Global regularity of optimal mappings in non-convex domains

TL;DR

This paper proves a global regularity result for optimal transport maps with quadratic cost in non-convex domains, extending previous results from convex domains using a perturbation approach.

Contribution

It introduces a novel perturbation method to establish regularity of optimal transport in non-convex domains, expanding the scope of existing regularity theory.

Findings

Global regularity established for non-convex domains

Perturbation technique effectively extends convex domain results

Supports broader applications of optimal transport theory

Abstract

In this paper, we establish a global regularity result for the optimal transport problem with the quadratic cost, where the domains may not be convex. This result is obtained by a perturbation argument, using a recent global regularity of optimal transportation in convex domains by the authors.

Equations

$${\int }_{\Omega }f={\int }_{{\Omega }^{\ast }}g.$$

$$\mathcal{C}(T)={\int }_{\Omega }c(x,T(x))f(x)dx$$

$${\int }_{T^{-1}(E)}f={\int }_{E}g.$$

$$c(x,y)=x\cdot y,$$

$$\text{mbox}det{D}^{2}u(x)=\frac{f(x)}{g(Du(x))},$$

$$Du(\Omega )={\Omega }^{\ast }.$$

$$c_0\le f,g\le c_1$$

$$\Vert u{\Vert }_{C^{2,\alpha }(\overline{\Omega })}\le C,$$

$$\Vert u{\Vert }_{C^{1,\beta }(\overline{\Omega })}\le C_{\beta }\forall \beta \in (0,1),\text{mbox}and\Vert u{\Vert }_{W^{2,p}(\overline{\Omega })}\le C_p\forall p\ge 1,$$

$$\Vert \Phi -I{\Vert }_{C^{1,1}(\overline{\Omega })}\le \delta$$

$$\partial \Omega \cap B_r=\{x=(x^{\prime },x_n)\in {\mathbb{R}}^n:x_n=\eta (x^{\prime })\},$$

$$\eta \ge 0,\eta (0)=0\text{and}D\eta (0)=0.$$

$$\Vert \rho -\eta {\Vert }_{C^{1,1}(B_r^{\prime })}\le \delta ,$$

$${\int }_{\Omega }\tilde{f}={\int }_{{\Omega }^{\ast }}\tilde{g}.$$

$$\|u-\tilde{u}\|_{\infty}=\sup_{x\in\Lambda}\big{\{}[u(x)-u(x_{0})]-[\tilde{u}(x)-\tilde{u}(x_{0})]\big{\}},$$

$$\Vert u-\tilde{u}{\Vert }_{\infty }\le \omega (\delta ).$$

$$\Vert u_k-{\tilde{u}}_k{\Vert }_{\infty }>{\delta }_0$$

$$\Vert u_{\infty }-{\tilde{u}}_{\infty }{\Vert }_{\infty }\ge {\delta }_0.$$

$$\partial \Lambda =\{x:x_n=\rho (x^{\prime })\}\text{with}\rho (0)=0,D\rho (0)=0.$$

$$\tilde{u}=\frac{1}{2}|x{|}^2+O(|x{|}^{2+\alpha }),$$

$$\left|u-\frac{1}{2}|x{|}^2\right|\le \omega (\delta )+O(|x{|}^{2+\alpha }),$$

$$\partial {\Lambda }^{\ast }=\{x:x_n={\rho }^{\ast }(x^{\prime })\}\text{with}{\rho }^{\ast }(0)\to 0\text{and}D{\rho }^{\ast }(0)\to 0,\text{ as}\delta \to 0.$$

$$\Vert u-\frac{1}{2}|x{|}^2{\Vert }_{L^{\infty }(U)}\le \omega (\delta )+C{ϵ}_0^{2+\alpha },$$

$$\{x:x_n>{\rho }^{\ast }(x^{\prime })\}\cap B_{{ϵ}_0/2}\subset U^{\ast }\subset \{x:x_n>{\rho }^{\ast }(x^{\prime })\}\cap B_{2{ϵ}_0}.$$

$$x\mapsto x/{ϵ}_0$$

$$u^{\prime }(x):=\frac{1}{{ϵ}_0^2}u({ϵ}_{0}x),\text{mbox}inthedomain{U}^{\prime }:=\frac{1}{{ϵ}_0}U.$$

$$\Vert u^{\prime }-\frac{1}{2}|x{|}^2{\Vert }_{L^{\infty }(U^{\prime })}<{\eta }_0.$$

$$\Vert u^{\prime }-\frac{1}{2}|x{|}^2{\Vert }_{L^{\infty }(U^{\prime })}\le \frac{\omega (\delta )}{{ϵ}_0^2}+C{ϵ}_0^{\alpha }<{\eta }_0$$

$$x_n>$$

$$B_{1/2}\cap \{x_n>\tilde{\rho }(x^{\prime })\}\subset U^{\prime }\subset B_2\cap \{x_n>\tilde{\rho }(x^{\prime })\},$$

Full text

This work was supported by ARC FL130100118 and ARC DP170100929.

1. Introduction

The regularity of optimal mappings is a core issue in optimal transport problem [5, 33], which can be described as follows: Suppose there is a source domain $\Omega\subset\mathbb{R}^{n}$ with density $f$ and a target domain $\Omega^{*}\subset\mathbb{R}^{n}$ with density $g$ satisfying the balance condition

[TABLE]

Given a cost function $c(x,y):\Omega\times\Omega^{*}\rightarrow\mathbb{R}$, one asks for the existence and regularity of an optimal mapping $T$ that minimises the transport cost

[TABLE]

among all measure preserving maps. A mapping $T:\Omega\to\Omega^{*}$ is called measure preserving, denoted as $T_{\sharp}f=g$, if for any Borel set $E\subset\Omega^{*}$

[TABLE]

The optimal transport problem was first introduced by Monge [26] with the natural cost function $c(x,y)=|x-y|$, and was extensively studied since after. When the cost function

[TABLE]

or equivalently the quadratic cost $c(x,y)=\frac{1}{2}|x-y|^{2}$, the existence and uniqueness of the optimal mapping were obtained by Brenier [2]. It was shown that the optimal mapping $T=Du$ is the gradient of a convex potential function $u$, which satisfies

[TABLE]

[TABLE]

In this paper we study the regularity of solutions to the above boundary value problem. The densities $f,g$ are always assumed to satisfy (1.1) and

[TABLE]

for two positive constants $c_{1}\geq c_{0}>0$, which makes the equation (1.4) elliptic.

