Obstructions to regularity in the classical Monge problem
OOBSTRUCTIONS TO REGULARITY IN THE CLASSICAL MONGEPROBLEM
MARIA COLOMBO AND EMANUEL INDREI
Abstract.
We provide counterexamples to regularity of optimal maps in the classicalMonge problem under various assumptions on the initial data. Our construction is basedon a variant of the counterexample in [10] to Lipschitz regularity of the monotone optimalmap between smooth densities supported on convex domains. Introduction
The classical optimal transportation problem appears in a 1781 article of Gaspard Monge[13]. In the modern formulation, one is given a cost function c : R n × R n → [0 , ∞ ] and apair of probability densities f and g defined on two domains Ω ⊂ R n , Ω ∗ ⊂ R n , respectively.The objective is to minimize the functional(1.1) T → ˆ Ω c ( x, T ( x )) f ( x ) dx among maps T : Ω → Ω ∗ which satisfy the “push-forward” condition T f = g , i.e. ˆ A g ( y ) dy = ˆ T − ( A ) f ( x ) dx for all Borel sets A ⊂ Ω ∗ .A minimizer of (1.1) is commonly referred to as an “optimal map” or “optimal transport”for the corresponding cost function. In the classical problem, one selects the Monge cost,i.e. c ( x, y ) = | x − y | . The existence of optimal maps was addressed by many authors [15,5, 6, 14, 1]. In general, minimizers are not unique; nevertheless, there is a unique optimaltransport which is monotone on transfer rays [7]. The regularity of optimal maps is stillwidely open; the only known result is in R : if the given densities are positive, continuous,and have compact, disjoint, convex support, then the monotone optimal map is continuousin the interior of the transfer rays [9]. Recently, it was shown in [10] that the monotoneoptimal transport between positive C ∞ densities supported on the same bounded, convexdomain fails to be Lipschitz continuous at interior points. In fact, the authors prove This work was supported by NSF grant DMS-0932078, administered by the Mathematical SciencesResearch Institute in Berkeley, California, while the authors were in residence at MSRI during the 2013program “Optimal Transport: Geometry and Dynamics.” The first author was also supported by the
Gruppo Nazionale per l’Analisi Matematica, la Probabilit`a e le loro Applicazioni (GNAMPA) of the Italian
Istituto Nazionale di Alta Matematica (INdAM) and by the
PRIN 2011 Calcolo delle variazioni of theitalian
Ministero dell’istruzione dell’Universit`a e della Ricerca . a r X i v : . [ m a t h . A P ] N ov MARIA COLOMBO AND EMANUEL INDREI something stronger: namely, they construct an example in which the monotone optimalmap does not belong to C + ε for any ε > ∗ = (0 , × (0 ,
1) and suppose that f and g are bounded away from zero and infinity.Let f x ( · ) := f ( x , · ) and g x ( · ) := g ( x , · ), and write F x and G x for their primitives,respectively. If F x (0) = G x (0) = 0, F x (1) = G x (1), and F x ( x ) ≥ G x ( x ) for all x , x , then the optimal potential is u ( x , x ) = − x and the monotone optimal transportis explicitly given by(1.2) T ( x , x ) = G − x ( F x ( x )) . We note that the primitives endow T with more regularity than the densities in the x variable; nevertheless, the map has the same regularity as f and g in the x variable. Forexample, continuous densities may produce a monotone optimal map which is not H¨oldercontinuous. Moreover, the same example shows that a discontinuous monotone optimalmap may arise from discontinuous, bounded initial data (see also [9, Example 4.5]).Our main result constructs a family of examples which shed further light on the nature ofthe monotone optimal transport map, see Theorem 2.1. More precisely, we build exampleswhere the modulus of continuity of the monotone optimal transport is worse than themodulus of continuity of the densities. Indeed, some corollaries of our result include thefollowing statements (Ω is the interior of a triangle, and the densities are positive on Ω): f, g ∈ C ∞ (Ω) (cid:54)⇒ T ∈ C , + εloc (Ω) , ∀ ε > f, g ∈ C ∞ (Ω) (cid:54)⇒ T ∈ C , + ε (Ω) , ∀ ε > f, g ∈ C , (Ω) (cid:54)⇒ T ∈ C , + εloc (Ω) , ∀ ε > f, g ∈ C ,α (Ω) (cid:54)⇒ T ∈ C , αα +2 + εloc (Ω) , ∀ ε > ∀ α > f, g ∈ W ,p (Ω) (cid:54)⇒ T ∈ C , p − p − + εloc (Ω) , ∀ ε > . (1.7)Note that (1.3) recovers the aforementioned example appearing in [10, § T by reducing α and improves the bound derived from (1.2) (i.e.that C α data implies a transport that is not more regular than C α ). This improvement isgenerated by choosing nonparallel transport rays, see § c ( x, y ) = | x − y | /
2, is of a completely differentnature. In this setting, if the densities are bounded away from zero and infinity on convexdomains, then Caffarelli [2, 3, 4] asserts that T is H¨older continuous, cf. [9, Example 4.5]; This example was communicated to us by Filippo Santambrogio.
