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Interior C2 Estimates for 2-Sum Hessian Equations
Abstract
This paper establishes an interior C2 estimate for solutions of σ2(D2u)+α(x)σ1(D2u)=f(x,u,Du)>0, on the positive branch Δu>0, under the assumptions α≥0 and D2α=βI. Using one extra variable and a quartic correction, the equation is reduced to a quadratic Hessian equation in one higher dimension. The terms involving D2u cancel, so the interior estimate of Li--Wu \cite{LW} applies. The gradient-independent case is also treated. Finally, an example shows that an L∞ bound on α alone is not enough.
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BibTeX
@misc{ren2026interior,
title = {{Interior $C^2$ Estimates for 2-Sum Hessian Equations}},
author = {C. Y. Ren and X. L. Sun},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.09490},
eprint = {2610.09490},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.09490},
}
RIS
TY - UNPB
TI - Interior C^2 Estimates for 2-Sum Hessian Equations
AU - C. Y. Ren
AU - X. L. Sun
PY - 2026
DA - 2026/10/07
PB - arXiv
DO - 10.48550/arxiv.2610.09490
UR - https://arxiv.org/abs/2610.09490
AB - This paper establishes an interior C^2 estimate for solutions of σ_2(D^2u)+α(x)σ_1(D^2u)=f(x,u,Du)>0, on the positive branch Δu>0, under the assumptions α≥0 and D^2α=βI. Using one extra variable and a quartic correction, the equation is reduced to a quadratic Hessian equation in one higher dimension. The terms involving D^2u cancel, so the interior estimate of Li--Wu \cite{LW} applies. The gradient-independent case is also treated. Finally, an example shows that an L^∞ bound on α alone is not enough.
ER -