Let $\Omega\subset\mathbb{R}^n$ be a bounded strictly convex domain and $\nu:\partial\Omega\rightarrow\mathbb{R}$ be a lower semi-continuous functions. It is well known that the function $\overline{\nu}:\overline{\Omega}\rightarrow\mathbb{R}$ defined by $$ \overline{\nu}:=\sup\Big\{\alpha|_{\overline{\Omega}}\,\Big|\,\alpha:\mathbb{R}^n\rightarrow\mathbb{R} \mbox{ is an affine function and }\alpha|_{\partial\Omega}\leq \nu\Big\} $$ is not $C^1$ in general but is the generalised solution of the Dirichlet problem of Monge-Ampère equation $$ \det D^2u=0,\quad u|_{\partial\Omega}=\nu. $$ For a problem I'm working on, it would be nice if there is a smooth alternative of $\overline{\nu}$:

Question. Is it true that for any $\Omega$, $\nu$ and any $\epsilon>0$ one can find a strictly convex function $u\in C^\infty(\Omega)$ satisfying $$ \det D^2u<\epsilon,\quad u|_{\partial\Omega}=\nu\, ? $$

Any comments or hints for reference are welcome.

Remark on boundary value. Here, for a convex function $u$ on $\Omega$, the boundary value $u|_{\partial\Omega}$ is defined in such a way that $u|_{\partial\Omega}(x_0)$ ($x_0\in\partial\Omega$) is the limit of $u(x)$ when $x$ tends to $x_0$ along a line segment contained in $\Omega$. This does not depend on the choice of the segment. In general, $u|_{\partial\Omega}$ is only lower semi-continuous.