BSTRUCTIONS TO REGULARITY IN THE CLASSICAL MONGE PROBLEM 3 moreover, if f, g ∈ C k,β , then T ∈ C k +1 ,ε for k = 0 , , . . . , and ε ∈ (0 , β ), cf. (1.3), (1.5),and (1.6) (see also Remark 2.6). For general cost functions satisfying a strong form of theso-called Ma-Trudinger-Wang condition [12], it was shown in [11] under mild conditionson the densities, that the transport map is H¨older continuous up to the boundary if thedomains satisfy a certain type of convexity condition called c -convexity (see also the interiorregularity result in [8] for cost functions satisfying a weak form of the Ma-Trudinger-Wangcondition). Indeed, the domain Ω above satisfies this convexity condition for a large classof cost functions. Thus, the Monge cost generates a significant loss of regularity comparedto other cost functions for which a regularity theory is known. Nevertheless, it remainsan open problem to determine sufficient conditions on the initial data that ensure thecontinuity of optimal maps in the classical Monge problem.2. Main result
For the reader’s convenience, we keep our notation as close as possible to [10, § ω : [0 , → [0 , /
2] be a function such that(2.1) ω ∈ C ([0 , ∩ C ∞ ((0 , , ω (cid:48) ( t ) > ∀ t ∈ (0 , , ω (0) = 0 , ω (1) = 12 . In our situation, we consider the following transport rays:(2.2) l a = { ( x , x ) ∈ R : x = ω ( a )( x + a ) , x ∈ ( − a, } a ∈ (0 , . It is clear that the segments l a do not mutually intersect. The domain representing bothsource and target will be ∆ ⊆ R , where(2.3) ∆ = interior of the triangle with vertices ( − , , , f ( x ) = 1 ∆ ( x ) , g ( x ) = 1 ∆ ( x ) (cid:0) c ( ζ ( x ) + η ( x )) (cid:1) , x = ( x , x ) ∈ R , where(2.5) ζ ( x ) = − x + 12 x − , x ∈ [ − , , η ∈ L ∞ (0 , ∩ C ∞ (0 , , and c > g is bounded away from zero:(2.6) c < (cid:0) (cid:107) ζ (cid:107) L ∞ ( − , + (cid:107) η (cid:107) L ∞ (0 , (cid:1) . Since we want the segments l a to be transport rays for the optimal map, the followingmass balance condition for the region in the domain below each l a must be satisfied:(2.7) ˆ Γ a f = ˆ Γ a g ∀ a ∈ [0 , , where Γ a is the subgraph of l a in ∆, namely the triangle formed by ( − a, , ω ( a )(1 + a )),and (1 , η . MARIA COLOMBO AND EMANUEL INDREI
Figure 1.
