Li-Chun Liang

Preprint Sep 2026

Interior Curvature Estimates of Semi-convex Solutions for the scalar curvature equation with Lipschitz Right-Hand Sides

In this paper, let $u\in C^4(B_{10})$ with $D^2u\geq -KI$ define a $2$-admissible graph $M=\{(x,u(x)):x\in B_{10}\}\subset\R^{n+1}$ satisfying \[ \sigma_2(\kappa[u])=f(x). \] We prove an interior curvature estimate depending on the Lipschitz norm of the right-hand sides. The proof combines a shifted Jacobi inequality for $b=\log(H+J_0)$, a parallel hypersurface transformation that makes the Newton tensor uniformly elliptic and local boundedness estimate then reduces the pointwise bound to a weighted $L^1$ estimate, which is completed using the Jacobi energy inequality and integration by parts.

Li-Chun Liang · 0 citations

Preprint Aug 2026

Interior Hessian Estimates for Semi-convex Solutions of the $\sigma_2/\sigma_1$ Equation with Lipschitz Right-Hand Sides

Let $n\ge2$ and let $u$ be a smooth 2-convex and semi-convex solution of \[ \frac{\sigma _2(D^2u)}{\sigma _1(D^2u)}=f(x). \] We prove an interior Hessian estimate depending on the Lipschitz norm of $f$. The proof combines the integral approach of Chen--Jian--Zhou with the algebraic reduction of the quotient equation to a $\sigma _2$ structure. The main new point is a shifted algebraic inequality that yields a shifted trace Jacobi inequality in divergence form for $\log(\Delta u+a)$. We work with the linearized operator $G=(\Delta u-f)I-D^2u$ of the equivalent equation $\sigma_2(D^2u)=f \Delta u$. The almost divergence-free identity $\partial_iG_{ij}=-f_j$ enables us to control the $\Delta f$ term by integration by parts solely in terms of the Lipschitz norm of $f$. A Legendre--Lewy transformation converts the resulting degenerate divergence-form equation into a uniformly elliptic one. The estimate then follows from a mean-value inequality together with a weighted energy argument. As an application, in dimension two we obtain interior $C^2$ regularity for convex viscosity solutions with positive Lipschitz right-hand side. Moreover, our counterexamples show that the Lipschitz regularity required of the right-hand side is optimal.

Ke Ji, Li-Chun Liang · 0 citations