Let $\Delta$ be an open triangle in $\mathbb{R}^2$ and $u\in C^0(\overline{\Delta})\cap C^\infty(\Delta)$ be the convex function satisfying $$ \det D^2u=1,\quad u|_{\partial\Delta}=0. $$ Classical results on Monge-Ampère equations imply that there exists a unique such $u$. I need informations on the boundary regularity of $u$, but could not find any literature on regularity of such Monge-Ampère equations near a boundary line segment.

Question 1.What can be said about the boundary regularity of $u$? In particular, is $\|\nabla u(x)\|$ bounded as $x\in\Delta$ approaches a point on $\partial \Delta$ which is not a vertex?

**Remarks.**

The second question is equivalent to asking whether $u$ has infinite slope at a non-vertex point $p\in\partial\Delta$, or more precisely, whether the limit $$ \lim_{t\rightarrow 0^+}\frac{u((1-t)p+tq)}{t}\in [-\infty,0) $$ equals $-\infty$ for some $q\in\Delta$ (which implies the same limit for every $q\in \Delta$).

It can be shown $u$ has finite slope at a vertex (namely, the above limit is finite if $p\in\partial\Delta$ is a vertex): assuming $p=0$ w.l.o.g., by Comparison Principle, $u$ is minorized by a convex function $v$ of the form $$ v(x)=c|x|^\alpha-L(x), $$ where the constant $\alpha\in (1,2)$ is arbitrary, $c>0$ is a constant to make $\det D^2v=c^2\alpha^2(\alpha-1)|x|^{2\alpha-4}$ bigger than $1$ on $\Delta$, and $L$ is a linear function on $\mathbb{R}^2$ to make $v|_{\partial\Delta}\leq 0$; whereas $v$ has value $0$ and finite slope at $x=0$.

Attempting to apply Comparison Principle near non-vertex boundary points, one is lead to the following problem:

Question 2.Let $D=\{x\in\mathbb{R}^2\mid |x|<1,\, x_1>0\}$ be the half-disk. Is there anonnegativeconvex function $u\in C^0(\overline{D})$ which vanishes on the boundary line segment $\{x_1=0,\, |x_2|\leq 1\}$, such that $\det D^2u\geq \lambda>0$?

I tried to construct such a $u$ of the form $u(x_1,x_2)=f(x_1)g(x_2)$ (with $f$ and $g$ nonnegative, convex and $C^2$) but was lead to the conclusion that such $u$ can never satisfy the hypotheses.