Interior Hessian Estimates for Semi-convex Solutions of the ${\sigma }_2/{\sigma }_1$ Equation with Lipschitz Right-Hand Sides

Abstract

Let $n\ge 2$ and let $u$ be a smooth 2-convex and semi-convex solution of \[ \frac{σ_2(D^2u)}{σ_1(D^2u)}=f(x). \] 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 $G=(\Delta u-f)I-D^{2}u$ of the equivalent equation ${\sigma }_2(D^{2}u)=f\Delta u$. The almost divergence-free identity \(\partial_iG_{ij}=-f_j\) enables us to control the \(Δf\) 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.