Abstract:We prove interior $C^{2}$ regularity result for convex viscosity solutions of the quadratic Hessian equation $\sigma_2(D^2u) = f(x)$, under the assumption that $f\in C^{0,1}$ with $\inf f>0$. The result is almost sharp: if $f$ are merely continuous, there exist convex viscosity solutions that fail to be $C^{1,1}$. When $f\in C^{\alpha}$ for some $\alpha\in (0,1)$, the corresponding interior regularity remains open.
From: Xingchen Zhou [view email]
[v1]
Fri, 29 May 2026 04:15:47 UTC (160 KB)
[v2]
Mon, 1 Jun 2026 03:14:19 UTC (162 KB)
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