2. Preliminaries [04TN]

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2. Preliminaries

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

Mv(A)=|\nabla u(A)|

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

|\nabla v(A)|=\int_{A}\det D^{2}v\,dx.

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

\det D^{2}v=\mu

if $Mv=\mu$ .

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

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

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

Lemma 2.1.

(John’s Lemma). If $K\subset\mathbb{R}^{n}$ is a bounded convex set with nonempty interior, and $0$ is the center of mass of $K$ , then there exists an ellipsoid $E$ and a dimensional constant $C(n)$ such that

E\subset K\subset C(n)E.

We call $E$ the John ellipsoid of $K$ . There is some linear transformation $A$ such that $A(B_{1})=E$ , and we say that $A$ normalizes $K$ .

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

Lemma 2.2.

Assume that $\det D^{2}u\geq 1$ in $\Omega\subset\mathbb{R}^{n}$ . Then if $S_{h,p}(x)$ is any section of $u$ , we have

|S_{h,p}(x)|\leq Ch^{n/2}

for some constant $C$ depending only on $n$ .

Proof.

Assume by translation that $0$ is the center of mass of $S_{h,p}(x)$ . By subtracting a linear function we can assume that

p=0,\quad u|_{\partial S_{h,0}(x)}\leq 0,\text{ and }\quad|\min_{S_{h,0}(x)}u|=h.

By John’s Lemma, there is a linear transformation $A$ that normalizes $S_{h,0}(x)$ . Let

\tilde{u}(x)=|\det A|^{-2/n}u(Ax).

It is easy to check that

\det D^{2}\tilde{u}\geq 1,\quad\tilde{u}|_{\partial\tilde{\Omega}}\leq 0

where $B_{1}\subset\tilde{\Omega}\subset B_{C(n)}$ . Then $\frac{1}{2}(|x|^{2}-1)$ is an upper barrier for $\tilde{u}$ , so

|\min_{\tilde{\Omega}}\tilde{u}|\geq\frac{1}{2}.

Since $|\det A|\geq c(n)|S_{h,0}(x)|$ , the conclusion follows. ∎

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

Lemma 2.3.

Assume

\det D^{2}u\geq 1

in $B_{1}\subset\mathbb{R}^{n}$ . Then $u$ cannot vanish on a subspace of dimension $\frac{n}{2}$ or higher.

Proof.

Suppose $u$ vanishes on

\{x_{k+1}=...=x_{n}=0\}.

By subtracting a linear function of the form $a_{k+1}x_{k+1}+...+a_{n}x_{n}$ we may assume that $u(te_{n})=o(t)$ . Then $S_{h,0}(0)$ has length $R(h)h$ in the $e_{n}$ direction, where $R(h)\rightarrow\infty$ as $h\rightarrow 0$ . Furthermore, $S_{h,0}(0)$ has length exceeding $\frac{1}{C}h$ in the $e_{n-k},...,e_{n-1}$ directions, where $C$ is the Lipschitz constant of $u$ in $B_{1/2}$ . Finally, $S_{h,0}(0)$ contains the unit ball in the subspace spanned by $\{e_{1},...,e_{k}\}$ . We conclude that

|S_{h,0}(0)|\geq C^{-k}R(h)h^{n-k},

which contradicts Lemma 2.2 as $h\rightarrow 0$ for $k\geq\frac{n}{2}$ . ∎

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.

Lemma 2.4.

Let $v$ be any convex function on $\Omega\subset\mathbb{R}^{n}$ with $v|_{\partial\Omega}=0$ . Then

Mv(\Omega)\,|\Omega|\geq c(n)|\min_{\Omega}v|^{n}.

Proof.

By translation assume that the center of mass of $\Omega$ is $0$ . Let $A$ normalize $\Omega$ and let

\tilde{v}(x)=(\det A)^{-2/n}v(Ax).

Then

M\tilde{v}(\tilde{\Omega})=(\det A)^{-1}Mv(\Omega)

with $B_{1}\subset\tilde{\Omega}\subset B_{C(n)}$ .

The maximum of $|\tilde{v}|$ is achieved at some point $\tilde{x}\in\tilde{\Omega}$ . Let $K$ be the function whose graph is the cone generated by $(\tilde{x},\tilde{v}(x))$ and $\partial B_{C(n)}$ . By convexity,

M\tilde{v}(\tilde{\Omega})\geq|\nabla K(\tilde{x})|.

Since $\nabla K(\tilde{x})$ is a ball of radius at least $c(n)|\min_{\tilde{\Omega}}\tilde{v}|$ , we have

|\nabla K(\tilde{x})|\geq c(n)|\min_{\tilde{\Omega}}\tilde{v}|^{n}\geq c(n)|\det A|^{-2}|\min_{\Omega}v|^{n}.

Finally, $|\det A|\leq C(n)|\Omega|$ so the conclusion follows. ∎