Brenier弱解和Aleksandrov弱解

Setting:

Consider the Monge-Ampere problem:
\begin{equation}
\left{
$\det D^{2}u=1$
$Du(\Omega )={\Omega }^{\ast }$
\right.
\end{equation}

Lemma 1

If $u:\Omega \to {\Omega }^{\ast }$ is a convex function, and $x\in \Omega$ is a Lebesgue point of $u$, then $u$ is continuous at $x$.

Lemma 2

If $u:\Omega \to {\Omega }^{\ast }$ is a convex function, and $Du(\Omega )\subset {\Omega }^{\ast }$, then we have$$\forall E\subset \mathcal{B}(\Omega ),\partial u(E)\subset \Gamma ({\Omega }^{\ast })$$where $\Gamma ({\Omega }^{\ast })$ is the convex hull of ${\Omega }^{\ast }$.

Lemma 3

If $u,\Omega ,{\Omega }^{\ast }$ as above, and under the following 3 conditions:

(i) $u(x)=\underset{\alpha }{\sup }{L}_{\alpha }(x),$ for each $x$, where $\{L_{\alpha }\}$ is a family of affine functions, satisfying $D{L}_{\alpha }(x)\in {\Omega }^{\ast }$ for each $\alpha$.

(ii) $\lambda (\Gamma ({\Omega }^{\ast })-{\Omega }^{\ast })=0$

(iii) $\forall h\in C({\mathbb{R}}^n)$ , one has the Brenier’s Monge-Ampere inequality:$$C_1{\int }_{\Omega }h(Du(x))dx\le {\int }_{{\Omega }^{\ast }}h(y)dy\le C_2{\int }_{\Omega }h(Du(x))dx$$
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Then, one has the Aleksandrov’s Monge-Ampere inequality:$$C_1{\chi }_{\Omega }\le \det D^{2}u\le C_2{\chi }_{\Omega }$$in ${\mathbb{R}}^n$.