The Monge problem on non-compact manifolds

The Monge Problem on Non-Compact Manifolds.

ALESSIOFIGALLI(*)

ABSTRACT- In this paper we prove the existence of an optimal transport map on non- compact manifolds for a large class of cost functions that includes the case c(x;y)ˆd(x;y), under the only hypothesis that the source measure is absolutely continuous with respect to the volume measure. In particular, we assume com- pactness neither of the support of the source measure nor of that of the target measure.

1. Introduction.

Monge transportation problem consists in moving a distribution of mass into another one in an optimal way, that is minimizing a given cost of transport. In a mathematical language, the problem can be stated as fol- lows: given two probability distributionsmandn, with respective support in measurable spacesXandY, find a measurable mapT:X!Ysuch that

T]mˆn;

…1†

i.e.

n(A)ˆm T ¹(A)

8AY measurable,

and in such a way thatTminimizes the transportation cost, that is Z

X

c(x;T(x))dm(x)ˆmin

S]mˆn

Z

X

c(x;S(x))dm(x) 8<

:

9=

;;

where c:XY!R is a given cost function. When condition (1) is sa- tisfied, we say thatTis atransport map, and ifTminimizes also the cost we call itoptimal transport map. The difficulties in solving such problem (*) Indirizzo dell'A.: Scuola Normale Superiore, Piazza dei Cavalieri, 7, 56126 Pisa - Italy

E-mail: a.figalli@sns.it

even in an Euclidean setting motived Kantorovich to find a relaxed for- mulation (see [11], [12]). He suggested to look forplansinstead of trans- port maps, that is probability measuresginXYwhose marginals arem andn, i.e.

(p_X)_]gˆm and (p_Y)_]gˆn;

wherepX :XY!XandpY :XY!Yare the canonical projections.

Denoting byP(m;n) the class of plans, the new minimization problem be- comes then the following:

g2P(m;n)min Z

X

c(x;y)dg(x;y) 8<

:

9=

;: …2†

Ifg is a minimizer for the Kantorovich formulation, we say that it is an optimal plan. Using weak topologies, the existence of solutions to (2) be- comes simple under the assumption thatXandYare Polish spaces andcis lower semicontinuous (see [15, Chapter 1]). The connection between the formulation of Kantorovich and that of Monge can be seen by noticing that any transport mapTinduces the plan defined by (IdT)_]mwhich is con- centrated on the graph ofT.

In a forthcoming paper with Fathi [8], we prove that, in the case XˆY ˆM withMa smooth manifold, if mgives zero mass to sets with s-finite (n 1)-dimensional Hausdorff measure and the cost function is induced by a C² Lagrangian that verifies some reasonable assumptions, then the Monge problem has a unique solution and this coincides with the solution of the Kantorovich problem, which turns out to be unique. In particular, this result covers all the casesc(x;y)ˆd(x;y)^pforp>1, butnot the limit case pˆ1. So this paper arises as an extension to non-compact manifolds of the results of Bernard and Buffoni proved in [5], where the authors showed the existence of optimal transport maps for a large class of costs (that includes in particular the case c(x;y)ˆd(x;y)) on compact manifolds without boundary. The existence of an optimal transport map in the casec(x;y)ˆd(x;y) on non-compactmanifolds has also been proved in [10] under the assumption of compactness of the supports of the two mea- sures (see the references in [10] for earlier works in the same spirit). In the present paper we extend this result at the case of non-compactly supported measures and, more generally, we prove the existence of an optimal transport for a larger class of cost functions, which is the class of ManÄe potentials associated to a supercritical Lagrangian, using in particular re- sults on weak KAM theory on non-compact manifolds (see [9]).

