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

Mooney, Connor

doi:10.1090/proc/15454

Strict $2$-convexity of convex solutions to the quadratic Hessian equation
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Proc. Amer. Math. Soc. 149 (2021), 2473-2477 Request permission

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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