Due to its applications in optimal transportation and in many other areas, the boundary value problem (1.4)–(1.5) has received huge attention and been studied intensively in recent years [18, 33]. Assuming both domains $\Omega,\Omega^{*}$ are convex Pogorelov [27] obtained a generalised solution in the sense of Aleksandrov. In [2], Brenier showed the existence and uniqueness of solutions in another weak sense, which is equivalent to Aleksandrov’s solution when $f,g$ satisfy (1.6) and the target domain is convex. But we also refer the reader to [1] for extension of Aleksandrov’s generalised solutions. The interior regularity was developed by many people, see for example [3, 4, 16, 22, 31], the books [19, 20, 21] and references therein. Very recently, a new proof was found in [35], using the Green function of the linearised Monge-Ampère equation.

For the global regularity, assuming both domains $\Omega,\Omega^{*}$ are uniformly convex and $C^{3,1}$ smooth, and the densities $f\in C^{1,1}({\overline{\Omega}})$, $g\in C^{1,1}(\overline{\Omega^{*}})$, the global smooth solution was first obtained by Delanoë [17] for dimension two and later extended to high dimensions by Urbas [32]. In a milestone work [6], Caffarelli proved that $u\in C^{2,\alpha^{\prime}}({\overline{\Omega}})$ for some $\alpha^{\prime}\in(0,\alpha)$, if $\Omega,\Omega^{*}$ are uniformly convex with $C^{2}$ boundary, and the densities $f,g\in C^{\alpha}$. The uniform convexity of domains plays a critical role in the above mentioned papers [6, 17, 32], which is also necessary for the global regularity of solutions to other boundary value problems such as in [24, 28, 30]. In a recent paper [13], the authors removed this condition for the problem (1.4)–(1.5) and obtained the following

Theorem 1.1** ([13]).**

Assume that $\Omega$ and $\Omega^{*}$ are bounded convex domains in $\mathbb{R}^{n}$ with $C^{1,1}$ boundaries. Let $u$ be a convex solution to (1.4)–(1.5). We have the following estimates:

• (i)

If $f\in C^{\alpha}({\overline{\Omega}})$, $g\in C^{\alpha}(\overline{\Omega^{*}})$, for some $\alpha\in(0,1)$, then

[TABLE]

where $C$ is a constant depending on $n,\alpha,f,g,\Omega$, and $\Omega^{*}$.

• (ii)

If $f\in C^{0}({\overline{\Omega}})$, $g\in C^{0}(\overline{\Omega^{*}})$, then

[TABLE]

where the constants $C_{\beta},C_{p}$ depend on $n,f,g,\Omega,\Omega^{*}$, and on $\beta,p$, respectively.

In this paper, we relax furthermore the convexity condition of the domains. For any given non-convex domain $\Omega^{*}$, it was shown that there exist smooth and positive densities $f,g$ such that the potential function $u$ is not $C^{1}$ [25]. However, for fixed positive and smooth densities $f,g$, by Theorem 1.1 and a perturbation argument, in this paper we can show that $u$ is smooth up to the boundary when the domains are small perturbations of convex domains, but may not be convex themselves.

Given a bounded domain $\Lambda\subset{\mathbb{R}}^{n}$, we say $\Lambda$ is $\delta$-close to $\Omega$ in $C^{1,1}$ norm, if there exists a bijective mapping $\Phi:\Omega\to\Lambda$ such that $\Phi\in C^{1,1}({\overline{\Omega}})$ and

[TABLE]

where $I:\Omega\to\Omega$ is the identity mapping. A localised definition of $\delta$-closeness is given in §2.1. Now we can state our main theorem.

Theorem 1.2**.**

Let $\Lambda$ and $\Lambda^{*}$ be $C^{1,1}$ domains that are $\delta$-close to $\Omega$ and $\Omega^{*}$ in $C^{1,1}$ norm, respectively, where $\Omega$ and $\Omega^{*}$ are bounded convex domains with $C^{1,1}$ boundaries. Suppose that $f,g$ satisfy (1.1) and (1.6), and $f\in C^{\alpha}(\overline{\Lambda}),g\in C^{\alpha}(\overline{\Lambda^{*}})$, for some $\alpha\in(0,1)$. Then there exists a small constant $\delta_{0}>0$ depending only on $\Omega,\Omega^{*},\alpha,c_{0},c_{1},\|f\|_{C^{\alpha}(\Lambda)}$ and $\|g\|_{C^{\alpha}(\Lambda^{*})}$, such that the potential function $u\in C^{2,\alpha}(\overline{\Lambda})$, provided $\delta<\delta_{0}.$

The proof of Theorem 1.2 is carried out in the following two sections: In §2, we prove that $u\in C^{1,\beta}(\overline{\Lambda})$ for any given $\beta\in(0,1)$, by using a localisation and iteration argument. Then in §3, by adapting a perturbation argument from [13, §5], we obtain $u\in C^{2,\alpha}(\overline{\Lambda})$. As a byproduct we also obtain the following global $W^{2,p}$ estimate.

Theorem 1.3**.**

Let $\Lambda,\Lambda^{*}$ be as in Theorem 1.2. Suppose $f,g$ satisfy (1.1) and (1.6), and $f\in C(\overline{\Lambda}),g\in C(\overline{\Lambda^{*}})$. Then $\forall\ p\geq 1$, $\exists$ a small constant $\delta_{0}>0$ depending only on $\Omega,\Omega^{*},p,c_{0},c_{1},f,g$, such that the potential function $u\in W^{2,p}(\overline{\Lambda})$, provided $\delta<\delta_{0}.$

In the last section §4, we give some interesting applications of Theorems 1.2, 1.3 in the free boundary problems, minimal Lagrangian diffeomorphism, and optimal transportation with general costs, and hope to motivate future study in these areas.

2. $C^{1,\beta}$ regularity

In this section, by using a perturbation and an iteration argument, we prove the $C^{1,\beta}$ estimate, for any given $\beta\in(0,1)$. Some of our arguments are inspired by those in [11, 7, 15]. In [15], De Philippis and Figalli [15] obtained a partial regularity result for optimal transport problem with general cost functions. In [11], Figalli and the first author obtained a global regularity under a small perturbation of the quadratic cost. In [7], the regularity of optimal transport is obtained for the cost $|x-y|^{p}$ when $p$ is close to $2$.