Figure (a) depicts the nature of a boundary singularity when ω ( a ) = a s : points on the vertical axis are mapped to points of the samecolor along the corresponding transport rays, and the horizontal componentof the images approach the origin in a H¨older way, see Corollary 2.5. Figure(b) illustrates how an interior singular point arises after reflecting the data.It is well known that the optimal map for the Monge cost is not unique. On the otherhand, there exists a unique monotone optimal map [7] which in our setting can be shownto satisfy T ( x ) ∈ l a ∀ x ∈ l a , a ∈ (0 , , and ( T ( x ) − T ( y )) · ( x − y ) ≥ ∀ x, y ∈ l a , a ∈ (0 , . From the construction, it follows that this map is pointwise defined in ∆, and we investigateits regularity. In particular, it is not difficult to see that the origin is a fixed point for asuitable continuous extension and that the regularity of this map at the origin stronglydepends on the behavior of ω (see Figure 1); indeed, in the next section we show howsuitable choices of ω will provide a collection of counterexamples to regularity. All of theseresults will be consequences of the following theorem. Theorem 2.1.
Let ω be as in (2.1) , ∆ ⊆ R as in (2.3) , and { l a } a ∈ (0 , the family of linesdefined in (2.2) . Let f, g, c, η, ζ be as in (2.4) , (2.5) , (2.6) and assume that (2.7) holds and lim t → η ( t ) = 0 . Then the monotone optimal transport T = ( T , T ) between f and g withrespect to the Monge cost satisfies (2.8) 0 < lim inf a → T ( ω ( a ) a e ) a ≤ lim sup a → T ( ω ( a ) a e ) a < ∞ . BSTRUCTIONS TO REGULARITY IN THE CLASSICAL MONGE PROBLEM 5
To obtain counterexamples to interior regularity, it suffices to reflect the picture acrossthe x -axis (see Figure 1); in this way we construct initial data for which the optimal mapfails to be regular at an interior fixed point. Let ∆ (cid:48) ⊆ R be defined as(2.9) ∆ (cid:48) := interior of the triangle with vertices ( − , , , − f ( x ) = 1 ∆ (cid:48) ( x )2 , ˜ g ( x ) = 1 ∆ (cid:48) ( x )2 (cid:0) c ( ζ ( x ) + η ( | x | )) (cid:1) , x = ( x , x ) ∈ R , where c, η, ζ are as in (2.5) and (2.6). By the uniqueness of monotone optimal mappings[7] it follows that the symmetric extension of T across the x -axis is the correspondingoptimal map between the densities ˜ f and ˜ g . Note that the regularity of ˜ g on the x -axis isaffected by the choice of ω (via η ).2.1. Obstruction to regularity with smooth data.
The first corollary appears in workof Li, Santambrogio, and Wang [10, § C function in the interior. The proofs of the corollaries will employ Theorem 2.1to describe the regularity of the transport map as well as an analysis lemma to guaranteethe regularity of the target density (see Lemma 3.1). Corollary 2.2.
Let ∆ (cid:48) be as in (2.9) . There exists g ∈ C ∞ (∆ (cid:48) ) bounded away from zerosuch that if T is the monotone optimal transport map between ∆ (cid:48) and g , then (2.11) T / ∈ C + ε ( B / (0)) for any ε > . Proof.
By selecting ω ( a ) = √ a/
2, we choose η ∈ C ∞ [ − ,
1] as in Lemma 3.1 and g as in(2.10). An application of Theorem 2.8 yields(2.12) 0 < lim inf t → T ( t e ) t ≤ lim sup t → T ( t e ) t < ∞ . Thus, T ( t e ) can be extended continuously to t = 0 by the second inequality in (2.12)with the value 0. Finally, (2.11) follows from the first inequality in (2.12). (cid:3) Next, we show that the monotone map between smooth data and convex domains is ingeneral not better than C up to the boundary. Corollary 2.3.
Let ∆ be as in (2.4) . There exists g ∈ C ∞ (∆) bounded away from zerosuch that if T is the monotone transport map between ∆ and g , then (2.13) T / ∈ C + ε ( B / (0) ∩ ∆) for any ε > . Proof.
