## Monge-Ampere equation and bounday behavior of domain

$\mathbf{Problem:}$ Suppose $\Omega\subset \mathbb{R}^n$ is a bounded domain and $\partial \Omega$ is $C^2$. If there exists a convex function $u\in C^2(\overline{\Omega})$ satisfies

$\displaystyle \begin{cases}\det D^2u=1 \text{ in }\Omega\\ u=0\quad \quad\text{ on }\partial \Omega\end{cases}$

Then $\Omega$ is uniformly convex. In other words, the principle curvature  of every point on $\partial \Omega$, namely $\kappa_1,\kappa_2,\cdots,\kappa_{n-1}$, are positive. Moreover, $\partial \Omega$ is connect.

$\mathbf{Proof:}$ For any boundary point of $\Omega$, we may suppose the point is origin. Since $\partial \Omega\in C^2$, there exists a neighborhood of 0 such that $\partial \Omega$ is represented by $x_n=\rho(x_1,x_2,\cdots,x_{n-1})$ with $\rho\in C^2$. Choosing the principle coordinate system, poositive $x_n$ axis is interior normal at origin and

$\displaystyle \rho(x')=\frac{1}{2}\sum\limits_{i=1}^{n-1}\kappa_ix_i^2+O(|x'|^3)$ here $x'=x_1,x_2,\cdots, x_{n-1}$.

Since $u(x',\rho(x'))=0$ near the origin, it following, on differentiation,

$\displaystyle u_{ij}=-u_n\rho_{ij}=-u_n\kappa_i\delta_{ij}$    for $i,j.

So at origin, we get

$\displaystyle D^2u=\left( \begin{array}{cccc} \kappa_1 & 0 &\cdots & u_{1n} \\ 0 & \kappa_2 & \ldots & u_{2n} \\ \vdots &\vdots & \ddots &\vdots\\ u_{1n}&u_{2n}& \cdots & u_{nn} \end{array} \right)$

which means

$\displaystyle 1=\det D^2u=|u_n|^{n-2}\prod\limits_{i=1}^{n-1}\kappa_i\left\{|u_n|u_{nn}-\sum\limits_{i=1}^{n-1}\frac{(u_{in})^2}{\kappa_i}\right\}.$

So $\kappa_i\neq 0$ on $\partial \Omega$ and for all $\, i=1,2,\cdots,n$. However, $\kappa_i$ is continuous function on $\partial \Omega$, which means $\kappa_i$ does not change sign on $\partial \Omega$.

Since $u$ is convex and $u=0$ on $\partial \Omega$, then $u\leq 0$ in $\Omega$. Also we know that $D^2u$ is positive definite matrix. Thus $\Delta u>0$ in $\Omega$, by the hopf lemma,

$u_n>0$ at $\partial \Omega$

So near origin $\kappa_iu_n$ is almost $u_{ii}$, which is positive by the property of $D^2u$. So $\kappa_i$ is positive.

For the argument $\partial \Omega$ is connect, see the paper in the following remark.

$\text{Q.E.D}\hfill \square$

$\mathbf{Remark:}$ Caffarellli, Nirenberg and Spruck. The Dirichlet problem for nonlinear second order elliptic equations,III:Functions of the eigenvalues of the Hessian.

Also refer to the formula [GT, p471].