ap.analysis of pdes - Solutions to a Monge-Ampère equation on the simplex

Let $Delta_k$ be the k-simplex and $mu$ a non-negative measure over $Delta_k$. I want to know if there exists a function $u : Delta_k to mathbb{R}$ such that $u$ is convex, $u(e_i) = 0$ for all vertices $e_i$ of $Delta_k$, and $M[u] = mu$ where
$M[u] = detleft(frac{partial^2 u}{partial x_j partial x_k}right)$ is the Monge-Ampère operator. Furthermore, I'd like to know if the solution is unique. Any techniques for how one might solve a specific instance of this problem would be a bonus.

My background is not in PDEs but the closest I've found to an answer seem to be in [1] and [2] where the boundary conditions are more restrictive and the domain is required to be strictly convex for uniqueness.

[1] "On the fundamental solution for the real Monge-Ampère operator", Blocki and Thorbiörnson, Math. Scand. 83, 1998

[2] "The Dirichlet problem for the multidimensional Monge-Ampère equation", Rauch and Taylor, Rocky Mountain Journal of Mathematics, 7(2), 1977.

Any other pointers to solving this type of problem would be greatly appreciated.