Author

Ruo-Si Chen

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Preprint Aug 2026

Interior $C^{2,\alpha}$ Regularity for the Quadratic Hessian Equation

We establish interior $C^{2,\alpha}$ regularity for admissible solutions of the quadratic Hessian equation with positive $C^\alpha$ right-hand side on the full positive branch. The main ingredients are a quantitative large-trace propagation argument and an adaptive Dirichlet comparison method.

Ruo-Si Chen, Xing-Chen Zhou, Rui-Xuan Zhu · 0 citations

Preprint Sep 2026

Gradient estimates for the Lagrangian mean curvature equation with Lipschitz critical phase

Let $n\ge3$. We prove a priori interior gradient estimate for the Lagrangian mean curvature equation \[ \sum_{i=1}^n\arctan\lambda_i(D^2u)=\theta(x) \] under the critical and supercritical phase condition $\theta\ge (n-2)\pi/2$, assuming only that the Lipschitz norm of the prescribed phase is bounded.

Ruo-Si Chen, Xing-Chen Zhou, Rui-Xuan Zhu · 0 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 $u$.

Ruo-Si Chen, Yan-Nan Liu, Xing-Chen Zhou · 0 citations