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

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

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

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

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

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

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.

We first recall the precise definition of Alexandrov solutions. Any convex function $v:\Omega\subset\mathbb{R}^{n}\rightarrow\mathbb{R}$ has an associated Borel measure $Mv$, called the Monge-Ampère measure, defined by

where $|\nabla u(A)|$ represents the Lebesgue measure of the image of the subgradients of $v$ in $A$ (see [Gut]). If $v\in C^{2},$ then

Given a Borel measure $\mu$, we say that $v$ is an Alexandrov solution to

if $Mv=\mu$.

For a convex function $v$ defined on $\Omega\subset\mathbb{R}^{n}$, we define a section $S_{h,p}(x)$ by

for some subgradient $p$ at $x$. We now present some results on the geometry of the sections.

The next lemma is an important observation about the volume growth of sections which may not be compactly contained in $\Omega$:

Caffarelli proved the next proposition in [C3]. We provide a short proof using a technique related to our proof of the main theorem.

In particular, every solution to $\det D^{2}u\geq 1$ in two dimensions is strictly convex.

We conclude the section with the following variant of Alexandrov’s maximum principle. In the following $c(n),C(n)$ denote small and large constants depending only on $n$, and their values may change from line to line.

In this section assume that

in $B_{1}\subset\mathbb{R}^{n}$. Fix $x\in\Sigma$ and a subgradient $p$ at $x$. By translation and subtracting a linear function assume that $x=p=0$. Then $\{u=0\}$ contains a line segment of some length $l$. By Lemma 2.2,

for all $h>0$.

Letting $v=u+\frac{1}{2}|x|^{2}$ and denoting the sections of $v$ by $S_{h,p}^{v}$, it follows that

for all $h$ small.

Theorem 1.1 thus follows from the following more general result:

We briefly discuss the main ideas of the proof. Fix $x\in\Sigma_{v}$ and a subgradient $p$ at $x$. In the following analysis $c,\,C$ will denote small and large constants depending on $n$ and the $C_{p}$. If $S_{h,p}^{v}(x)\subset\subset B_{1}$ then the definition of $\Sigma_{v}$ and Lemma 2.4 give

for all $h$ small.

An important technique of the proof is to replace $v$ by $v+\frac{1}{2}|x|^{2}$. Then all of the sections are compactly contained in $B_{1}$ for $h$ small, and the diameter of sections is at most $h^{1/2}$. By replacing the sections $S_{h,p}^{v}(x)$ by $B_{\sqrt{h}}(x)$ and using a covering argument, we easily obtain that $\Sigma_{v}$ has Hausdorff dimension at most $n-1$.

Lemmas 3.2 and 3.3 improve this result as follows. We aim to rule out behavior like

which has a singular hyperplane. For this example, the sections at $\{x_{n}=0\}$ have the correct growth when we take supporting slopes with no $x_{n}$-component, but the sections are too large when we take supporting slopes with $x_{n}$-component $1$.

In the first lemma we use that the sections are small for all supporting planes at $x\in\Sigma_{v}$ to show that $v$ must grow much faster than quadratically in at least two directions, unlike the example above. In the second lemma we use the above observation about the Monge-Ampère mass of $v$ in the directions where $v$ grows much faster than quadratically from $x$. Since we replaced $v$ by $v+\frac{1}{2}|x|^{2}$ we also know that $v$ grows at least quadratically in the remaining directions. This allows us to cover $\Sigma_{v}$ with balls in which the Monge-Ampère mass of $v$ is much larger than the radius to the $n-1$, giving the desired improvement.

We can complete the proof of theorem 3.1 with a covering argument.

In this section we construct examples of solutions to $\det D^{2}u=1$ in $\mathbb{R}^{3}$ such that $\Sigma$ has Hausdorff dimension as close to $2$ as we like. A small modification produces the analagous examples in $\mathbb{R}^{n}$.

For this section, fix $\delta>0$ small. We construct our examples in several steps, which we briefly describe:

1. (i)

   First, we construct functions $w$ with

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

   in $\mathbb{R}^{3}$ that degenerate along $\{x_{1}=x_{2}=0\}$ and behave like $x_{1}^{2-\delta}$ along the $x_{1}$ axis.

