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.



Solutions to a Monge-Ampère equation on the simplex 
Any set of values of $u$ at the vertices of $\Delta\_k$ can be attained just by adding an affine function to $u$, which does not change $M[u]$. To see that the solution of your problem is not unique, consider $u(x,y)=ax^2+a^{-1}y^2+\mathrm{(affine\ terms)}$ with $a>0$. Clearly $M[u]=4$ for any $a$. 


On the other hand, Theorem 1.6.2 in the book *The Monge-Ampère equation* by C. Gutierréz states that there is a unique convex solution of $M[u]=\mu$ (with $M[u]$ properly understood) with prescribed continuous boundary values in a **strictly convex** domain $\Omega$. Without strict convexity we can't allow arbitrary continuous boundary data; it must be at least consistent with some convex function in $\Omega$. (But I don't know if that's enough). The uniqueness part holds without strict convexity; see Corollary 1.4.7 in the same book. 


