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 $u$.
We prove that for each $3\le k\le n-1$ a convex $C^2$ entire solution of $\sigma_k$ equation in $\mathbb{R}^n$ with positive constant right hand side is a quadratic polynomial, without a growth assumption on the solution itself.
We prove that continuous $2$-convex solutions of the $\sigma_{2}$ equation with a density bounded above and below by positive constants belong to $W^{2,1}_{\mathrm{loc}}$ in every dimension. If, in addition, the density is continuous, the solutions belong to $W^{2,p}_{\mathrm{loc}}$ for every $1
We prove that the gradient graph of any $W^{2,1}$ semi-convex viscosity solution to the three-dimensional subcritical special Lagrangian equation is area-minimizing and smooth. Furthermore, we prove the smoothness of $W^{2,1}\cap C^{1,1/3+}$ semi-convex viscosity solutions. This yields progress on the conjecture propos...
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.
We prove a concavity inequality for the Hessian quotient operator $\sigma_k/\sigma_{k-1}$ using a change of basis for symmetric polynomials. This removes an additional structural concavity assumption previously imposed in the interior $C^2$ estimate for convex solutions of the $\sigma_k/\sigma_{k-1}$ equation. The meth...
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.
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