We remark that we do not assume, as usual in the standard theory of optimal transportation, that the cost function is bounded from below. In fact such assumption would be quite non-natural for a ManÄe potential and, also in particular cases, it would not be simple to check its validity. So, in order to apply the standard duality result that gives us an optimal pair for the dual problem, the idea will be to add to our cost a null-Lagrangian, so that the cost becomes non-negative and still satisfies the triangle in- equality, and the minimization problem does not change (see Section 2.2).

1.1 ±The main result

LetMbe a smoothn dimensional manifold,ga complete Riemannian metric onM. We fixL:TM!RaC²Lagrangian onM, that satisfies the following hypotheses:

(L1) C²-strict convexity: 8(x;v)2TM, the second derivative along the fibersr²_vL(x;v) is positive strictly definite;

(L2) uniform superlinearity: for every K0 there exists a finite constantC(K) such that

8(x;v)2TM; L(x;v)Kkvk_x‡C(K);

wherek k_xis the norm onTxMinduced byg;

(L3) uniform boundedness in the fibers:for everyR0, we have A(R):ˆsup

x2MfL(x;v)j kvk_xRg<‡ 1:

We define the cost function c_T(x;y):ˆ inf

g(0)ˆx;g(T)ˆy

Z^T

0

L(g(t);g(t))_ dt:

The assumptions on the Lagrangian ensure that the inf in the definition of cT(x;y) is attained by a curve of classC². We now define the cost

c(x;y):ˆinf

T cT(x;y):

In the theory of Lagrangian Dynamics, this function is usually called ManÄe potential. We now make the last assumption onL:

(L4) supercriticality: for each x6ˆy2M, we have c(x;y)‡

‡c(y;x)>0.

This assumption ensures that also the inf in the definition ofc(x;y) is attained by a curve of classC². We will consider the Monge transportation problem for the costc. Our main resultis the following:

THEOREM1.1. Assume that c is the cost function associated to a su- percritical Lagrangian that satisfies all the assumption above. Suppose that

Z

MM

d(x;y)dm(x)dn(y)<‡ 1;

where d is the distance associated to the Riemannian metric. Ifmis ab- solutely continuous with respect to the volume measure, then there exists an optimal transport map T:M!M for the Monge transportation pro- blem betweenmandn. This map turns out to be optimal for the Kantorovich problem. More precisely, the plan associated to this map is the unique minimizer of the secondary variational problem

min Z

MM



1‡(c(x;y) U(y)‡U(x))² q

dg(x;y)

among all optimal plans for (2),where U is a strict subsolution of the Hamilton Jacobi equation(see Proposition2.2).

We recall that the idea of using a secondary variational problem in order to select a ``good'' optimal plan was first used in [2] and refined in [3].

REMARK1.2. We observe that, by the triangle inequality for the dis- tance, the condition Z

MM

d(x;y)dm(x)dn(y)<‡ 1 is equivalent to the existence of a point x₀2M such that

Z

M

d(x;x0)dm(x)<‡ 1;

Z

M

d(y;x₀)dn(y)<‡ 1:

In fact, fixed x₀;x₁2M, since d(x;x₀) d(x₀;x₁)d(x;x₁)d(x;x₀)‡

‡d(x0;x1),

x7!d(x;x0) is integrable if and only if x7!d(x;x1) is integrable:

In particular all Lipschitz functions on M are integrable with respect to bothmandn.

We remark that the Lagrangian

L(x;v)ˆ1‡ kvk²_x 2

satisfies all the hypotheses of the above theorem and, in this case, we obtain

c(x;y)ˆd(x;y):

2. Definitions and preliminary results 2.1 ±Preliminaries in Lagrangian Dynamics

We recall some results of Lagrangian Dynamics that will be useful in the sequel (see [6], [7], [13]) and that shows the naturality of the super- criticality assumption.

