Regularity of optimal mapping between hypercubes

1 Introduction

Let Ω , Ω * be two bounded domains in R n . Assume that ρ , ρ * are two positive density functions supported on Ω , Ω * , respectively, and satisfy the balance condition ‖ ρ ‖ L 1 ( Ω ) = ‖ ρ * ‖ L 1 ( Ω * ) . The optimal mapping T : Ω → Ω * is the minimiser of the functional

among all measure-preserving maps s : Ω → Ω * such that s # ρ = ρ * , [15,20,21].

In [2], Brenier obtained the existence and uniqueness of the optimal mapping T that is the gradient of a convex function u , which is called potential function and satisfies a natural boundary condition of the Monge-Ampère equation:

Regularity of the optimal mapping T (equivalently, of the potential function u ) is a fundamental problem in the theory of optimal transportation. For interior regularities, C 1 , α , W 2 , p , and C 2 , α estimates for u have been obtained by Caffarelli [3,4] under appropriate assumptions. For regularity near the boundary, if Ω , Ω * are smooth, uniformly convex, and ρ , ρ * > 0 are smooth, Delanoë [10] and Urbas [19] proved that u ∈ C ∞ ( Ω ¯ ) . If Ω , Ω * are C 2 smooth and uniformly convex, and ρ , ρ * are Hölder continuous, Caffarelli [6] proved that D 2 u are Hölder continuous up to the boundary.

In applications such as in machine learning [9], computer vision [1] and computer graphics [16,18], the domains may fail to be uniformly convex or smooth. When the domains Ω , Ω * are convex and the densities ρ , ρ * are bounded from zero and infinity, Caffarelli [5] proved that u ∈ C 1 , α ( Ω ¯ ) for some α ∈ ( 0 , 1 ) . When Ω , Ω * are C 1 , 1 and convex, recently in [8], we obtained the regularity u ∈ C 2 , α ( Ω ¯ ) if ρ , ρ * ∈ C α ; and u ∈ W 2 , p ( Ω ¯ ) for all p > 1 if ρ , ρ * ∈ C 0 . In dimension two, for constant densities, Savin and Yu [17] obtained the global W 2 , p estimate for arbitrary bounded convex domains Ω , Ω * ⊂ R 2 .

In practice, a typical domain in medical image processing like the magnetic resonance imaging and computed tomography images is the hypercube Q ≔ ( 0 , 1 ) n . Optimal transport between the hypercubes was also used by Caffarelli [7] in proving the FKG type inequalities. An interesting question is whether one can obtain higher regularity for this special case. The techniques used in previous works [6,8,10, 19] do not apply to the case when the domain Ω = Q , due to the loss of regularity of ∂ Q at the corners. Very recently, Jhaveri constructed a counterexample showing that there exist smooth densities such that T ∈ C 2 , α ( Q ¯ ) for every α < 1 but not C 3 ( Q ¯ ) , see [12, Theorem 3.7]. Hence, the best possible regularity one can expect is T ∈ C 2 , α ( Q ¯ ) .

By the symmetry of Q , we can make even extensions for the densities. Hence, by Caffarelli’s interior regularity [3,5], we see that for any positive ρ , ρ * ∈ C α ( Q ¯ ) satisfying ‖ ρ ‖ L 1 ( Q ) = ‖ ρ * ‖ L 1 ( Q ) , the optimal mapping T ∈ C 1 , α ( Q ¯ ) [7, Corollary 3]. Moreover, T maps each face of Q to itself correspondingly. In dimension two, by using the partial Legendre transform, Jhaveri [12, Theorem 3.3] proved that if further ρ , ρ * ∈ C 1 , α ( Q ¯ ) , then T ∈ C 2 , α ( Q ¯ ) .

In this article, we establish the optimal global C 2 , α regularity of T in higher dimensions.

We remark that in dimension two, Jhaveri [12] used the partial Legendre transform to change the Monge-Ampère equation (1.1) to a quasi-linear, uniformly elliptic equation due to the fact u ∈ C 2 , α ( Q ¯ ) , then he further obtained u ∈ C 3 , α ( Q ¯ ) by an estimate for uniformly elliptic equations. However, this method no longer works when the dimension n > 2 . In higher dimensions, to obtain u ∈ C 3 , α ( Q ¯ ) , we shall adopt the result of [13] for the regularity of solutions along a given direction. The proof of Theorem 1.1 is contained in §2. For the Dirichlet problem of the Monge-Ampère equation in convex polygonal domains in R 2 , Le and Savin [14] recently obtained global C 2 , α estimates of solution u by assuming there exists a globally C 2 , convex, strict subsolution.

