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 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.
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.
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_{...
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 $u$.
