Finding a smooth convex function with prescribed boundary value and small Monge-Ampère measure

Question

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 $$ \overline{\nu}(x):=\sup\Big\{\alpha(x)\,\Big|\,\alpha:\mathbb{R}^n\rightarrow\mathbb{R} \mbox{ is an affine function and }\alpha|_{\partial\Omega}\leq \nu\Big\} $$ is not $C^1$ in general but is the generalised convex solution to 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 $\epsilon>0$ there is a strictly convex smooth function $u$ on $\Omega$ 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.

Answer 1

In the case $n \geq 3$ it is false. One way to see this is to use the Pogorelov example $$w(x', \,x_n) = |x'|^{2-2/n}f(x_n),$$ which for the appropriate choice of $f$ solves $\det D^2w = 1$ in $\{|x_n| < \rho\}$. This example is $C^{1,1-2/n}$ and is $0$ on the $x_n$-axis. (This example arises from the affine invariance of the Monge-Ampere equation; $w$ is invariant under the scaling $w(x',\,x_n) = \frac{1}{\lambda^{2-2/n}}w(\lambda x', x_n)$, which preserves the Monge-Ampere measure).

Take $\nu = w$ on $\partial B_{\rho}$. By the comparison principle, $u \geq w$ in $B_{\rho}$, and by convexity $u = 0$ on the $x_n$ axis. Thus, $u$ is not strictly convex, and its regularity is at best $C^{1,1-2/n}$. (One can use analogues of the Pogorelov example of the form $|x'| + |x'|^{n/2}g(x_n)$ in the same way to generate Lipschitz singularities for $u$ along a line segment when the boundary data are Lipschitz).

This example is in a sense optimal; Caffarelli showed that if $\nu$ is $C^{1,\beta}$ for $\beta > 1-2/n$ then solutions to $\det D^2u = \epsilon, \quad u|_{\partial \Omega} = \nu$ are strictly convex and smooth in $\Omega$.

In the case $n = 2$, solutions to $\det D^2u = \epsilon$ are locally strictly convex by a classical result of Alexandrov. By taking smooth approximations $\nu_k$ of the boundary data and solving $\det D^2u_k = \epsilon, \quad u_k|\partial \Omega = \nu_k$ one can hope to obtain the desired approximation in the limit.