Solutions to a Monge-Ampère equation on the simplex - MathOverflow

Solutions to a Monge-Ampère equation on the simplex

2010-02-16T23:35:27Z

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.

Answer by Deane Yang for Solutions to a Monge-Ampère equation on the simplex

2010-02-17T02:03:09Z

This is not an answer, but...

I don't know of any previous work on this, but it appears to be well worth studying. Techniques inspired by convex geometry, like those used to solve the Minkowski problem, might work. The idea would be to first solve the equation for a finite discrete measure using a piecewise linear function.

You might want to ask Luis Caffarelli.

I am also very interested in how you came to be interested in this question.

Answer by 002 for Solutions to a Monge-Ampère equation on the simplex

2010-02-23T22:46:50Z

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.

Answer by fible for Solutions to a Monge-Ampère equation on the simplex

2011-05-30T15:43:52Z

Its useful to see the paper" [1] S.Y. Cheng, S.T. Yau. On the regularity of the solutions of the Monge-Amp$\grave{e}$re equation $\det(\frac{\partial^2 u}{\partial x_i\partial x_j})=F(x,u)$. Comm Pure Appl. Math, 1977, 30: 41-68."

For special $\mu$, the hyperbolic affine sphere is a solution in a simplex with zero boundary value.