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.

  • $\begingroup$ Thank you so much! A slight issue with the interior smoothness result you quoted: Both of the two papers impose exterior sphere condition on the convex domain. It is correct to say that the condition can be easily removed? (Maybe because the condition is only used to get paraboloid barriers, and for non-strictly convex domain we still have other barriers instead?) $\endgroup$ – Xin Nie Apr 5 at 4:46
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    $\begingroup$ 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). $\endgroup$ – Connor Mooney Apr 5 at 4:58
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    $\begingroup$ 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 $S_h$. $\endgroup$ – Connor Mooney Apr 5 at 5:24
  • $\begingroup$ Got it, thanks a lot! BTW, in a paper (arxiv.org/abs/1903.08139, accepted in APDE) with my collaborator Andrea, we used your answer to a question of mine on MO two years ago, and acknowledged that at the end of the introduction. Should have let you know earlier. $\endgroup$ – Xin Nie Apr 5 at 6:12
  • $\begingroup$ Thank you for sharing your paper- those are nice results. $\endgroup$ – Connor Mooney Apr 5 at 15:13

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