Let ω ( a ) = a/ η ∈ C ∞ [0 ,
1] as in Lemma 3.1. We select the smoothtarget density g as in (2.4), and apply Theorem 2.8 to obtain(2.14) 0 < lim inf t → T ( t e ) t / ≤ lim sup t → T ( t e ) t / < ∞ . Thus, T ( t e ) can be extended continuously to t = 0 by the second inequality in (2.14)with the value 0. Finally, (2.13) follows from the first inequality in (2.14). (cid:3) MARIA COLOMBO AND EMANUEL INDREI
When ω ( a ) = a/
2, it can be seen from Lemma 3.1 that η ∈ C ∞ [0 , η is quadratic at the origin with a bounded left andright third derivative; thus, its even reflection is C , , and this implies the following interiorresult: Corollary 2.4.
Let ∆ (cid:48) be as in (2.9) . There exists g ∈ C , (∆ (cid:48) ) bounded away from zerosuch that if T is the monotone transport map between ∆ (cid:48) and g , then T / ∈ C + ε ( B / (0)) for any ε > . Obstruction to regularity with continuous data.
In the next corollary, we showthat one may degenerate the possible H¨older continuity of the monotone optimal transportwith (degenerate) H¨older data.
Corollary 2.5.
Let ∆ (cid:48) be as in (2.9) . For every < α ≤ , there exists g ∈ C α (∆ (cid:48) ) bounded away from zero such that if T is the monotone optimal transport map between ∆ (cid:48) and g , then T / ∈ C α α + ε ( B / (0)) for any ε > . Proof.
Let ω ( a ) = a s / s ≥ η as in Lemma 3.1, so that η ∈ C , /s [0 , η (0) = 0. Since the even reflection is Lipschitz, it follows that η ( | x | ) ∈ C , /s [ − , α := 2 /s and select the C ,α target density g as in (2.10). An application of Theorem2.8 yields 0 < lim inf t → T ( t e ) t α α ≤ lim sup t → T ( t e ) t α α < ∞ . Thus, T ( t e ) can be extended continuously to t = 0 with the value 0, and this finishes theproof. (cid:3) Remark 2.6.
In Corollary 2.5 one could also take α ∈ (0 , ∞ ) . In fact, it follows fromRemark 3.2 and Theorem 2.1 that for every k ∈ N , β ∈ (0 , there exists g ∈ C k,β (∆ (cid:48) ) bounded away from zero such that the monotone optimal transport map T between ∆ (cid:48) and g is not C k + β k + β + ε for any ε > . Remark 2.7.
One may not hope for a H¨older continuous map with only C data. Indeed,this fact can readily be inferred from the example giving rise to (1.2) , and it can also bededuced from Theorem 2.1 and Lemma 3.1 by selecting the function ω ( a ) = e − /a / . Remark 2.8.
It is not difficult to see that the function g constructed in the proof ofCorollary 2.5 is W ,p (∆ (cid:48) ) for every p > (1 − α ) − . In particular, this shows that for every p > , f, g ∈ W ,p (∆ (cid:48) ) (cid:54)⇒ T ∈ C , p − p − + εloc (∆ (cid:48) ) , ∀ ε > . If p = ∞ , it follows that the monotone optimal map between Lipschitz densities is not moreregular than C (this can also be deduced from Corollary 2.5 by choosing α = 1 ). BSTRUCTIONS TO REGULARITY IN THE CLASSICAL MONGE PROBLEM 7 Proof
In the following lemmas, we determine the target density by imposing the mass balancecondition (2.7) and build a Kantorovich potential. This construction yields the existenceof an optimal transport map for the Monge cost between f and g whose transport rays aregiven by (2.2). Lemma 3.1.
