# Regularity of Aleksandrov solution to Monge-Ampère equation with infinite slope at boundary

I'm interested in the regularity of solutions to Monge-Ampère equations in a bounded convex domain $$\Omega\subset\mathbb{R}^n$$. It seems that the following statement can be deduced from known results:

Proposition. Let $$u\in C^0(\overline{\Omega})$$ be a convex function satisfying the following conditions:

(a) the Monge-Ampère measure of $$u$$ in $$\Omega$$ is a positive smooth function times the Lebesgue measure;

(b) $$u$$ has infinite slope at every boundary point (or equivalently, the graph of $$u$$ does not have non-vertical supporting planes at any boundary point).

Then $$u$$ is smooth in $$\Omega$$.

However, since my understanding of the known results is rather limited and superficial, I'm not sure whether this is correct. Here's my argument: First, by Evans-Krylov theory, if $$u$$ is strictly convex , then condition (a) would imply the smoothness (see e.g. Theorem 3.1 in this survey of Trudinger-Wang). If $$u$$ is not strictly convex, then there is a supporting affine function $$a$$ with $$u\geq a$$ such that the closed convex set $$\{u=a\}$$ is not a single point. But it follows from "balancing of sections" that this set cannot have extremal points in $$\Omega$$, hence must meet $$\partial\Omega$$ (see e.g. Theorem 7 in these notes of Connor Mooney, which violates condition (b). Q.E.D.

So my question is: Are the above proposition and argument correct? If yes, what are the references to cite if I want to quote the proposition in a paper?

The proposition is true. One can argue as follows: for $$x \in \Omega$$, let $$L$$ be a supporting linear function to $$u$$ at $$x$$. Then $$L < u$$ on $$\partial \Omega$$ by (b). For $$h > 0$$ small we conclude that $$S_h := \{u < L + h\}$$ is compactly contained in $$\Omega$$, that $$u|_{\partial S_h}$$ is affine, and that $$\det D^2u$$ is positive and smooth on $$\overline{S_h}$$. By the interior smoothness of solutions to the Dirichlet problem with affine boundary data (first proven by Cheng-Yau using geometric methods in this paper and later proven by Lions using PDE techniques in this paper) we conclude that $$u$$ is smooth.
• Yes, $u$ itself serves as a barrier. Alternatively, one can approximate $S_h$ by a sequence of smooth uniformly convex domains and solve the Dirichlet problem with RHS $\det D^2u$ and boundary data $L$. These functions are smooth and enjoy uniform derivative estimates of all orders in a neighborhood of $x$ by Pogorelov's interior $C^2$ estimate (which, by Evans-Krylov and Schauder estimates, implies interior derivative estimates of all higher orders). – Connor Mooney Apr 5 at 4:58
• To be clear: solutions to $\det D^2w = f$ in $E$ convex, $w|_{\partial E}$ linear and $f$ positive and smooth in $\overline{E}$ are smooth in $E$ by the approximation argument, but in general will have infinite derivative on flat" parts of $\partial E$. In the specific case considered above, with $E = S_h$, this doesn't happen because $|\nabla u|$ is bounded on $S_h$. – Connor Mooney Apr 5 at 5:24