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<n.

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].

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