Lemma 2.1**.**

Under the hypotheses of Theorem 1.2, for any given $\beta\in(0,1)$, there exits a small constant $\delta_{0}>0$ depending only on $\Omega,\Omega^{*},\alpha,\beta,c_{0},c_{1},\|f\|_{C^{\alpha}(\Lambda)}$ and $\|g\|_{C^{\alpha}(\Lambda^{*})}$, such that the potential function $u\in C^{1,\beta}(\overline{\Lambda})$, provided $\delta<\delta_{0}.$

By the interior regularity of the Monge-Ampère equation [3, 22], it suffices to prove Lemma 2.1 near the boundary. The proof is divided into four subsections following the strategy that: First in §2.1, we show that since the domains are small perturbations of convex domains, the potential $u$ is also a small perturbation of a $C^{2,\alpha}$ potential function $\tilde{u}$ for convex domains. Then in §2.2, we localise the problem by rescaling it near a boundary point, and show that $u$ is close to the parabola $\frac{1}{2}|x|^{2}$ (given by the second order Taylor expansion of $\tilde{u}$ at the origin), the densities are close to constants, and the boundaries of domains are close to be flat. Next in §2.3, we prove that $u$ is close to a convex function $w$ solving an optimal transport problem with constant densities. In addition, $w$ is smooth, and thus $u$ is even closer to a parabola (given by the second order Taylor expansion of $w$ at the origin) inside a small sub-level set $S_{h_{0}}[u]$. Last in §2.4, by rescaling $S_{h_{0}}[u]$ at scale $1$ and iterating the above steps, we obtain that $u$ is $C^{1,\beta}$ at the origin for any given $\beta\in(0,1)$, and thus prove Lemma 2.1.

2.1. Comparison with a solution over convex domains

Let $\Omega$ be a $C^{1,1}$ convex domains in ${\mathbb{R}}^{n}$. For any given point $x_{0}=0\in\partial\Omega$, there exists a small ball $B_{r}=B_{r}(x_{0})$, of which the radius $r$ is independent of $x_{0}$, such that after a rotation of the coordinates, locally the boundary can be expressed as

[TABLE]

where $x^{\prime}=(x_{1},\cdots,x_{n-1})$, and $\eta$ is a $C^{1,1}$ convex function satisfying

[TABLE]

Let $\Lambda\subset{\mathbb{R}}^{n}$ be a $C^{1,1}$ domain and is $\delta$-close to $\Omega$ in $C^{1,1}$ norm. From the definition (1.9), locally $B_{r}\cap\partial\Lambda$ can be represented as the graph of a $C^{1,1}$ function $\rho$ such that

[TABLE]

for all $x_{0}\in\partial\Omega$, where $B^{\prime}_{r}$ is a ball in $\mathbb{R}^{n-1}$ with the radius $r>0$ independent of $x_{0}$. In fact, the bijection $\Phi$ in (1.9) can be defined such that for $x\in\Omega$ close to ${\partial\Omega}$, $\Phi(x)=(x^{\prime},x_{n}+(\rho-\eta)(x^{\prime}))$. Then (1.9) is equivalent to (2.1) due to a finite covering and the compactness of ${\partial\Omega}$.

Recall that $f,g$ are the given densities supported on $\Lambda,\Lambda^{*}$, respectively, and $u$ is the potential function of the optimal transport from $\Lambda$ to $\Lambda^{*}.$ Under the hypotheses of Theorem 1.2, let $f_{1},g_{1}\in C^{\alpha}(\mathbb{R}^{n})$ be the extensions of $f,g$ with the same Hölder exponent and satisfy (1.6) for some positive constants $c_{0},c_{1}$ (that may be different to the constants in (1.6)). Let $\tilde{f},\tilde{g}$ be the restriction of $f_{1},g_{1}$ on $\Omega,\Omega^{*}$. Let $\tilde{u}$ be the potential function of the optimal map from $(\Omega,\tilde{f})$ to $(\Omega^{*},\lambda\tilde{g}),$ where the constant $\lambda$ is chosen such that $\int_{\Omega}\tilde{f}=\int_{\Omega^{*}}\lambda\tilde{g}.$ Apparently $\lambda\rightarrow 1$ as $\delta\rightarrow 0$. Without loss of generality we may assume directly that $\lambda=1$. Replacing $\Omega$ by $\hat{x}+(1+C\delta)(\Omega-\hat{x})$ for some interior point $\hat{x}\in\Omega$, (similarly to $\Omega^{*}$), we may also assume that $\Lambda\subset\Omega$ and $\Lambda^{*}\subset\Omega^{*}$.

Therefore, in the following we always have $\Lambda\subset\Omega$, $\Lambda^{*}\subset\Omega^{*}$, and $\Lambda,\Lambda^{*}$ are $\delta$-close to the convex domains $\Omega,\Omega^{*}$ respectively, and that

[TABLE]

Let $\tilde{u}$ be the potential function of the optimal transport from $(\Omega,\tilde{f})$ to $(\Omega^{*},\tilde{g})$. For simplicity, we introduce the notation

[TABLE]

where $x_{0}$ is the mass centre of $\Lambda$. By adding a suitable constant to $u$ such that $u(x_{0})=\tilde{u}(x_{0})$, one can see that $\|u-\tilde{u}\|_{\infty}=\|u-\tilde{u}\|_{L^{\infty}(\Lambda)}$.

Lemma 2.2**.**

There exists a positive function $\omega:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$, depending only on $c_{0},c_{1}$, the inner and outer radii of $\Omega,\Omega^{*}$, with the property $\omega(\delta)\rightarrow 0$ as $\delta\rightarrow 0$, such that

[TABLE]

Proof.

Suppose to the contrary that there exist $\{\Lambda_{k},\Lambda_{k}^{*},f_{k},g_{k}\}$, and the convex approximations $\{\Omega_{k},\Omega_{k}^{*},\tilde{f}_{k},\tilde{g}_{k}\}$ satisfying the above conditions such that the associated potential functions satisfy

[TABLE]

for a fixed small constant $\delta_{0}$ independent of $k$. Passing to a subsequence and taking the limit we have

• •

$\Lambda_{k},\Lambda_{k}^{*}$ (and also $\Omega_{k},\Omega^{*}_{k}$) converge to convex domains $\Omega_{\infty}$, $\Omega^{*}_{\infty}$ in Hausdorff distance respectively, as $k\rightarrow\infty$.

• •

$f_{k}\chi_{\Lambda_{k}},g_{k}\chi_{\Lambda^{*}_{k}}$ (and also $\tilde{f}_{k}\chi_{\Omega_{k}},\tilde{g}_{k}\chi_{\Omega^{*}_{k}}$) converge weakly to $f_{\infty}\chi_{\Omega_{\infty}}$, $g_{\infty}\chi_{\Omega^{*}_{\infty}}$, where $f_{\infty},g_{\infty}$ satisfy (1.6) in $\Omega_{\infty}$, $\Omega^{*}_{\infty}$.

• •

$u_{k}\rightarrow u_{\infty},\tilde{u}_{k}\rightarrow\tilde{u}_{\infty}$ for some convex functions $u_{\infty},\tilde{u}_{\infty}$.