PROPOSITION2.1. Let L be a Lagrangian that satisfies assumption (L1), (L2) and (L3). For k2R, let us define ck the ManÄe potential associated to the LagrangianL‡k. Then there exists a constantk₀such that

(i) for k<k₀, c_k 1and the Lagrangian is called subcritical;

(ii) for kk₀, c_k is locally Lipschitz on MM and satisfies the triangle inequality

c_k(x;z)c_k(x;y)‡c_k(y;z) 8x;y;z2M;

in addition c_k(x;x)ˆ08x2M;

(iii) for k>k₀, the Lagrangian L is supercritical, that is c(x;y)‡

‡c(y;x)>0for each x6ˆy2M.

The following proposition is a simple corollary of the results proved in [9]:

PROPOSITION 2.2. The Lagrangian L is supercritical if and only if there existd>0and aC¹ functionU such that

H(x; dxU) d; 8x2M;

or equivalently

L(x;v) d_xU(v)d; 8(x;v)2TM;

where H is the Hamiltonian associated to the Lagrangian L, that is 8(x;p)2TM; H(x;p):ˆ sup

v2TxMfhp;vi L(x;v)g:

PROOF. The value k₀ is the so called critical value of L, and is the smallest value for which there exists a globalC¹subsolution of

H(x;dxu)ˆk

(under the assumptions made on the Lagrangian, this value exists and is unique). Then it suffices to apply the following approximation result, also proven in [9]:

THEOREM 2.3. If u:M!Ris locally Lipschitz, with its derivative dxu satisfying H(x;dxu)k almost everywhere, then for eache>0there exists a C¹ function u_e:M!R such that H(x;d_xu_e)k‡e and ju(x) u_e(x)j efor each x2M.

In fact, ifLis supercritical, thenk0is strictly negative, and it suffices to apply the theorem above with eˆjk0j

2 . On the other hand, the inverse implication follows by the characterization ofck0given above. p

We observe that, in the caseL(x;v)ˆ1

2 1‡ kvk²_x

, it suffices to take U0,dˆ1

2.

2.2 ±Duality and Kantorovich potential

It is well-known that a linear minimization problem with convex con- straints, like (2), admits a dual formulation. Before stating the duality for- mula, we recall the definition ofc-transform (see [2], [4], [5], [8], [15], [16]).

DEFINITION 2.4 (c-transform). Given a pair of Borel functions W:X!R[ f 1g,c:Y!R[ f‡ 1g, with

Z

X

jWjdm<‡ 1 and Z

Y

jcjdn<‡ 1;

we say thatcis thec-transformofWif 8(x;y)2XY; c(y)ˆsup

x2MW(x) c(x;y);

and we writecˆW^c.

Observe that, ifcˆW^c, then one obviously hasW(x) c(y)c(x;y) for all (x;y)2XY. Integrating this inequality on the productXY, one obtains

8g2P(m;n);

Z

X

Wdm Z

Y

cdnˆ Z

XY

(W(x) c(y))dg(x;y)

Z

XY

c(x;y)dg(x;y):

THEOREM 2.5 [Duality formula]. Let X and Y be Polish spaces equipped with probability measuresmandnrespectively, c:XY!Ra lower semicontinuous cost function bounded from below such that

Z

XY

c(x;y)dm(x)dn(y)<‡ 1:

Then

g2P(m;n)min Z

XY

c(x;y)dg(x;y) 8<

:

9=

;ˆ max

(W;c)2L¹(m)L¹(n)

Z

X

Wdm Z

Y

cdnjcˆW^c 8<

:

9=

;: For a proof of this theorem see [1, Theorem 6.1.5], [2, Theorems 3.1 and 3.2], [16, Theorem 5.9].

