Liouville theorem for the free boundary Monge–Ampère equation

Let $l,m,k$ be parameters and let $\varphi:(\mathbb R_+)^l\times \mathbb R^{m}\to \mathbb R_{\geq 0}$ solve

$$\left(\sum_{i = 1}^l\frac{\partial \varphi}{\partial x_i}\right)^k\det D^2\varphi = 1$$

in $(\mathbb R_+)^l\times \mathbb R^{m}$, with

$$\sum_{i = 1}^l\frac{\partial \varphi}{\partial x_i} = 0$$

on $\partial (\mathbb R_+)^l\times \mathbb R^{m}$. Here $\mathbb R_+=(0,\infty)$, and let $\varphi_{l,k}$ denote the homogeneous solution of the corresponding equation on $(\mathbb R_+)^l$ obtained from the stated construction. Liouville theorem. Then

$$\varphi(x)=c+\sum_{i=l+1}^{l+m}v_i x_i+\sum_{i,j=1}^lP_{ij}x_i x_j+p\varphi_{l,k}(x_1,\ldots,x_l),$$

where $p>0$, $c$ and the $v_i$ are constants, and $P_{ij}$ is a positive-definite $l\times l$ matrix. The asserted classification would give a rigidity theorem for global solutions of the degenerate Monge–Ampère equation and is identified in the source as a key step toward regularity near lower-dimensional faces and the associated free boundary. Its status is unresolved in the supplied material; it is presented as a conjectural Liouville theorem generalizing results of Jhaveri and Savin.