Let ∆ be as in (2.4) , ω as in (2.1) , { l a } a ∈ (0 , the family of lines defined in (2.2) . Moreover, let f, g, c, η, ζ be as in (2.4) , (2.5) , (2.6) . Then the following statementshold:(i) (2.7) holds if and only if (3.1) η ( t ) = d dt (cid:0) t a ( t ) (cid:1) ∀ t ∈ (0 , , where a ( t ) ∈ C [0 , ∩ C ∞ (0 , is the unique solution of (3.2) t = ω ( a )(1 + a ) . (ii) If ω ( a ) = a s / for a ∈ [0 , and some s ∈ [2 , ∞ ) , then η ∈ C , s [0 , with η (0) = 0 .Moreover, if s = 1 /n for n ∈ N then η ∈ C ∞ [0 , and if s = 1 / (2 n ) for n ∈ N ,then η has a C ∞ even extension to [ − , .(iii) If ω ( a ) = e − /a / for every a ∈ (0 , , ω (0) = 0 , then η ∈ C [0 , with η (0) = 0 .Proof.Proof of statement (i) . Note that (2.7) can be rewritten as(3.3) − ˆ Γ a ζ ( x ) dx dx = ˆ Γ a η ( x ) dx dx a ∈ (0 , . Let N ∈ C ∞ (0 ,
1) be such that N (cid:48)(cid:48) ( t ) = η ( t ), N (0) = N (cid:48) (0) = 0. To compute the secondintegral, we integrate first with respect to x ˆ Γ a η ( x ) = ˆ − a ˆ ω ( a )( x + a )0 η ( x ) dx dx = ˆ − a N (cid:48) ( ω ( a )( x + a )) dx = 1 ω ( a ) ˆ ω ( a )(1+ a )0 N (cid:48) ( x ) dx = N ( ω ( a )(1 + a )) ω ( a ) . (3.4)Similarly, we compute the left integral in (3.3) by first integrating with respect to x andthen with respect to x . By noting that the function Z ( t ) = − t ( t − satisfies Z (cid:48)(cid:48) ( t ) = ζ ( t )and Z (1) = Z (cid:48) (1) = 0, it follows that(3.5) − ˆ Γ a ζ ( x ) dx dx = − ˆ ω ( a )( a +1)0 ˆ x /ω ( a ) − a ζ ( x ) dx dx = ω ( a ) a ( a + 1) . Thanks to (3.4) and (3.5), (3.3) can be rewritten as N ( ω ( a )(1 + a )) = ω ( a ) a ( a + 1) , a ∈ (0 , MARIA COLOMBO AND EMANUEL INDREI i.e.,(3.6) N ( t ) = t a ( t ) , t ∈ (0 , , which is equivalent to (3.1). Moreover, by (2.1) and the implicit function theorem, thefunction a : [0 , → [0 ,
1] which solves (3.2) is monotone, continuous, and belongs to C ∞ (0 , Proof of statement (ii) . Let ω ( a ) = a s / s ∈ (2 , ∞ ). We are left to prove η (0) = 0and that η is H¨older continuous at 0. By the implicit function theorem, there exists a C ∞ function b defined in a neighborhood of 0 so that r = b ( r )2 /s (1 + b ( r )) /s for every r ≥ b ( t /s ) is a solution to t = b s ( t /s )2 (1 + b ( t /s )) , and hence a ( t ) = b ( t /s ) for every t sufficiently small. Therefore the function t a ( t ) = t b ( t /s ) belongs to C , /s [0 , t /s through the formula:(3.7) d dt [ t b ( t /s )] = 2 b ( t /s ) + (cid:16) s + 2 s (cid:17) t /s b ( t /s ) b (cid:48) ( t /s ) + 2 s t /s (cid:0) [ b (cid:48) ( t /s )] + b ( t /s ) b (cid:48)(cid:48) ( t /s ) (cid:1) . By computing the third derivative and integrating it between any two points, it followsthat each term in this expression is C /s . We show this regularity for the first term in (3.7)(the argument is similar for the other terms). Since b is Lipschitz and b (0) = 0, we havethat b ( r ) ≤ r (cid:107) b (cid:48) (cid:107) L ∞ ((0 , for every r ∈ [0 , t , t ∈ [0 ,