2 Proof of Theorem

First, we do the following even reflections around the origin. Let Q ˜ ≔ ( − 1 , 1 ) n ,

If ρ , ρ * ∈ C α ( Q ¯ ) for some α ∈ ( 0 , 1 ) , then ρ ˜ , ρ * ˜ ∈ C α ( Q ˜ ¯ ) . By the assumption of Theorem 1.1, we further have ρ ˜ , ρ * ˜ ∈ C 0 , 1 ( Q ˜ ¯ ) .

Let u ˜ be the potential function of optimal transportation from ( Q ˜ , ρ ˜ ) to ( Q ˜ , ρ * ˜ ) . By symmetry and the uniqueness of optimal mapping, we see that D u ˜ = D u in Q ¯ . We recall some known regularities as follows:

1. By Caffarelli’s C 1 , σ regularity [5], we have u ˜ ∈ C 1 , σ ( Q ˜ ¯ ) for some σ ∈ ( 0 , 1 ) , provided ρ ˜ , ρ * ˜ are positive and bounded.

2. Furthermore, since ρ ˜ , ρ * ˜ ∈ C β ( Q ˜ ¯ ) for all β ∈ ( 0 , 1 ) , by the interior regularity [3,4], we have u ˜ ∈ C 2 , β ( Q ˜ ) for all β ∈ ( 0 , 1 ) , and thus, u ∈ C 2 , β ( B 3 ∕ 4 ( 0 ) ∩ Q ¯ ) for all β ∈ ( 0 , 1 ) .

3. By doing the same argument for each corner of Q and using a covering argument, we can obtain u ∈ C 2 , β ( Q ¯ ) for all β ∈ ( 0 , 1 ) .

Hence, under the assumption of Theorem 1.1 that ρ , ρ * ∈ C 1 , α ( Q ¯ ) , for simplicity, we may write (1.1) as follows:

where f = ρ ρ * ∘ D u ∈ C 1 , α ( Q ¯ ) . To prove Theorem 1.1, it suffices to prove u ∈ C 3 , α ( Q ¯ ) .

By the even reflections (2.1), we have

Similarly, u ˜ satisfies

As mentioned in ( i i ) , by [3–5], we have

Note that f ˜ ∈ C 0 , 1 ( Q ˜ ) but is not C 1 ( Q ˜ ) in general. Denote x = ( x 1 , x ′ ) , where x ′ = ( x 2 , … , x n ) . From the definition and symmetry of f ˜ , the partial derivative ∂ 1 f ˜ = ∂ f ˜ ∂ x 1 is well-defined in { x ∈ Q ˜ : x 1 ≠ 0 } . Let’s assign ∂ 1 f ˜ = 0 on the interface { x ∈ Q ˜ : x 1 = 0 } so that ∂ 1 f ˜ is defined in Q ˜ . Let v ≔ ∂ 1 u ˜ . By (2.5) and approximation, it is easy to see that v ∈ W 2 , p ( Q ˜ ) for any p > 1 , and v is a strong solution of

where { a i j } is the cofactor matrix of D 2 u ˜ . Let B r = B r ( 0 ) be the ball with radius r and centre at the origin. In B 9 ∕ 10 ⋐ Q ˜ , { a i j } is Hölder continuous and uniformly positive definite. In fact, from (2.5) and equation (2.4), there is a positive constant λ > 0 depending on n , ‖ u ˜ ‖ C 2 ( B 9 ∕ 10 ) , inf Q ˜ f ˜ such that in B 9 ∕ 10 ,

in the sense of matrix, where I is the n × n identity matrix. Equation (2.6) is satisfied almost everywhere in Q ˜ , [11].

Now we recall a useful partial directional regularity result from [13]. We say a function h ∈ L ∞ ( Ω ) is C α in x ′ = ( x 2 , … , x n ) for some α ∈ ( 0 , 1 ) , if

where C > 0 is a constant, and denote