# ap.analysis of pdes – 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 $$Omegasubsetmathbb{R}^n$$. It seems that the following statement can be deduced from known results:

Proposition. Let $$uin C^0(overline{Omega})$$ 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 $$ugeq a$$ 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 $$partialOmega$$ (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?

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