Since (2.4) is independent of $k$, we obtain

[TABLE]

On the other hand, $u_{k}$ are potential functions of the optimal transport from $(\Lambda_{k},f_{k})$ to $(\Lambda_{k}^{*},g_{k})$, $u_{\infty}$ is the potential function of the optimal transport from $(\Omega_{\infty},f_{\infty})$ to $(\Omega^{*}_{\infty},g_{\infty})$. Hence $Du_{\infty}$ and $D\tilde{u}_{\infty}$ are both optimal maps from $(\Omega_{\infty},f_{\infty})$ to $(\Omega^{*}_{\infty},g_{\infty})$. It follows that $Du_{\infty}=D\tilde{u}_{\infty}$ a.e. and thus $u_{\infty}=\tilde{u}_{\infty}+c$ for some constant $c$, which implies that $\|u_{\infty}-\tilde{u}_{\infty}\|_{\infty}=0$ contradicting with (2.5). The lemma is proved. ∎

2.2. Localisation near a boundary point

Let $0\in\partial\Lambda$ be a boundary point, and locally the boundary is given by a $C^{1,1}$ function $\rho$ such that

[TABLE]

Recall that $\tilde{u}$ is the potential function of the optimal transport from $C^{1,1}$ convex domains $\Omega$ to $\Omega^{*}$. From Theorem 1.1 (i), $\tilde{u}\in C^{2,\alpha}(\overline{\Omega})$. Hence by subtracting a linear function and performing an affine transformation, we may assume that

[TABLE]

near the origin. By adding a constant, we also assume that $u(0)=\tilde{u}(0)=0$. Then from Lemma 2.2, near the origin, one has

[TABLE]

which implies that $\partial u(0)$ converges to [math] as $\delta\rightarrow 0$, where $\partial u$ is the sub-differential of $u$. Hence, up to an affine transformation (converging to identity as $\delta\rightarrow 0$) there is a $C^{1,1}$ function $\rho^{*}$ such that locally

[TABLE]

Denote $U=B_{\epsilon_{0}}\cap\Lambda$ and $U^{*}=\partial u(U)\cap\Lambda^{*}$, where $\epsilon_{0}>0$ is a small constant to be determined later. By subtracting a linear function $\ell$ from $u$ with $|D\ell|\rightarrow 0$ as $\delta\rightarrow 0$, we may assume that $u(0)=0$ and $0\in\partial u(0)$. By Lemma 2.2 and (2.6), we have

[TABLE]

and by the convexity of $u$,

[TABLE]

Make a rescaling

[TABLE]

such that $B_{\epsilon_{0}}$ becomes $B_{1}$. Denote

[TABLE]

Lemma 2.3**.**

For any given $\eta_{0}>0$ small, by choosing $\delta,\epsilon_{0}$ sufficiently small, one has

[TABLE]

Moreover, under the rescaling (2.9), the densities $f,g$ tend to be constant and boundaries $\partial\Lambda,\partial\Lambda^{*}$ tend to be flat, as $\epsilon_{0}\to 0$.

Proof.

From (2.7), (2.9) and the fact that $U\subset B_{\epsilon_{0}}$, it is straightforward to see that for any given $\eta_{0}>0$ small,

[TABLE]

provided $\delta,\epsilon_{0}$ are sufficiently small.

By the rescaling (2.9), the domains become $U^{\prime}=\frac{1}{\epsilon_{0}}U$, ${U^{*}}^{\prime}=\frac{1}{\epsilon_{0}}U^{*}$, which are locally given respectively by

[TABLE]

From (2.8) it is easy to see that

[TABLE]

[TABLE]

For any given small $\eta_{0}>0$, by direct computation one can check that

[TABLE]

provided $\epsilon_{0}$ are sufficiently small, where $B^{\prime}_{r}$ indicates a ball in $\mathbb{R}^{n-1}$ with radius $r$. Therefore, under the rescaling (2.9) the boundaries $\partial\Lambda$ and $\partial\Lambda^{*}$ tend to be flat as $\epsilon_{0}\to 0$.

Correspondingly, the density functions become $f_{1}(x)=f(\epsilon_{0}x)$ in $U^{\prime}$, and $g_{1}(x)=g(\epsilon_{0}x)$ in ${U^{*}}^{\prime}$. Similarly as (2.13) one has for any small $\eta_{0}>0$,

[TABLE]

provided $\epsilon_{0}$ are sufficiently small. Note that without loss of generality, we can always assume that $f(0)=g(0)=1.$ ∎

2.3. Approximation by a smooth solution

By the rescaling in Lemma 2.3, having (2.11)–(2.14) we show that $u^{\prime}$ can be approximated by a smooth convex function $w$ solving an optimal transport problem with constant densities. And then we deduce that $u$ is even closer to another parabola (comparing with (2.10)) in a small sub-level set

[TABLE]

Let

[TABLE]

and denote $U^{\prime}_{-}$ (resp. ${U^{*}}^{\prime}_{-}$) the reflection of $U^{\prime}$ (resp. ${U^{*}}^{\prime}$) with respect to the hyperplane $\{x_{n}=-\delta_{1}\}$. Let $z=(0,\cdots,0,-\delta_{1})$,

[TABLE]

where the constant $\lambda$ is chosen such that $|\mathcal{D}_{1}|=|\mathcal{D}_{2}|$. Note that $\delta_{1}\rightarrow 0$ as $\delta\rightarrow 0$, both $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$ are symmetric with respect to $\{x_{n}=-\delta_{1}\}$, and

[TABLE]

Note also that $\lambda\rightarrow 1$ as $\delta\rightarrow 0$.

Let $w$ be the convex function solving $(\partial w)_{\sharp}\chi_{\mathcal{D}_{1}}=\chi_{\mathcal{D}_{2}}$ with $w(0)=u^{\prime}(0)$. Namely, $w$ is a solution to

[TABLE]

By the symmetry of the data and the uniqueness of optimal transport maps, we see that $w^{\prime}$, the restriction of $w$ on to $\mathcal{D}_{1}\cap\{x_{n}>-\delta_{1}\}$, solves $(\partial w^{\prime})_{\sharp}\chi_{\mathcal{D}_{1}\cap\{x_{n}>-\delta_{1}\}}=\chi_{\mathcal{D}_{2}\cap\{x_{n}>-\delta_{1}\}}$. Namely, $\partial w^{\prime}(\mathcal{D}_{1}\cap\{x_{n}>-\delta_{1}\})=\mathcal{D}_{2}\cap\{x_{n}>-\delta_{1}\}$. By a compactness argument similar to that of Lemma 2.2, we have

[TABLE]

By (2.10) we then have

[TABLE]

By the symmetry of $w$, we also have that

[TABLE]

Noting that by (2.20) and symmetry of $w$, we have $\partial w(B_{1/5})\subset B_{1/4}$, provided $\delta,\eta_{0}$ are sufficiently small. Since $w$ is a solution to (2.17), by the interior estimates [20] we obtain

[TABLE]

for a universal constant $C>0$.

