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 proposed by Wang--Yuan [WY13]. The corresponding results in higher dimensions are also established.
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 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.
We construct a continuous function $u$ on the closed unit ball in $\mathbb{R}^4$ and a smooth autonomous Hamiltonian $H:\mathbb{R}^4\to[1,2]$ for which $u$ solves $H(Du)=c$ in the unit ball in the viscosity sense for every $c\in[1,2]$. This reveals a striking and distinctive pathology of viscosity solutions: a fixed fu...
We prove the strict log-concavity of the positive first eigenfunction \(-u\) of the \(2\)-Hessian equation and the strict $1/2$-convexity of the solution for the corresponding torsion problem in smooth bounded uniformly convex domains in $\mathbb{R}^{n}$. As applications, we establish the associated Brunn--Minkowski in...
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 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.
Xing-Chen Zhou, Rui-Xuan Zhu· 0 citations
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