1] with t < t b ( t /s ) − b ( t /s ) = ˆ t t ddt [ b ( t /s )] dt ≤ s ˆ t t b ( t /s ) b (cid:48) ( t /s ) t /s − dt ≤ | b (cid:48) (cid:107) ∞ s ˆ t t t /s − dt = (cid:107) b (cid:48) (cid:107) ∞ (cid:0) t /s − t /s (cid:1) ≤ (cid:107) b (cid:48) (cid:107) ∞ (cid:0) t − t (cid:1) /s . Since b (0) = 0, (3.7) implies η (0) = 0. Moreover, if s = 1 /n for some n ∈ N , then a ( t ) = b ( t n ) is C ∞ up to t = 0. Similarly, if s = 1 / (2 n ), then t a ( t ) = t b ( t n ) ∈ C ∞ [0 , t ∈ [ − ,
1] (since the function t a ( t ) depends smoothly on t ). Proof of statement (iii) . We apply the same line of reasoning as before by letting ω ( a ) = e − /a /
2: consider the equation t = e − /a a ) , which is equivalent to r = b ( r )1 − b ( r ) (log(1 + b ( r )) + 1 − log 2) , BSTRUCTIONS TO REGULARITY IN THE CLASSICAL MONGE PROBLEM 9 where r = − t ) . By the implicit function theorem, there exists a smooth solution b = b ( r )in a neighborhood of the origin. Thus, a ( t ) = b ( r ( t )) = b ( − / log( t )), and this impliesthat the function t a ( t ) = t b ( − / log( t )) belongs to C [0 , b (0) = 0, we deduce η (0) = 0. (cid:3) Remark 3.2.
In Lemma 3.1 (ii), one could also take s ∈ (0 , in order to obtain a moreregular η . More precisely, if ω ( a ) = a s / with s ∈ (0 , ∞ ) , then η ( | · | ) ∈ C k,β ([ − , , where k is the integer part of s and β = s − k . Indeed, the regularity of η is determined by theregularity of N as in (3.6) ; thanks to Lemma 3.1 (i), it is enough to check the regularity of η at the origin. This can be made explicit by noting that a ( t ) = O ( t /s ) close to t = 0 , see (3.2) . It follows that N ( t ) = O ( t /s ) . To make this argument rigorous, one can rewrite N ( t ) = t b ( t /s ) with b smooth up to the origin and explicitly compute the derivatives ofthis expression noting that many vanish at the origin. We also remark that in Lemma 3.1(iii), a modulus of continuity of η is given by the function − / log( · ) (up to a constant). Remark 3.3.
Slight generalizations of the construction in this paper do not lead to bettercounterexamples. For instance, the particular choice of ζ is made in such a way that ζ (0) (cid:54) = 0 , a fact which is needed in the proof of Theorem 2.1. This implies that the function Z ∈ C ∞ (0 , satisfying Z (cid:48)(cid:48) ( t ) = ζ ( t ) and Z (1) = Z (cid:48) (1) = 0 , is at most quadratic at theorigin. On the other hand, this limits the regularity of η in Lemma 3.1. Similarly, altering f to have the same structure as g , namely f ( x ) = 1 ∆ ( x ) (cid:0) c ( ˜ ζ ( x ) + ˜ η ( x )) (cid:1) , x = ( x , x ) ∈ R , does not lead to better counterexamples, as can be seen from the proof of Lemma 3.1. Lemma 3.4.