Since $\|w\|_{C^{3}(B_{1/5})}\leq C$ and $w(0)=0$, we have the Taylor expansion

[TABLE]

From (2.18) and by a compactness argument, we claim that $|Dw(0)|\rightarrow 0$ as $\delta\rightarrow 0$. Indeed, in the limit profile as $\delta\rightarrow 0$ we have $w=u^{\prime}$ on $B_{\frac{1}{3}}\cap\{x_{n}>{\tilde{\rho}}(x^{\prime})\}$ which implies that $Dw(0)=Du^{\prime}(0)=0$. Therefore, for a given small $h_{0}>0$, up to an affine transformation,

[TABLE]

In the following lemma, we show that the sub-level sets of $u^{\prime}$ and their images are close to ellipsoids with controlled eccentricity, and $u^{\prime}$ is close to a parabola given by the second order Taylor expansion of $w$ in a small sub-level set $S_{h_{0}}[u^{\prime}]$ defined by (2.15).

Lemma 2.4**.**

For any given constant $\tilde{\eta}_{0}>0$ small, there exist small positive constants $h_{0}=h_{0}(\tilde{\eta}_{0})$, $\delta_{0}=\delta_{0}(h_{0},\tilde{\eta}_{0})$, and a symmetric matrix $A$ with $\|A\|$, $\|A^{-1}\|\leq K$ (a universal constant) and $\mbox{det}\ A=1$ such that

[TABLE]

[TABLE]

provided $\delta<\delta_{0}$. Moreover

[TABLE]

and $A^{-1}(\textbf{e}_{n})$ is parallel to $A(\textbf{e}_{n})$.

Proof.

The proof is similar to that in [11], and the main steps are outlined as follows. Denote by $\omega_{0}$ the term $\omega(\delta_{0})$ in (2.18). Then, from (2.18) and (2.22) we have

[TABLE]

where $E_{h_{0}}:=\{x:\frac{1}{2}D^{2}w(0)x\cdot x\leq h_{0}\}.$ Therefore, by taking $h_{0},\delta_{0}$ sufficiently small we can obtain

[TABLE]

Note that $\frac{1}{C}I\leq D^{2}w(0)\leq CI$ for some universal constant $C$. By symmetry, $w_{n}=0$ on $\mathcal{P}:=\{x\in B_{1/5}:x_{n}=-\delta_{1}\}$, and thus $w_{ni}=0$ on $\mathcal{P}$ for all $1\leq i\leq n-1$. Since $w\in C^{3}(B_{1/5})$ and $\delta_{1}\to 0$ as $\delta\to 0$, we obtain at the origin $D^{2}w(0)=D^{2}w(z)+O(\delta)$, where $z=(0,\cdots,0,-\delta_{1})$, and

[TABLE]

Hence, one can find a symmetric matrix $A$ satisfying $\|A-[D^{2}w(0)]^{-1/2}\|=O(\delta)$ such that $A^{-1}(e_{n})$ is parallel to $A(e_{n})$. This implies that $\|A^{-1}\|,\|A\|\leq K$, for a universal constant $K$. And from (2.26) we obtain

[TABLE]

which gives (2.23) and the second inclusion of (2.24). We refer the reader to [11, Lemma 4.3] for more detailed computation on the matrix $A$.

To prove the first inclusion of (2.24), we need to use the Legendre transform of $u^{\prime}$, namely let $u^{*}:B_{2\sqrt{h_{0}}}\cap\{x_{n}>{\tilde{\rho}^{*}}(x^{\prime})\}\rightarrow\mathbb{R}$ be the convex function defined by

[TABLE]

Then one can verify that

[TABLE]

By the standard property of Legendre transform $B\subset\partial u^{\prime}(\partial u^{*}(B))$ for any Borel set $B$, we can easily get the desired inclusion from the previous estimate. ∎

2.4. Iteration argument

Let $u_{1}=u^{\prime}$, $U_{1}=U^{\prime}$ and $U^{*}_{1}={U^{*}}^{\prime}$. Then we have the initial setting in Lemma 2.4 for $u_{1}$, which is the potential function of the optimal transport from $(U_{1},f_{1})$ to $(U^{*}_{1},g_{1})$, where $f_{1},g_{1}$ are the rescaled densities in Lemma 2.3. Let $A_{1}=A$ be the symmetric matrix in Lemma 2.3. Now make the rescaling

[TABLE]

and define

[TABLE]

Moreover, let $U_{2}=S_{1}[u_{2}]$ and $U^{*}_{2}=\partial u_{2}(S_{1}[u_{2}])$. Thanks to Lemma 2.4, we have $u_{2}$ is the potential function of the optimal transport from $(U_{2},f_{2})$ to $(U^{*}_{2},g_{2})$ satisfying all conditions of Lemma 2.3. Therefore, we can apply the argument in §2.3 to $u_{2}$.

Similarly, for $k=1,2,\cdots$, by the rescaling $x\mapsto\frac{1}{\sqrt{h_{0}}}A_{k}^{-1}x$, letting

[TABLE]

and from Lemma 2.4 we can find a symmetric matrix $A_{k+1}$ satisfying

[TABLE]

where $K$ and $h_{0}$ are the same as in Lemma 2.4, while $\tilde{\rho},\tilde{\rho}^{*}$ are the rescaled boundary functions, which tend to be flat as $k\to\infty$.

Let

[TABLE]

We obtain a sequence of symmetric matrices satisfying

[TABLE]

From the above iteration we have

[TABLE]

Hence,

[TABLE]

For any given $\beta\in(0,1)$, by choosing $h_{0}$ and $\delta_{0}$ small enough we can show that $u_{1}$ is $C^{1,\beta}$ at the origin, and thus obtain Lemma 2.1.

Proof of Lemma 2.1.

