of Theorem \[t101\] is contained in Section 5, where the $C^{1,\alpha}$ and $C^{1,1}$ regularities are obtained, respectively.

Preliminaries
=============

Monge-Ampère measure
--------------------

Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ and $u$ be a convex function in $\Omega$. The [*normal mapping*]{} of $u$, $N_u$, is a set-valued mapping defined as follows. For any point $x\in\Omega$, $N_u(x)$ is the set of slopes of supporting hyperplanes of $u$ at $x$, that is, $$\label{e201}
   N_u(x)=\{p\in\mathbb{R}^n\,:\,u(y)\geq u(x)+p\cdot(y-x),\quad\forall y\in\Omega\}.$$ For any Borel set $E\subset\Omega$, $N_u(E)=\bigcup_{x\in E}N_u(x)$. If $u$ is $C^1$, the normal mapping $N_u$ is exactly the gradient mapping $Du$.

From the normal mapping we define the [*Monge-Ampère measure*]{} $\mu[u]$ by $$\label{e202}
   \mu[u](E)=|N_u(E)|$$ for any Borel set $E\subset\Omega$, where the right hand side is the Lebesgue measure of $N_u(E)$. If $u$ is $C^2$ smooth, we have $\mu[u]=({\mbox{det}}\,D^2u)dx$. In the non-smooth case, the Monge-Ampère measure $\mu[u]$ is a Radon measure, and is weakly continuous with respect to the convergence of convex functions, namely if a sequence of convex functions $\{u_i\}$ converges to a convex function $u$ in $L^\infty_{loc}$, then for any closed $E\subset\Omega$, $$\label{e203}
   \lim_{i\to\infty}\sup\mu[u_i](E)\leq\mu[u](E).$$

Existence and uniqueness of maximizer
-------------------------------------

Note that the functional $J_\alpha$ in is well defined on the set of $C^2$-smooth, convex 