functions. To study the maximization problem, we extend the functional $J_\alpha$ to the set $\mathcal{S}[\varphi,\psi]$ in , which is closed under the locally uniform convergence of convex functions. It is clear that the linear part in $J_\alpha$ is naturally defined. It suffices to extend the nonlinear part $A_\alpha$ to $\mathcal{S}[\varphi,\psi]$. If $u$ is a convex function, $u$ is almost everywhere twice-differentiable, i.e., the Hessian matrix $(D^2u)$ exists almost everywhere. Denote the extended Hessian matrix by $\partial^2u(x)=D^2u(x)$ when $u$ is twice differentiable at $x\in\Omega$ and $\partial^2u(x)=0$ otherwise. As a Radon measure, $\mu[u]$ can be decomposed into a regular part and a singular part as follows, $$\mu[u]=\mu_r[u]+\mu_s[u].$$ It was proved in [@TW05] that the regular part $\mu_r[u]$ can be given explicitely by $\mu_r[u]={\mbox{det}}\, \partial^2u\, dx$ and hence ${\mbox{det}}\, \partial^2u$ is a locally integrable function. Therefore for any $u\in{ S[\varphi,\psi]}$, we can define $$\label{e204}
   A_\alpha(u)=\left\{\begin{array}{ll}
                       \int_\Omega \left[{\mbox{det}}\,\partial^2u\right]^{\alpha}, & \alpha>0, \\[5pt]
               \int_\Omega \log{\mbox{det}}\,\partial^2u, & \alpha=0.
                      \end{array}\right.$$

\[l201\] Suppose $0\leq\alpha\leq \frac{1}{n+2}$. $J_\alpha$ is upper semi-continuous, bounded and concave in $\mathcal{S}[\varphi,\psi]$. It follows that there exists a unique maximizer $u_0$ of .

The proof for the cases $\alpha=\frac{1}{n+2}$ and $\alpha=0$ can be found in [@TW05; @Z2], respectively. One can check that the proof also holds for 