Fix $\beta\in(0,1)$, and let $r_{0}:={\sqrt{h_{0}}}/({\sqrt{3}K})$. From (2.29) we have

[TABLE]

provided $h_{0}$ (and so $r_{0}$) is sufficiently small. This implies the $C^{1,\beta}$ regularity of $u_{1}$ at the origin. By rescaling back to the original solution and the arbitrariness of the boundary point $x_{0}\in\partial\Lambda$, we obtain $u\in C^{1,\beta}(\overline{\Lambda})$ and finish the proof. ∎

3. Proof of Theorems 1.2 and 1.3

3.1. $C^{2,\alpha}$ estimate

We can adapt a perturbation argument from [13, §5] to prove Theorem 1.2. By changing coordinates and subtracting a linear function, we assume $0\in\partial\Lambda$, $u\geq 0$, $u(0)=0$ and $Du(0)=0$. From Lemma 2.1, we see that for any fixed $\epsilon>0$ small,

[TABLE]

provided $\delta_{0}$ is sufficiently small. Now we construct an approximate solution of $u$ in $S_{h}[u]$ as follows. Denote

[TABLE]

When $h>0$ is sufficiently small, we have $D_{h}^{+}\Subset\Lambda.$ Let $u^{*}$ be the dual potential function, that is the Legendre transform of $u$. The proof of Lemma 2.1 applies also to $u^{*}$, namely $u^{*}\in C^{1,\beta}(\overline{\Lambda^{*}})$ for any given $\beta\in(0,1)$. Hence, for any $x\in D_{h}^{+}$, we have $u_{n}(x)\geq 0$. Otherwise, one has $\mbox{dist}(Du(x),\partial\Lambda^{*})\lesssim h^{1-2\epsilon}$, but from the $C^{1,\beta}$ estimate of $u^{*}$,

[TABLE]

provided $h$ is sufficiently small, which contradicts to the definition of $D_{h}^{+}$.

Let $D_{h}^{-}$ be the reflection of $D_{h}^{+}$ with respect to the hyperplane $\{x_{n}=h^{1-3\epsilon}\}$. Denote $D_{h}=D_{h}^{+}\cup D_{h}^{-}$. By the property that u_{n}\big{|}_{D_{h}^{+}}\geq 0, we see that $D_{h}$ is a convex set. Now, let $w$ be the solution of

[TABLE]

Our proof relies on the following lemma.

Lemma 3.1**.**

Assume that

[TABLE]

Then we have

[TABLE]

Proof.

The proof uses a similar idea as that of Theorem 1.1 (i) in [13, §5]. Divide $\partial D_{h}^{+}=\mathcal{C}_{1}\cup\mathcal{C}_{2}$ into two parts, where $\mathcal{C}_{1}\subset\{x_{n}>h^{1-3\epsilon}\}$ and $\mathcal{C}_{2}\subset\{x_{n}=h^{1-3\epsilon}\}$. On $\mathcal{C}_{1}$ we have $u=w$. On $\mathcal{C}_{2}$, by symmetry we have $D_{n}w=0$. We claim that $0\leq D_{n}u\leq C_{1}h^{1-4\epsilon}$ on $\mathcal{C}_{2}$, for any given small $\epsilon>0$.

To see this, for any $x=(x^{\prime},x_{n})\in\mathcal{C}_{2}$, let $z=(x^{\prime},\rho(x^{\prime}))$ be the point on $\partial\Lambda$. Since $Du(\partial\Lambda)\subset\partial\Lambda^{*}$ and $u\in C^{1,1-\epsilon}(\bar{\Lambda})$, for any $\epsilon\in(0,1)$ (by Lemma 2.1), it is straightforward to compute that $|D_{n}u(z)|\leq Ch^{2(\frac{1}{2}-\epsilon)(1-\epsilon)}$. On the other hand $|D_{n}u(x)-D_{n}u(z)|\leq Ch^{(1-3\epsilon)(1-\epsilon)}$. Hence $0\leq D_{n}u(x)\leq C_{1}h^{1-4\epsilon}$, provided $\epsilon$ is sufficiently small.

Let

[TABLE]

and

[TABLE]

Then

[TABLE]

By comparison principle, we have $\hat{w}\geq u\geq\check{w}$ in $D_{h}^{+}$.

Since $h>0$ is small, $\tau<1/2$, and $\epsilon>0$ is small, we obtain

[TABLE]

Next, we estimate $|u-w|$ in $D_{h}\cap\Lambda$. For $x=(x^{\prime},x_{n})\in D_{h}^{-}\cap\Lambda$, let $z=(x^{\prime},2h^{1-3\epsilon}-x_{n})\in D_{h}^{+}$. Then $|x-z|\leq Ch^{1-3\epsilon}$. From (3.2), $|u(z)-w(z)|\leq Ch^{1+\tau}$. Since $w$ is symmetric with respect to $\{x_{n}=h^{1-3\epsilon}\}$, we have $w(x)=w(z)$. Since $u\in C^{1,1-\epsilon}(\bar{\Lambda})$, we obtain

[TABLE]

Therefore, for the given constant $\tau\in(0,\frac{1}{2})$,

[TABLE]

Combining with (3.2) we thus obtain the desired $L^{\infty}$ estimate

[TABLE]

∎

Having Lemma 3.1 in hand, we can prove Theorem 1.2 by following the proof of Theorem 1.1 (i) as in [13, §5]. Here, we outline the main steps as follows.

Proof of Theorem 1.2.

Let $D_{k}=D_{h_{k}},$ where $h_{k}=4^{-k}$, $k=0,1,2,\cdots.$ To obtain the $C^{1,1}$ estimate of $u$ is equivalent to show that $D_{k}$ has good shape for all $k$, namely the ratio of the largest radius and the smallest radius of its minimal ellipsoid is uniformly bounded. This is done by an induction argument.

Let $u_{k}$, $k=0,1,2,\cdots$, be the convex solution of

[TABLE]

Suppose $D_{k}$ has good shape for all $k\leq N.$ Then by Lemma 3.1 and Schauder estimate (see [13, Lemma 5.4]), we have

[TABLE]

for $x\in D_{i+2}$ and $0\leq i\leq N$, where the constant $\tau=\frac{\alpha}{2}\in(0,\frac{1}{2})$. Therefore,

[TABLE]

for some universal constant $C_{1}.$ Then, by [13, Lemma 5.3] we have that $D_{N+1}$ also has good shape. Hence by induction on $k$ we see that $D_{k}$ has good shape for all $k.$

To obtain the $C^{2,\alpha}$ estimate, for any given point $z\in\overline{\Lambda}$ near the origin such that $4^{-k-4}\leq u(z)\leq 4^{-k-3}$ we only need to estimate

[TABLE]

Since $f\in C^{\alpha}(\overline{\Lambda})$, $g\in C^{\alpha}(\overline{\Lambda^{*}})$, similarly as in [13, §5] we can obtain

[TABLE]

which gives the Hölder continuity of $D^{2}u$ at the boundary. Combining with the interior $C^{2,\alpha}$ estimates in [3, 22], we have the global $C^{2,\alpha}$ regularity in Theorem 1.2. ∎

3.2. $W^{2,p}$ estimate

When the densities $f,g$ satisfy (1.1), (1.6) and are continuous, we can also obtain the global $W^{2,p}$ estimate in Theorem 1.3. As seen in §2, for the Hölder continuous densities, the solution $u$ can be approximated by $\tilde{u}\in C^{2,\alpha}$ that is a potential function over convex domains, and thus one has (2.7) at the initial step. For continuous densities, we do not have the initial estimate (2.7) since such an approximate solution may not even be $C^{1,1}$.

