Continuity and injectivity of optimal maps

Abstract

Figalli–Kim–McCann proved in (2009) the continuity and injectivity of optimal maps under the assumption (B3) of nonnegative cross-curvature. In the recent Figalli et al. (Arch Ration Mech Anal 209:747–795, 2013), Figalli et al. (Methods Appl Anal 2013), they extend their results to the assumption (A3w) of Trudinger–Wang (Ann Sc Norm Super Pisa Cl Sci 8:143–174, 2009), and they prove, moreover, the Hölder continuity of these maps. We give here an alternative and independent proof of the extension to (A3w) of the continuity and injectivity of optimal maps based on the sole arguments of Figalli et al. (2009) and on new Alexandrov-type estimates for lower bounds.

Appendix

In this appendix, we gather some results on optimal transportation and convex sets that we use in this paper.

1.1 The maximum principle for cost functions

In Theorem 6.1 below, we state a maximum principle which was first established by Loeper [29], see also Kim–McCann [24], Trudinger–Wang [33, 35], and Villani [38] for other proofs and extensions. For any $x\in \overline{\varOmega ^+}$ and $y,y'\in \overline{\varOmega ^*}$, letting $D_xc\left( x,y\right) =-p$ and $D_xc\left( x,y'\right) =-p'$, i.e. $Y\left( x,p\right) =y$ and $Y\left( x,p'\right) =y'$, we say that $t\mapsto Y\left( x,\left( 1-t\right) p+tp'\right)$ is the $c^*$ -segment connecting $y$ and $y'$ with respect to $x$. Loeper’s maximum principle states as follows. Needless to say, a dual result can be obtained by exchanging the roles of $x$ and $y$.

Theorem 6.1

Let $U$, $\varOmega ^+$, $\varOmega ^*$, $c$ satisfy (A0)–(A3w) and (Bw). Let $x,x'\in \overline{\varOmega ^+}$ and $y,y'\in \overline{\varOmega ^*}$. Then for any $t\in \left[ 0,1\right]$, there holds $f\left( t\right) :=c\left( x,y_t\right) -c\left( x',y_t\right) \le \max \left( f\left( 0\right) ,f\left( 1\right) \right)$, where $t\mapsto y_t$ is the $c^*$-segment connecting $y$ and $y'$ with respect to $x$.

For any $x\in \varOmega ^+$, we let $\partial _c u\left( x\right)$ be as in (2.3) and $\partial u\left( x\right)$ be the subdifferential of $u$ at $x$ defined by

$$\begin{aligned} \partial u\left( x\right) :=\left\{ y\in \mathbb {R}^n;\,\quad u\left( x'\right) \ge u\left( x\right) +\left<y,x'-x\right>+\mathrm{o }\left( \left| x'-x\right| \right) \text { as }x'\rightarrow x\right\} . \end{aligned}$$

(6.1)

We give two corollaries below of Theorem 6.1 which we use in this paper. These results were obtained by Loeper [29]. Here again, we mention the related references by Kim–McCann [24], Trudinger–Wang [33, 35], and Villani [38].

Corollary 6.2

Let $U$, $\varOmega ^+$, $\varOmega ^*$, $c$, $u$ satisfy (A0)–(A3w), (Bw), and (C). For any $x\in \varOmega ^+$, there holds $\partial u\left( x\right) =-D_xc\left( x,\partial _cu\left( x\right) \right)$, where $\partial u\left( x\right)$ and $\partial _cu\left( x\right)$ are as in (2.3) and (6.1). In particular, $D_xc\left( x,\partial _cu\left( x\right) \right)$ is convex.

Corollary 6.3

Let $U$, $\varOmega ^+$, $\varOmega ^*$, $c$, $u$ satisfy (A0)–(A3w), (Bw), and (C). For any $y\in \overline{\varOmega ^*}$, any local minimum point of the function $x\mapsto u\left( x\right) +c\left( x,y\right)$ is a global minimum point in $\varOmega ^+$.

1.2 On the image and inverse image of boundary points

The assumption (B) of strong $c$-convexity of domains is used in this paper in order to apply Theorem 6.4 below, which is due to Figalli–Kim–McCann [14].

Theorem 6.4

Let $\varOmega \subset \varOmega ^+$ and $\varOmega ^*$ be bounded open subsets of $\mathbb {R}^n$. Let $c$ and $u$ satisfy (A0)–(A2) and (C).

• (i) If $\left| \partial _c u\right| \ge \varLambda _1\mathcal {L}^n$ in $\varOmega$ for some $\varLambda _1>0$ and $\varOmega ^*$ is strongly $c^*$-convex with respect to $\varOmega$, then interior points of $\varOmega$ cannot be mapped by $\partial _c u$ to boundary points of $\varOmega ^*$, i.e. $\partial _cu^{-1}\left( \partial \varOmega ^*\right) \cap \varOmega =\emptyset$.

• (ii) If $\left| \partial _c u\right| \le \varLambda _2\mathcal {L}^n$ in $\overline{\varOmega ^+}$ for some $\varLambda _2>0$ and $\varOmega ^+$ is strongly $c$-convex with respect to $\varOmega ^*$, then boundary points of $\varOmega ^+$ cannot be mapped by $\partial _c u$ into interior points of $\varOmega ^*$, i.e. $\partial _cu\left( \partial \varOmega ^+\right) \cap \varOmega ^*=\emptyset$.

1.3 Two results on convex sets

We also use in this paper the following two results on convex sets. The first one is John’s Theorem [23]. We use this result in both the Alexandrov-type estimates for lower and upper bounds in Sects. 3 and 4.

Theorem 6.5

For any compact convex subset $K$ of $\mathbb {R}^n$ with nonempty interior, there exists a $n$-dimensional ellipsoid such that $E\subset K\subset nE$, where $nE$ is the dilation of $E$ with respect to its center of mass.

Another result on convex sets, Theorem 6.6 below, is used in the proof of the Alexandrov-type estimates for upper bounds Corollary 4.3. This result was established by Figalli–Kim–McCann [14].

Theorem 6.6

Let $K$ be a convex subset of $\mathbb {R}^n=\mathbb {R}^{n'}\times \mathbb {R}^{n''}$. Let $\pi ''$ be the projection of $\mathbb {R}^n$ onto $\mathbb {R}^{n''}$. Let $x''\in \pi ''\left( K\right)$ and $K'=\left( \pi ''\right) ^{-1}\left( x''\right) \cap K$. Then there holds

$$\begin{aligned} \mathcal {H}^{n'}\left( K'\right) \mathcal {H}^{n''}\left( \pi ''\left( K\right) \right) \le C\mathcal {L}^n\left( K\right) \end{aligned}$$

for some $C=C\left( n',n''\right) >0$, where $\mathcal {H}^d$ is the $d$-dimensional Haussdorf measure and $\mathcal {L}^n$ is the $n$-dimensional Lebesgue measure.