In this paper we investigate the Hausdorff dimension of the set where Alexandrov solutions (see Section $2$ for the precise definition) to

$$\det D^{2}u\geq 1$$

are not strictly convex. Recall that we say that a convex function $u$ is strictly convex at $x_{0}$ if there exists $L_{x_{0}}$, a supporting tangent plane at $x_{0}$, such that

$$\{u=L_{x_{0}}\}=x_{0}.$$

Our main theorem is:

We show this is optimal by constructing solutions to $\det D^{2}u=1$ with singular set of Hausdorff dimension as close as we like to $n-1$. This result is interesting especially for $n\geq 3$ since it is well-known that in two dimensions solutions to $\det D^{2}u\geq 1$ are strictly convex.

Theorem 1.1 has several applications to the regularity theory for singular solutions to the Monge-Ampère equation with bounded right hand side, which we now describe.

Caffarelli developed a regularity theory of solutions to

$$\det D^{2}u=f\quad\text{in }\Omega,\quad\quad\lambda\leq f\leq\Lambda$$

at points where $u$ is strictly convex. We briefly summarize the main results. We define a section of $u$ at $x$ with height $h$ and slope $p$ by

$$S_{h,p}(x)=\{y\in\Omega:u(y)<u(x)+p\cdot(y-x)+h\}$$

for some subgradient $p$ at $x$. If $u$ is strictly convex at $x$ then we can find a subgradient $p$ such that the supporting plane of this slope touches only at $x$, and then take $h$ small enough that $S_{h,p}(x)\subset\subset\Omega$. In this setting, Caffarelli ([C1],[C2]) showed that

1. (i)

   $u$ is strictly convex in $S_{h,p}(x)$ and $u\in C^{1,\alpha}_{loc}(S_{h,p}(x))$,

2. (ii)

   If $f\in C^{\alpha}(\Omega)$ then $u\in C^{2,\alpha}_{loc}(S_{h,p}(x))$, and

3. (iii)

   For every $p>1$ there is some $\epsilon(p)>0$ such that if $|f-1|<\epsilon$ then $u\in W^{2,p}_{loc}(S_{h,p}(x))$.

However, these regularity theorems fail at points where $u$ is not strictly convex. Consider the famous Pogorelov examples on $B_{1}\subset\mathbb{R}^{n},\quad n\geq 3$ which degenerate along $x^{\prime}=(x_{1},...,x_{n-1})=0$. One constructs these examples by seeking solutions of the form $|x^{\prime}|+|x^{\prime}|^{\beta}g(x_{n})$ and $|x^{\prime}|^{\alpha}f(x_{n})$. The first is

$$|x^{\prime}|+|x^{\prime}|^{n/2}(1+x_{n}^{2}),$$

which solves $\lambda\leq\det D^{2}u\leq\Lambda$ but is merely Lipschitz. The second is

$$|x^{\prime}|^{2-2/n}(1+x_{n}^{2}),$$

which solves $\det D^{2}u=f$ with $f$ strictly positive and smooth, but is only $C^{1,\alpha}$ for $\alpha=1-2/n$ and $W^{2,p}$ for $p<\frac{n(n-1)}{2}$.

In [C3], Caffarelli generalizes these examples to solutions that degenerate along subspaces of any dimension less than $\frac{n}{2}$, and shows that it is not possible to find solutions degenerating on subspaces of dimension $\frac{n}{2}$ or higher. We provide a short proof in the next section (see Lemma 2.3). If $u$ agrees with a linear function $L$ on a $k$-dimensional set, we say that $\{u=L\}$ is a $k$-dimensional singularity. Our proof of Theorem 1.1 in fact shows that the collection of $k$-dimensional singularities has Hausdorff $n-k$ dimensional measure zero (see Remark 3.4).

Since we cannot hope for $C^{1}$ regularity or $W^{2,p}$ regularity of singular solutions to $\lambda\leq\det D^{2}u\leq\Lambda$ for large $p$, it is natural to ask what we can show about the integrability of the second derivatives. De Philippis, Figalli and Savin ([DFS],[DF]) recently showed $W^{2,1+\epsilon}$ regularity of strictly convex solutions to $\lambda\leq\det D^{2}u\leq\Lambda$, where $\epsilon$ depends only on $\lambda,\Lambda$ and $n$. Our main theorem rules out the possibility that the second derivatives concentrate on $\Sigma$:

We also show that Theorem 1.2 is optimal by proving that the examples giving optimality of Theorem 1.1 are not in $W^{2,1+\epsilon}$ for $\epsilon$ as small as we like.

A second consequence of Theorem 1.1 is that the points of strict convexity for $u$ form a connected set when $f$ is bounded away from $0$. If $f$ is sufficiently regular we obtain unique continuation for the Monge-Ampère equation:

The paper is organized as follows. In section $2$ we present basic geometric properties of the sections of solutions to $\det D^{2}u\geq 1$. In particular, we present an important estimate on the volume growth of sections that are not compactly contained and relate the volume of compactly contained sections to the Monge-Ampère mass of these sections. In section $3$ we use these results at singular points together with the useful technique of replacing $u$ by $u+\frac{1}{2}|x|^{2}$ to prove Theorem 1.1. In section $4$ we construct, for any $\delta$, a solution to $\det D^{2}u=1$ with a singular set of Hausdorff dimension $n-1-\delta$, which shows that our main theorem is optimal. In section $5$ we use Theorem 1.1 to prove Theorem 1.2 and we show that the examples constructed in section $4$ are not in $W^{2,1+\epsilon}$ for $\epsilon$ as small as we like, which shows that $W^{2,1}$ regularity is optimal. Finally, in section $6$ we prove Theorem 1.3 by applying a classical unique continuation theorem in the set of strict convexity.

In future work we intend to present a more precise, quantitative version of our main theorem to obtain $L\log L$ estimates for the second derivatives of singular solutions to $\lambda\leq\det D^{2}u\leq\Lambda$.

Acknowledgements: This work is part of my forthcoming doctoral dissertation at Columbia University. I am very grateful to my thesis advisor Ovidiu Savin for his patient guidance and for his feedback on the drafts of this paper. I would also like to thank Nam Le and Yu Wang for helpful conversations about the subject.

The author was partially supported by the NSF Graduate Research Fellowship Program under grant number DGE 11-44155.