To overcome this difficulty we use a similar technique as in [9] by directly exploiting the sub-level sets. Eventually we can also establish Lemma 2.1 for continuous densities. Once having $u\in C^{1,\beta}(\overline{\Lambda})$ for all $\beta\in(0,1)$, the global $W^{2,p}$ estimate follows from the argument as in [29], (see also the proof of Theorem 1.1 (ii) in [13]).

Proof of Theorem 1.3.

In order to obtain Lemma 2.1 for continuous densities, we follow the four steps as in §2. First, let $\tilde{u}$ be the potential function of optimal transport from $(\Omega,\tilde{f})$ to $(\Omega^{*},\tilde{g})$, where $\Omega,\Omega^{*}$ are convex domains and $\tilde{f},\tilde{g}$ are the extended continuous densities. Let $0\in\partial\Lambda$, without loss of generality we may assume that $0=u(0)=\tilde{u}(0)$ and $D\tilde{u}(0)=0$. By a compactness argument as in Lemma 2.2, we have

[TABLE]

for a nondecreasing function $\omega:{\mathbb{R}}_{+}\to{\mathbb{R}}_{+}$ with $\omega(\delta)\to 0$ as $\delta\to 0$.

Next we localise the problem by normalising a small sub-level set $S_{h}[\tilde{u}]$ of $\tilde{u}$ at the origin, where $h>0$ is a fixed small constant. From Theorem 1.1 (ii), there is a universal constant $C$ and a unimodular matrix $A$ such that

[TABLE]

where $\epsilon>0$ can be as small as we want. Make the rescaling $x\mapsto\frac{1}{\sqrt{h}}A^{-1}x$ and define

[TABLE]

and accordingly

[TABLE]

where $A^{t}$ is the transpose of $A$. Correspondingly, the domains become $\Omega_{1}=\frac{1}{\sqrt{h}}A^{-1}\Omega$ and $\Omega^{*}_{1}=\frac{1}{\sqrt{h}}A^{t}\Omega^{*}$. From the proof of Lemma 2.3, one has the scaled densities $f_{1},g_{1}$ tend to the constant and the domains $\Omega_{1},\Omega^{*}_{1}$ tend to be flat near the origin, as $h\to 0$. Moreover, for any given $\eta_{0}>0$ small, one has

[TABLE]

provided $\delta$ is sufficiently small.

Then we construct an optimal transport problem with constant densities. Similarly as in §2.3, define the domains $\mathcal{D}_{1}$ and $\mathcal{D}_{2}$. Let $w$ be the convex function satisfying $w(0)=u_{1}(0)=0$ and

[TABLE]

Analogous to (2.18) and (2.20), we can then obtain

[TABLE]

and thus

[TABLE]

From [9, Lemma 3.4] and $\tilde{u}_{1}\in C^{1,\beta}$ for all $\beta\in(0,1)$, one has for any $x\in B_{{1}/{5}}\cap\{x_{n}>{\tilde{\rho}}(x^{\prime})\}$ and for any $p\in\partial w(x)$,

[TABLE]

Hence, by the symmetry of $w$, we have $\partial w(B_{1/5})\subset B_{1/4}$, provided $\delta,\eta_{0}$ are sufficiently small. Since $w$ is a solution to (3.7), by the interior estimates [20] we obtain

[TABLE]

for a universal constant $C>0$.

Once having the smooth approximate solution $w$, we can similarly obtain Lemma 2.4, namely $u_{1}$ is close to a parabola given by the second order Taylor expansion of $w$ in a small sub-level set $S_{h_{0}}[u_{1}]$. By the iteration argument, we then have (2.29), which implies Lemma 2.1, namely for any given $\beta\in(0,1)$, there is a small constant $\delta_{0}>0$ such that the original solution $u\in C^{1,\beta}(\overline{\Lambda})$ provided $\delta<\delta_{0}$. Finally, the global $W^{2,p}$ estimate can be obtained by using a covering argument from [29], see [12, 13] for more details. ∎

4. Some applications

In the last section we give some interesting applications of our global regularity of optimal mappings in non-convex domains.

4.1. Free boundary problem

As in [10] we discuss a model of free boundary problem arising in optimal transportation.

Let $\Lambda$ and $\Lambda^{*}$ be two bounded domains in ${\mathbb{R}}^{n}$, associated with densities $f$ and $g$, respectively. Let $m$ be a positive number satisfying

[TABLE]

Let the cost be the quadratic cost. The optimal partial transport problem asks for the optimal mapping that minimising the cost transporting mass $m$ from $\Lambda$ to $\Lambda^{*}$. The portion $U\subset\Lambda$ been transported is called the active region. In [8], Caffarelli and McCann proved that the free boundary $\partial U\cap\Lambda$ is $C^{1,\alpha}$. Assuming $\Lambda,\Lambda^{*}$ are $C^{2}$, uniformly convex, and the distance $\mbox{dist}(\Lambda,\Lambda^{*})$ is sufficiently large, the first author [10] obtained the $C^{2,\alpha}$ regularity of the free boundary $\partial U\cap\Lambda$.

The key observation is that when $\mbox{dist}(\Lambda,\Lambda^{*})$ is sufficiently large, for any $x\in\Lambda,y\in\Lambda^{*}$, $\frac{y-x}{|y-x|}$ is uniformly close to some unit vector $e$. It is known that for $x\in\partial U\cap\Lambda$, the unit inner normal of the free boundary $\partial U$ is given by

[TABLE]

Therefore, when $\mbox{dist}(\Lambda,\Lambda^{*})$ is sufficiently large, the active region $U$ is a small perturbation of a convex domain. By applying our Theorems 1.2 and 1.3, we can obtain the following:

Corollary 4.1**.**

Let $\Lambda$ and $\Lambda^{*}$ be $C^{1,1}$ domains that are $\delta$-close to $\Omega$ and $\Omega^{*}$ in $C^{1,1}$ norm, respectively, where $\Omega$ and $\Omega^{*}$ are bounded convex domains with $C^{1,1}$ boundaries. Let $m$ satisfying (4.1) be the mass to transport, and $U$ be the active region. Then:

• (i)

when $f,g$ are continuous, for any given $\beta\in(0,1)$, there exists a small constant $\delta_{0}>0$ and a large constant $L$ such that $\partial U\cap\Lambda$ is $C^{1,\beta}$, provided $\delta<\delta_{0}$ and $\mbox{dist}(\Lambda,\Lambda^{*})\geq L$;

• (ii)

when $f,g$ are $C^{\alpha}$ for some $\alpha\in(0,1)$, there exists a small constant $\delta_{0}>0$ and a large constant $L$ such that $\partial U\cap\Lambda$ is $C^{2,\alpha}$, provided $\delta<\delta_{0}$ and $\mbox{dist}(\Lambda,\Lambda^{*})\geq L$.

