TITLE: Solutions to a Monge-Ampère equation on the simplex 
QUESTION [10 upvotes]: 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 
$M[u] = \det\left(\frac{\partial^2 u}{\partial x_j \partial x_k}\right)$ 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.

REPLY [2 votes]: In the spirit of 002's answer, the existence is shown by the following argument. Choose a uniformly convex domain $\Omega$ such that the vertices of $\Delta_k$ belong to the boundary $\partial\Omega$. Extend $\mu$ as a non-negative measure $\bar\mu$ over $\Omega$. Choose a continuous function $\phi:\partial\Omega\rightarrow\mathbb R$ which vanishes at the vertices. Then solve the Dirichlet boundary value problem for the Monge-Ampère equation in $\Omega$, with data $(\bar\mu,\phi)$. The restriction to $\Delta_k$ of the unique convex solution solves your problem.
The non-uniqueness is almost obvious, because of the freedom in the extension of $\mu$ and in the choice of $\phi$.