2. (ii)

   Next, we construct a standard $S\subset[-1,1]$ with Hausdorff dimension close to $1$ and a convex function $v$ on $[-1,1]$ such that for any $x\in S$, there is a tangent line such that $v$ separates from this line faster than $r^{2-\delta}$.

3. (iii)

   Finally, we get our example by solving the Dirichlet problem

   $$\det D^{2}u=1\quad\text{ in }\Omega=\{|x^{\prime}|<1\}\times[-1,1],\quad\quad u|_{\partial\Omega}=C(\delta)(v(x_{1})+|x_{2}|)$$

   and comparing with $w$ at points in $S\times\{0\}\times\{\pm 1\}$.

In the following analysis $c$ and $C$ will denote small and large constants depending on $\delta$.

Construction of $w$: We look for a convex function $w(x_{1},x_{2},x_{3})$ with the homogeneity

where $\alpha$ and $\beta$ satisfy $1<\alpha,\beta<2$ and

(It is easy to check that $\geq 3/2$ is necessary for such a function to have $\det D^{2}w$ bounded below). Note that this rescaling preserves the curves $x_{2}=mx_{1}^{\alpha/\beta}$.

Let $f(x)$ denote $1+x^{2}$. An obvious candidate for $w$ is

One checks that

Take $\alpha=2-\delta$. Then for $|x_{3}|$ small depending on $\delta$ we have

Along the curves $x_{2}=mx_{1}^{\alpha/\beta}$, we compute

since $1<\beta<2$.

Thus, up to rescaling the $x_{3}$-axis and multiplying by a constant, we have

Construction of $S$: Let $\epsilon>0$ be a small constant we will choose shortly depending on $\delta$. Construct a self-similar set in $[-1/2,1/2]$ as follows: First, remove an open interval of length $\gamma=1-2^{-3\epsilon}$ from the center. Proceed inductively by removing intervals a fraction $\gamma$ of each of those that remains. Denote the centers of the intervals removed at stage $k$ by $\{x_{i,k}\}_{i=1}^{2^{k}}$, and the intervals by $I_{i,k}$. Finally, let

It is easy to check that $|I_{i,k+1}|=\gamma 2^{-(1+3\epsilon)k}$ and that $S$ has Hausdorff dimension $\frac{1}{1+3\epsilon}$.

Construction of $v$: Let

We add rescalings of $w$ together to produce the desired function:

We now check that $v$ satisfies the desired properties:

1. (i)

   v is convex, as the sum of convex functions. Furthermore,

   	$\displaystyle|v(x)|$	$\displaystyle\leq C\sum_{k=1}^{\infty}\sum_{i=1}^{2^{k}}2^{-(1+\epsilon)k}$
   		$\displaystyle\leq C\sum_{k=1}^{\infty}2^{-\epsilon k},$

   so $v$ is bounded.

2. (ii)

   Let $x\in S$. We aim to show that $v$ separates from a tangent line more than $r^{2-\delta}$ a distance $r$ from $x$. By subtracting a line assume that $v(x)=0$ and that $0$ is a subgradient at $x$. Assume further that $x+r<1/2$ and that $2^{-(1+3\epsilon)k}<r\leq 2^{-(1+3\epsilon)(k-1)}$. There are two cases to examine:

   Case 1: There is some $y\in(x+r/2,x+r)\cap S$. Then by the construction of $S$ it is easy to see that there is some interval $I_{i,k}$ such that $I_{i,k}\subset(x,x+r)$. On this interval, $v$ grows by

   $$2^{-2(1+2\epsilon)k}\geq cr^{2\frac{1+2\epsilon}{1+3\epsilon}}=cr^{2-\delta},$$

   where we choose $\epsilon$ so that

   $$\delta=\frac{2\epsilon}{1+3\epsilon}.$$

   Case 2: Otherwise, there is an interval $I_{i,j}$ of length exceeding $r/2$ such that $(x+r/2,x+r)\subset I_{i,j}$. Then at the left point of $I_{i,j}$, the slope of $v$ jumps by at least $2^{-(1+\epsilon)k}$. It follows that at $x+r$, $v$ is at least

   $$\frac{r}{2}2^{-(1+\epsilon)k}\geq cr^{2-\delta}.$$

   Thus, $v$ has the desired properties.

Construction of $u$: We recall the following lemma on the solvability of the Monge-Ampère equation (see [Gut]).