Note that the above regularity is interior regularity, namely for any $\Lambda^{\prime}\Subset\Lambda$, the $C^{1,\beta}$ (or $C^{2,\alpha}$) norm of $\partial U\cap\Lambda^{\prime}$ depends also on the domain $\Lambda^{\prime}$.

4.2. A singularity model

Consider an optimal transport problem from a source domain $\Lambda$ with density $f$ to the target $\Lambda^{*}=\Lambda^{*}_{1}\cup\Lambda^{*}_{2}$ with density $g$, where $\Lambda^{*}_{1}$ and $\Lambda^{*}_{2}$ are two domains separated by a hyperplane $H$, and the densities satisfy $C^{-1}\leq f,g\leq C$ and

[TABLE]

Let the cost be the quadratic cost, $u$ be the convex potential of the optimal transport from $(\Lambda,f)$ to $(\Lambda^{*},g)$. Then its Legendre transform $u^{*}$ is the convex potential of the optimal transport from $(\Lambda^{*},g)$ to $(\Lambda,f)$. In [10], it was proved that the domains $U_{1}:=\partial u^{*}(\Lambda^{*}_{1})$ and $U_{2}:=\partial u^{*}(\Lambda^{*}_{2})$ are separated by the free boundary $\mathcal{F}\subset\Lambda$, and when $\mbox{dist}(\Lambda^{*}_{1},\Lambda^{*}_{2})$ is sufficiently large, the free boundary $\mathcal{F}$ is close to a hyperplane. See also some related discussion in [23]. Hence, by applying our Theorems 1.2 and 1.3, we have:

Corollary 4.2**.**

Let $\Lambda,\Lambda^{*}_{1},\Lambda^{*}_{2}$ be $C^{1,1}$ domains that are $\delta$-close to $\Omega,\Omega^{*}_{1},\Omega^{*}_{2}$ in $C^{1,1}$ norm, respectively, where $\Omega,\Omega^{*}_{1},\Omega^{*}_{2}$ are bounded convex domains with $C^{1,1}$ boundaries. Let $\Lambda^{\prime}\Subset\Lambda$, then:

• (i)

when $f,g$ are continuous, for any given $\beta\in(0,1)$, there exists a small constant $\delta_{0}>0$ and a large constant $L$ such that $\mathcal{F}\cap\Lambda^{\prime}$ is $C^{1,\beta}$, provided $\delta<\delta_{0}$ and $\mbox{dist}(\Lambda,\Lambda^{*})\geq L$;

• (ii)

when $f,g$ are $C^{\alpha}$ for some $\alpha\in(0,1)$, there exists a small constant $\delta_{0}>0$ and a large constant $L$ such that $\mathcal{F}\cap\Lambda^{\prime}$ is $C^{2,\alpha}$, provided $\delta<\delta_{0}$ and $\mbox{dist}(\Lambda,\Lambda^{*})\geq L$.

4.3. Minimal Lagrangian diffeomorphisms

In [36], Wolfson studied minimal Lagrangian diffeomorphisms between simply connected domains in ${\mathbb{R}}^{2}$. The problem is as follows: given $D_{1}$ and $D_{2}$ two simply connected domains in ${\mathbb{R}}^{2}$ with smooth boundaries and with equal areas, find a diffeomorphism $\psi:D_{1}\to D_{2}$ smooth up to the boundaries such that the graph of $\psi$ is a minimal Lagrangian surface in ${\mathbb{R}}^{4}$. Such a mapping $\psi$ is called a minimal Lagrangian diffeomorphism from $D_{1}$ to $D_{2}$.

An equivalent statement is that there is a solution $u\in C^{\infty}(\overline{D}_{1})$ of the second boundary problem for the Monge-Ampère equation

[TABLE]

The equivalency can be seen that by choosing a suitable Lagrangian angle, the diffeomorphism $\psi=Du$ mapping from $D_{1}$ to $D_{2}$.

Under the assumption that both $\partial D_{1}$ and $\partial D_{2}$ have positive curvatures, the existence of global smooth solutions was proved by Delanoë [17]. The higher dimensional analogue of Delanoë’s result was proved by Caffarelli [6] and Urbas [32]. By our recent result Theorem 1.1 in [13], we show that both $\partial D_{1}$ and $\partial D_{2}$ have non-negative curvatures guarantees the existence of minimal Lagrangian diffeomorphisms $\psi$. By applying Theorem 1.2 we are able to further relax the assumption to the following:

Corollary 4.3**.**

Assume that $D_{1},D_{2}$ are $\varepsilon$-close to smooth convex domains $\Omega_{1},\Omega_{2}$, respectively. Then there exists a small constant $\varepsilon_{0}>0$ such that there exists a minimal Lagrangian diffeomorphim $\psi:D_{1}\to D_{2}$, provided $\varepsilon<\varepsilon_{0}$.

4.4. Optimal transportation with general costs

The regularity of an optimal transport map with general costs has been studied by many researchers. In [25], Ma, Trudinger, and Wang found a fourth order condition, the so-called MTW condition, of the cost function, which ensures the smoothness of the map. When the cost does not satisfy the MTW condition, but is a small perturbation of the quadratic cost, various regularity results have been obtained in [9, 12, 11], see also [7, 15].

We remark that the proofs of our Theorems 1.2 and 1.3 also allow a small perturbation of the cost function. To be specific, assume that the cost function $c=c(x,y)$ satisfies

• (C0)

The cost function is of class $C^{3}$ with $\|c\|_{C^{3}(\Lambda\times\Lambda^{*})}<\infty$.

• (C1)

$\forall x\in\Lambda$, the map $\Lambda^{*}\ni y\mapsto D_{x}c(x,y)\in{\mathbb{R}}^{n}$ is injective.

• (C2)

$\forall y\in\Lambda^{*}$, the map $\Lambda\ni x\mapsto D_{y}c(x,y)\in{\mathbb{R}}^{n}$ is injective.

• (C3)

$\mbox{det}(D_{xy}c)(x,y)\neq 0$ for all $(x,y)\in\Lambda\times\Lambda^{*}$,

and

[TABLE]

The conclusion of Theorems 1.2 and 1.3 remains true provided $\delta+\delta_{1}<\delta_{0}$ is sufficiently small, where $\delta$ is the perturbation of domains $\Lambda,\Lambda^{*}$ from convex domains, and $\delta_{1}$ is the perturbation of cost from the quadratic cost.