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Strict $2$-convexity of convex solutions to the quadratic Hessian equation


Author: Connor Mooney
Journal: Proc. Amer. Math. Soc. 149 (2021), 2473-2477
MSC (2020): Primary 35J60, 35B65
DOI: https://doi.org/10.1090/proc/15454
Published electronically: March 25, 2021
MathSciNet review: 4246798
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Abstract: We prove that convex viscosity solutions to the quadratic Hessian inequality \begin{equation*} \sigma _2(D^2u) \geq 1 \end{equation*} are strictly $2$-convex. As a consequence we obtain short proofs of smoothness and interior $C^2$ estimates for convex viscosity solutions to $\sigma _2(D^2u) = 1$, which were proven using different methods in recent works of Guan-Qiu, McGonagle-Song-Yuan, and Shankar-Yuan.


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Additional Information

Connor Mooney
Affiliation: Department of Mathematics, University of California Irvine, Irvine, California 92697
MR Author ID: 1092639
Email: mooneycr@math.uci.edu

Keywords: Sigma-2 equation, regularity
Received by editor(s): June 13, 2020
Published electronically: March 25, 2021
Additional Notes: This research was supported by NSF grant DMS-1854788.
Communicated by: Ryan Hynd
Article copyright: © Copyright 2021 American Mathematical Society