Existence and uniqueness conjecture for prescribed curvature equations with a subsolution

Let $2 \leq k \leq n-1$ satisfy $2k>n$. Let $\Omega \subseteq \mathbb{R}^n$ be a convex $(k-1)$-admissible bounded domain with smooth boundary. Suppose $\psi(x,u,Du)$ is a smooth positive function satisfying $\psi_u\geq 0$, and let $\varphi\in C^4(\overline{\Omega})$ be spacelike. Assume there is a $k$-admissible subsolution $\underline{u}$ satisfying

$$\sigma_k[\underline{u}]\geq \psi(x,\underline{u},D\underline{u}) \quad\text{in }\Omega,$$

and

$$\underline{u}=\varphi \quad\text{on }\partial\Omega.$$

Existence and uniqueness conjecture. There exists a unique $k$-admissible solution $u$ satisfying

$$\sigma_k[u]=\psi(x,u,Du) \quad\text{in }\Omega,$$

and

$$u=\varphi \quad\text{on }\partial\Omega,$$

with $u\in C^{3,\alpha}(\overline{\Omega})$ for every $\alpha\in(0,1)$. The conjecture would extend solvability of the Dirichlet problem to the range $2k>n$ under the stated subsolution hypotheses; the paper says it is motivated by the expected curvature estimate, while verification of the preceding Ren–Wang concavity conjecture is still absent.