Let ∆ be as in (2.4) , ω as in (2.1) , and { l a } a ∈ (0 , the family of lines definedin (2.2) . Then there exists u ∈ C , (∆) ∩ C (∆) such that | u ( x ) − u ( y ) | ≤ | x − y | , x, y ∈ ∆ , with equality if and only if x, y ∈ l a for some a ∈ (0 , .Proof. For every x ∈ ∆, let a ( x ) be such that the unique line passing through x is l a ( x ) .Note that a ( x ) is obtained by solving(3.8) x = ω ( a )( x + a );the assumptions on ω imply a ∈ C (∆) (via the implicit function theorem). Next, considerthe unit vector field b ∈ C (∆; R ) which points in the direction of l a ( x ) for each x ∈ ∆, i.e. b ( x ) = 1(1 + ω ( a ( x ))) / ( e + ω ( a ( x )) e ) ∀ x ∈ ∆ . The statement of the lemma is equivalent to the existence of a function u ∈ C (∆) suchthat(3.9) Du ( x ) = b ( x ) ∀ x ∈ ∆ . Indeed, if (3.9) holds, then || Du || L ∞ ((0 , ≤ ∂ t [ u ( − a e + tb ( x ))] = | b ( x ) | = 1 forevery x ∈ ∆ and t >
0; since − a e + tb ( x ) is a parametrization of the line l a ( x ) with < t < ( a + 1)(1 + ω ( a )) / , this shows that u is linear with slope 1 when restricted to l a ( x ) . On the other hand if the statement of the lemma holds, then | Du | ≤ u must point in the direction of l a ( x ) for every x ∈ ∆.As ∆ is simply connected, (3.9) is equivalent to showing that b is an irrotational vectorfield, i.e. 0 = ∂ b ( x ) − ∂ b ( x ) ∀ x ∈ ∆ , or equivalently(3.10) 0 = ω ( a ( x )) ω (cid:48) ( a ( x ))(1 + ω ( a ( x ))) / (cid:16) ∂ a ( x ) + ∂ a ( x ) ω ( a ( x )) (cid:17) ∀ x ∈ ∆ . By applying ∂ and ∂ to (3.8), it follows that(3.11) 0 = ω (cid:48) ( a ( x )) x ω ( a ( x )) ∂ a ( x ) + ω ( a ( x ))( ∂ a ( x ) + 1) ∀ x ∈ ∆ , (3.12) 1 = ω (cid:48) ( a ( x )) x ω ( a ( x )) ∂ a ( x ) + ω ( a ( x )) ∂ a ( x ) ∀ x ∈ ∆ . Multiplying (3.12) by ω ( a ( x )) and adding (3.11) yields0 = ∂ a ( x ) + ∂ a ( x ) ω ( a ( x )) ∀ x ∈ ∆ , which proves (3.10). To conclude, we note that u admits a Lipschitz extension to ∆. (cid:3) Proof of Theorem 2.1.
By the construction in [5, 15] and [7], there exists a unique measurepreserving map T : ∆ → ∆ between f and g such that x and T ( x ) are contained on acommon line l a for all x ∈ ∆. Moreover, by Lemma 3.4 (which exploits the constructionof the Kantorovich potential), one readily obtains that T is an optimal mapping in theMonge problem, see e.g. [10, Lemma 7]. Since T is measure preserving, the constructionin [5, 15] implies for every a ∈ (0 , δ → δ ˆ (Γ a + δ \ Γ a ) ∩{ x < } f ( x ) dx = lim δ → δ ˆ (Γ a + δ \ Γ a ) ∩{ x To prove the second inequality in (2.8), note that by (2.6), the second term of the integrandon the right-hand side of (3.14) is greater than 1 / x ∈ ( − , ˆ z ( a ) − a (cid:16) x + a + ω ( a ) ω (cid:48) ( a ) (cid:17) dx ≤ ˆ − a (cid:16) x + a + ω ( a ) ω (cid:48) ( a ) (cid:17) dx . By evaluating the two integrals we obtain12 (cid:16) ( z ( a ) + a ) ω ( a )( z ( a ) + a ) ω (cid:48) ( a ) (cid:17) ≤ a ω ( a ) aω (cid:48) ( a ) ≤ (cid:16) (2 a ) ω ( a )2 aω (cid:48) ( a ) (cid:17) . Since the function t → t / tω ( a ) /ω (cid:48) ( a ) is increasing on [0 , ∞ ), it follows that z ( a ) + a ≤ a, which proves that the lim sup in (2.8) is bounded by 1. Next, we prove the first inequalityin (2.8). By the properties of ζ and η it follows that there exists a constant 0 < c < ζ ≤ − c