We now consider a costfunctionc(x;y) as in Theorem 1.1. By hypoth- esis (L3), c is Lipschitz. In fact, given x;y2M, we consider a geodesic g_x;y:[0;d(x;y)]!Mfromxtoywithkg__x;yk ˆ1. Then

c(x;y) Z

d(x;y)

0

L(g_x;y(t);g__x;y(t))dtA(1)d(x;y);

and so we have

jc(x;y) c(z;w)j jc(x;y) c(z;y)j ‡ jc(z;y) c(z;w)j

maxfjc(x;z)j;jc(z;x)jg ‡maxfjc(y;w)j;jc(w;y)jg A(1)[d(x;z)‡d(y;w)]:

Moreovercsatisfiesc(x;x)0 and the triangle inequality c(x;z)c(x;y)‡c(y;z)

(see Proposition 2.1). Fix nowz2Mand consider the auxiliar cost c(x;y):ˆc(x;y)‡a(y) a(x);

with a(x):ˆc(x;z). Obviously c still satisfies the triangle inequality.

Moreover, sincecis Lipschitz and satisfies the triangle inequality, we have 0c(x;y)c(x;y)‡c(y;x)2A(1)d(x;y):

…3†

Thusc(x;y) is integrable with respect tomnif so itisd(x;y), and in this case we can apply Theorem 2.5 to prove the following:

THEOREM2.6. Given two probability measuresmandnon M such that Z

MM

d(x;y)dm(x)dn(y)<‡ 1;

let c be a cost function as above. Then there exists a Lipschitz function u:M!Rthat satisfies

u(y) u(x)c(x;y) 8x;y2M

and Z

M

u d(n m)ˆ Z

MM

c dg

for each g optimal transport plan between m and n. In particular, this implies

u(y) u(x)ˆc(x;y) for g-a:e:(x;y)2MM;

that is

(u a)(y) (u a)(x)ˆc(x;y) for g-a:e:(x;y)2MM;

The Lipschitz function u:ˆu a is called a Kantorovich potential.

PROOF. Let(W;c)2L¹(m)L¹(n) be a pair that realizes the maximum in the dual problem, withcˆW^c. We want to prove that it suffices to take uˆ c.

Fixx;y2M. SincecˆW^candcsatisfies the triangle inequality, we have c(x)ˆsup

z2MW(z) c(x;z)sup

z2Mc(y)‡c(z;y) c(x;z)c(y)‡c(x;y):

Letnowx0be a pointofMsuch thatc(x0)2R(such a pointexists, being c2L¹). Choosing in the inequality above firstxˆx0and afteryˆx0, we obtain thatcis finite everywhere. So we can subtractc(y) to the two sides, obtaining

( c)(y) ( c)(x)c(x;y):

Thus, if we defineu:ˆ c, by (3) we have

u(y) u(x)c(x;y)2A(1)d(x;y) 8x;y2M:

This inequality tells us that uis 2A(1)-Lipschitz, and so, by Remark 1.2, u2L¹(m)\L¹(n).

Observe now that

0ˆc(x;x)W(x) c(x) ) u(x)W(x):

Thus, ifgis an optimal transport plan betweenmandn, we have Z

MM

c…x;y†dg…x;y†

Z

MM

…u…y† u…x††dg…x;y†

ˆ Z

M

u d…n m†

Z

M

Wdm Z

M

cdnˆ Z

MM

c…x;y†dg…x;y†;

as wanted. p

2.3 ±Calibrated curves

Fix aC¹ functionUand ad>0 given by Proposition 2.2, and a Kan- torovich potential ugiven by Theorem 2.6. Following [5], we recall some useful definitions.

DEFINITION2.7 [u-calibrated curve]. We say that a continuous piece- wise differentiable curveg:I!Misu-calibratedif

u(g(t)) u(g(s))ˆ Z^t

s

L…g(t);g(t)_ †dtˆc(g(s);g(t)) 8st2I;

whereIRis a nonempty interval ofR(possibly a point). Au-calibrated curve is callednon-trivialifIhas non-empty interior.

