Subsection 2.5 (00B3) · ScalingStacks

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2.5 Regularity theory for real Monge-Ampère

There is an extensive literature on the local regularity theory for the real Monge-Ampère equation, largely due to the Caffarelli school. The author thanks C. Mooney for bringing some of these results to his attention. All results surveyed here can be found in [35].

Any convex function on an open set $v:\Omega\subset\mathbb{R}^{n}\to\mathbb{R}$ has an associated Borel measure called the Monge-Ampère measure, defined by

MA_{\mathbb{R}}(v)(E)=|\partial v(E)|,

where $|\partial v(E)|$ denotes the Lebesgue measure of the image of the subgradient map on $E\subset\Omega$ . Given a Borel measure $\mu$ , a solution to $MA(v)=\mu$ is called an Aleksandrov solution to $\det(D^{2}v)=\mu;$ if $v\in C^{2}$ , this is the classical real Monge-Ampère equation. We shall assume a two-sided density bound

\det(D^{2}v)=f\text{ in }B_{1},\quad 0<\Lambda_{1}\leq f\leq\Lambda_{2}.

Let $B_{1}\setminus\Sigma$ be the set of strictly convex points of $v$ , namely there is a supporting hyperplane touching the graph of $v$ only at one point. Then Caffarelli [8][9][10] shows

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If $f\in C^{\gamma}(B_{1})$ , then $v\in C^{2,\gamma}_{loc}(B_{1}\setminus\Sigma)$ . Then by Schauder theory, if $f$ is smooth, then $v$ is smooth in $B_{1}\setminus\Sigma$ .
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If $L$ is a supporting affine linear function to $v$ , such that the convex set $\{v=L\}$ is not a point. Then $\{v=L\}$ has no extremal point in the interior of $B_{1}$ .
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The above affine linear set $\{v=L\}$ has dimension $k<n/2$ .

Mooney [35] shows further that

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The singular set $\Sigma$ has $(n-1)$ -Hausdorff measure zero. Consequently $B_{1}\setminus\Sigma$ is path connected (because a generic path joining two given points does not intersect a subset of zero $(n-1)$ -Hausdorff measure).
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The solution $v\in W^{2,1}_{loc}(B_{1})$ even if $\Sigma$ is nonempty.

009R

Remark 2.9. A classical counterexample of Pogorelov shows that for $n=3$ , the singular set $\Sigma$ can contain a line segment. This is generalised by Caffarelli [10], who for any $k<n/2$ constructs examples where $f$ is smooth but $\Sigma$ contains a $k$ -plane. A surprising example of Mooney [35] shows that the Hausdorff dimension of $\Sigma$ can be larger than $n-1-\epsilon$ for any small $\epsilon$ . This means the local regularity theory surveyed above is essentially optimal.