hessian equation. As an application, we prove that every smooth entire solution of σ_k(D^2u)=1 satisfying u(x)≥ b|x|^2−B for some constants b,B with b>0 is a quadratic polynomial. For smooth solutions, admissibility is forced by the growth condition. The results also hold for viscosity solutions and for weak solutions in Trudinger's sense. Admissibility is required in the latter case.
ER -