Author

Xingchen Zhou

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Preprint Aug 2026

Interior $C^{2,\alpha}$ Regularity for Convex Solutions of the 2-Hessian Equation

We consider convex admissible viscosity solutions of $\sigma_2(D^2u)=f>0$ in an open subset of $\mathbb R^n$, where $n\ge2$, $0<\alpha<1$, and $f\in C_{\mathrm{loc}}^{0,\alpha}$. We prove that every such solution belongs to $C_{\mathrm{loc}}^{2,\alpha}$. For solutions in $B_2$, we also prove a uniform $C^{2,\alpha}(B_{1/4})$ estimate under an $L^\infty$ bound for $u$, a positive lower bound for $f$, and a $C^{0,\alpha}$ bound for $f$.

Xingchen Zhou, Ruixuan Zhu · 2 citations

Preprint Aug 2026

Regularity for convex viscosity solutions of $\sigma_3$ equation

We prove interior $C^2$ regularity for convex viscosity solutions of the $3$-Hessian equation $\sigma_3(D^2u)=f(x)$ with $f\in C^{0,1}, \inf f>0$, under a strict $3$-convexity condition on $u$.

Ruosi Chen, Yan-Nan Liu, Xingchen Zhou · 0 citations