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.

  • $\begingroup$ I also found Jerrard's "Some Remarks on Monge-Ampère functions" (2008) useful but not conclusive for my question. $\endgroup$ – Mark Reid Feb 17 '10 at 4:21

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.

  • $\begingroup$ Thanks for the demonstration of non-uniqueness. I had already realised that M was invariant under affine transformations which is why I had required u to vanish at the vertices. Another natural constraint in my problem is that $\sup_{x\in{\Delta_k}} |u(x)|=1$. Would that give uniqueness? I suspect, at least, that the construction you give would only satisfy this extra constraint for a single value of $a$. I'll definitely have a look at the book you suggested too. $\endgroup$ – Mark Reid Feb 24 '10 at 3:17
  • $\begingroup$ By a similar degrees of freedom argument I see that specifying exactly where |u(x)|=1 won't help either. My geometric intuition about the problem has completely failed me now. I originally thought that specifying the function's values at the vertices, its "curvature" (roughly speaking) at each point, as well as its height would have determined it. However, your construction shows that there are, in fact, a whole family that satisfy those constraints when M[u] is constant. I suspect M throws away too much information about the function and my intuition is confusing it with the Hessian. $\endgroup$ – Mark Reid Feb 24 '10 at 5:17
  • $\begingroup$ myweb.facstaff.wwu.edu/hartend/documents/… This paper answers the question raised in your response. In fact as long as the boundary data has convex extension. There is a unique solution. $\endgroup$ – lemega Jul 6 '10 at 20:02

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.

  • 1
    $\begingroup$ Thanks for the pointers. I work in statistical machine learning and have recently been investigating properties of loss functions for probability estimation and their associated entropies. Many properties of losses in the binary case (i.e., k=1) can be explained by the curvature of their entropy function. The question above is an attempt to generalise some results beyond the standard binary prediction problem. $\endgroup$ – Mark Reid Feb 17 '10 at 4:15

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.


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 $\phi$.


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