uniformly convex function in $\Omega$. If $\psi$ is a convex function in $\Omega$ satisfying $\psi<\varphi$ on $\partial\Omega$, then there exists a unique maximizer of which is strictly convex and $C^{1,\alpha}$ in $\Omega$. Furthermore, if $\psi$ is uniformly convex $\Omega$, then the maximizer of is $C^{1,1}$ in $\Omega$.

We remark that in higher dimensions, the problem is more complicated since Lemma \[l401\] does not hold. Furthermore, in the case of $\alpha=0$, the interior estimate in Lemma \[l203\] remains open when $n>2$. We will consider the higher dimensional cases and more general forms of the Monge-Ampère type functionals with $f=f(x,u,Du)$ is our forthcoming work.

This paper is organized as follows: In Section 2 we recall some preliminary results that will be used in subsequent sections. In addition, we show that how the functionals and equations change under a rotation in $\mathbb{R}^{n+1}$ and obtain the a priori determinant estimates under the rotation transform, where the functionals have more general forms . In Section 3 we show that the maximizer of $J_\alpha$ can be approximated by a sequence of smooth maximizers of appropriate penalized functionals. In Section 4 we prove that the maximizer is strictly convex by an observation in [@TW05; @TW10]. The proof 