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 establish interior Hessian estimates for $k$ convex solutions of $\frac{\sigma_k(D^2u)}{\sigma_l(D^2u)}=1$, where $1 \leq l<k<n$, $k-l \in \{1,2\}$, and $2k>n$.The proof combines a concavity inequality with a pointwise doubling argument.
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 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...
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.
In this note, we prove a Liouville type rigidity result for smooth entire solutions of the $k$-Hessian equation. The proof makes use of a concavity inequality for the $k$-Hessian operator.
We give a compactness proof of interior Hessian estimates for convex solutions of $\sigma_n(D^2u)/\sigma_{n-2}(D^2u)=1$ and $\sigma_n(D^2u)/\sigma_{n-1}(D^2u)=f(x,u,Du)>0$, with $1/f$ concave in the gradient variable in the latter case, revisiting results of Lu and Fung.
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