There is a article http://pages.iu.edu/~nqle/MA_EVP.pdf of NAM Q.LE state a conjecture of when the eigenvalue of Monge-Ampere equation will arrive the maximum.
It divide into two part:
Conjecture 1 1. Among all bounded open convex sets in $\mathbb R^n$ having a fixed positive volume, the n-dimensional regular simplex (that is the interior of the convex hull of (n + 1) equally spaced points in $\mathbb R^n$) has the smallest Monge-Ampere eigenvalue. 2. Among all open bounded centrally symmetric convex sets in $\mathbb R^n$ having a fixed positive volume, the n-dimensional cube has the smallest Monge-Ampère eigenvalue.
The story is more or less easier for the minimum, the minimum should arrive with the ball $B_{n-1}$ due to the Brunn-Minkowski inequality, which claim the eigenvalue is a convex function on convex body space.
I need explain the basic set first, we consider the Dirichlet eigenvalue problem of Monge-Ampere equation, $$\det(D^2u)=\lambda |u|^{n}\ in\ \Omega$$ $$u=0 \ on \ \partial \Omega$$ This equation is natural to consider due to a rescaling argument, just consider rescaling $u_{\epsilon}(x)=\epsilon\cdot u(x)$, then $\det(D^2 u_{\epsilon})={\epsilon}^n\det(D^2u)=\epsilon^n\lambda|u|^n=\lambda|u_{\epsilon}|^n$.
Lions showed there is a unique constant $\lambda>0$ such that the eigenvalue function have a solution. And there is a variational characterization of $\lambda$, $$\lambda(\Omega)=\inf_{u\in C_0^1(\bar \Omega)\cap C^2(\Omega), u \ convex, u|_{\partial \Omega}=0}\frac{\int_{\Omega}(-u) \cdot \det(D^2u)dx}{\int_{\Omega}(-u)^{n+1}dx}$$
The point is, I think this $\lambda$ should have a geometric characterization, related to some quality of convex body $\Omega$ itself, I do not know if it is right, but I expect the following result:
Conjecture 2 There is a geometric characterization of $\lambda$, that is, for convex body $\Omega$, $$\lambda=\inf_{T }\mu_{n-1}( T(\partial\Omega))$$ Where $T$ runs in all affine map fix the original point and the Lebesgue measure, i.e. under a orthogonal basis, the determination of the matrix of $T$ is 1.
Motivation Let me explain why I believe this conjecture could be true. we know, for equation $\det(D^2u)=\lambda|u|^n$, look it locally, by area formula with the map $F:\mathbb R^n\to \mathbb{TR}^n$, under a basis, $F:(e_1,...,e_n)\to u_{e_1},...,u_{e_n}$, any subset $A\subset \Omega$ we have $$\int_{A} u\cdot \det(D^2u)dx\overset{area \ formula}=\int_{\nabla A}ude$$ where $\nabla A=\{\nabla u(x),x\in A\}$. So the geometric meaning of locally solution of $\det(D^2u)=\lambda|u|^n\ \in \Omega$ is $\forall A\in \Omega$, $|\nabla A|=\int_A \lambda|u|^n$, so $\lambda=\sup_{A\to \Omega}\inf_u \frac{\int_{\nabla A}ude}{\int_{\Omega}(-u)^{n+1}dx} (*)$.
So what is the meaning of $(*)$ by intuition, it should be the optimal choice of a sires of level set, with the zero level set ecocide with $\partial \Omega$. because at once you fix the fibration of level set, the choice of $u$ is more or less the same. So this eigenvalue characterize the geometry of $\Omega$, but I can not go further, due to three obstacles:
- The first one is I do not know what is the universal choice of $u$ to arrive minimum $\lambda$ when fix a fiber of level set, I think it is come from a equal condition of holder inequality, but I can not work it out.
- The second one is I do not know what is the best choice of level set fibration, I think it come from the fibration center at the "center of mass of $\Omega$" and then do the most natural one, but I do not know how to proof this.
- If we can proof 1,2. Then how to explain the product $\lambda$ by this thing could coincide with the statement of $\lambda$ in conjecture 2.
I wish this could be a toy model of "hear the shape from the spectrum".
May be this approaches is useless. In any case, I wish we can say some thing about it, whatever positive answer or negative answer. I will appreciate to any interesting argument, thank you for your attention!