and η ≤ c in a neighborhood of the origin. Hence, since z ( a ) → a → 0, for a sufficiently small, ˆ − a (cid:16) x + a + ω ( a ) ω (cid:48) ( a ) (cid:17) dx ≤ (1 − c ) ˆ z ( a ) − a (cid:16) x + a + ω ( a ) ω (cid:48) ( a ) (cid:17) dx . Evaluating the two integrals yields a ω ( a ) aω (cid:48) ( a ) ≤ (1 − c ) (cid:16) ( z + a ) ω ( a )( z ( a ) + a ) ω (cid:48) ( a ) (cid:17) . On the other hand, note that(1 − c ) (cid:16) a − c ) + ω ( a ) aω (cid:48) ( a )(1 − c ) / (cid:17) ≤ a ω ( a ) aω (cid:48) ( a ) . Since the function t → t / tω ( a ) /ω (cid:48) ( a ) is increasing on [0 , ∞ ), we deduce a (1 − c ) / ≤ z ( a ) + a. Subtracting a from both sides yields that the lim inf in (2.8) is bounded below by1(1 − c ) / − > . (cid:3) Acknowledgments. We thank Filippo Santambrogio for his remarks on a preliminaryversion of the paper and for pointing out the example leading to (1.2). This work wascompleted at the Mathematical Sciences Research Institute in Berkeley, California, whilethe first author was a Program Associate and the second author a Huneke PostdoctoralScholar. The warm hospitality of the institute is kindly acknowledged. References [1] Ambrosio, L.: Lecture notes on optimal transport problems, Mathematical Aspects of Evolv-ing Interfaces, Lecture Notes in Math., 1812 (2003), 1-52.[2] Caffarelli, L.: Some regularity properties of solutions of Monge Amp`ere equation, Comm.Pure Appl. Math. 44 (1991), 965-969.[3] Caffarelli, L.: The regularity of mappings with a convex potential, J. Amer. Math. Soc., 5(1992), 99-104.[4] Caffarelli, L.: Boundary regularity of maps with convex potentials II, Ann. of Math. 144(1996), 453-496.[5] Caffarelli, L., Feldman, M., McCann, R.J.: Constructing optimal maps for Monge’s transportproblem as a limit of strictly convex costs, J. Amer. Math. Soc., 15 (2002), 1-26.[6] Evans, L.C., Gangbo, W.: Differential equations methods for the Monge-Kantorovich masstransfer problem, Mem. Amer. Math. Soc., 653 (1999).[7] Feldman, M., McCann, R.J.: Uniqueness and transport density in Monge’s mass transporta-tion problem, Cal. Var. PDE, 15 (2002), 81-113.[8] Figalli, A., Kim, Y.-H., McCann, R.J.: H¨older continuity and injectivity of optimal maps,Arch. Ration. Mech. Anal.,209 (2013), no.3, 1812-1824.[9] Fragal`a, I., Gelli, M.S., Pratelli, A.: Continuity of an optimal transport in Monge problem,J. Math. Pures Appl., 84 (2005), 1261-1294.[10] Li, Q.-R., Santambrogio, F., Wang, X.-J.: Regularity in Monge’s mass transfer problem,Preprint 2013.[11] Liu, J.: H¨older regularity of optimal mappings in optimal transportation, Calc. Var. PDE,34 (2009), 435-451.[12] Ma, X.-N., Trudinger, N. S., and Wang, X.-J.: Regularity of potential functions of theoptimal transportation problem, Arch. Ration. Mech. Anal. 177 (2005), 151-183.[13] Monge, G.: M´emoire sur la th´eorie des d´eblais et des remblais, Histoire de l’Acad´emie Royaledes Sciences de Paris (1781), 666-704.[14] Sudakov, V. N.: Geometric problems in the theory of infinite dimensional probability distri-butions, Proc. Steklov Inst., 141 (1979), 1-178.[15] Trudinger, N.S., Wang, X.J.: On the Monge mass transfer problem, Calc. Var. PDE, 13(2001), 19-31. Maria ColomboScuola Normale Superiorep.za dei Cavalieri 7, I-56126Pisa, Italyemail: [email protected]