Obviously a non-trivial u-calibrated curve is a minimizing extremal ofL, and hence is of classC². In addition, we observe that each u-ca- librated curvegcan be ext ended t o a maximal one, t hat is a curve~gthat can't be extended on an interval that strictly containsIwithout losing the calibration property (this follows by the fact that, fixed the initials position and the velocity, the minimizer is unique; thus, if two u-cali- brated curves locally coincide, they must coincide in the intersection of their domains of definition, and so one can use this fact to find an un- ique maximal extension ofg). We observe that, if gis maximal, then I mustbe closed. In the sequel, also in the case IˆR, Iˆ[a;‡ 1) or ( 1;b], for simplicity of notation we will always write the interval on which a maximal curve is defined as [a;b].

DEFINITION2.8 [transport ray]. Atransport rayis the image of a non- trivialu-calibrated curve.

In [7], it is proved that Kantorovich potentials are viscosity subsolutions of the Hamilton-Jacobi equation, that is equivalent to say thatuis locally Lipschitz and satisfies

H(x;dxu)0; for a.e.x2M;

or equivalently

L(x;v) d_xu(v)0; for a.e.x2M; 8v2T_xM:

We recall that, ifg:[a;b]!Ris au-calibrated curve, then for allt2(a;b) the functionuis differentable atg(t) (see [7]). Then we have the following:

LEMMA 2.9. Let g:[a;b]!R be a u-calibrated curve. Then for all t2(a;b)the function u is differentable atg(t)and we have

dg(t)(u U)(_g(t))d;

where U anddare given byProposition 2.2. This implies thatgis an em- bedding and transport rays are non-trivial embedded arcs.

PROOF. Asg(t) isu-calibrated, we have

u(g(t)) u(g(s))ˆ Z^t

s

L…g(t);g(t)_ †dt 8st; s;t2[a;b];

that implies, recalling Proposition 2.2, u…g…t†† u…g…s††

t s ˆ 1

t s Z^t

s

L…g…t†;g…t†_ †dt

)d_g…t†u…_g…t†† ˆL…g…t†;g…t†_ † d_g…t†U…_g…t†† ‡d: p

We now define the functionsa:M!Randb:M!Ras follows:

± a(x) is the supremum of all timesT0 such that there exists a u-calibrated curveg:[ T;0]!Msuch thatg(0)ˆx;

± b(x) is the supremum of all timesT0 such that there exists a u-calibrated curveg:[0;T]!Msuch thatg(0)ˆx.

LEMMA2.10. aandbare Borel functions.

PROOF. LetKⁱMbe a countable increasing sequence of compact set such thatS

i KⁱˆM. Then we can define the auxiliary functionsa_i(x) as t he supremum of all times T0 such that there exists au-calibrated curve g:[ T;0]!Msuch thatg(0)ˆxandg( T)2Kⁱ. We will prove thata_i is upper semicontinuous for eachi, and this will implies the measurability of aasa(x)ˆsup

i a_i(x) for allx2M(the case ofbis analogous).

Fixi2Nand let(x_j)M be a sequence converging to a limitx2M such thatai(xj)Tfor allj. Then we know that there exists a sequence g_j :[ T;0]!M of u-calibrated curves such that g_j(0)ˆxj and g_j( T)2Kⁱ. Asg_j( T)2Kⁱ, we know that there exists a constantAsuch that kg__j(0)k_gj(0)A for allj (see for example the Appendix in [8]). Then, taking a subsequence if necessary, we can assume thatg_j converges uni- formly on [ T;0] to a curveg:[ T;0]!Mwhich is stillu-calibrated, as it is easy to see, and satisfiesg(0)ˆx,g( T)2Kⁱ. Thenai(x)T. p

We now can define the following Borel sets:

DEFINITION 2.11. We define the set T given by the union of all the transport rays as

T :ˆ fx2Mja(x)‡b(x)>0g:

Fore0, we define the sets

Te:ˆ fx2Mja(x)>e; b(x)>eg:

ClearlyT_e T for alle0 and the setE:ˆ T T₀is the set of ray ends.

We now recall the following:

THEOREM2.12. The function u is differentiable at each point ofT₀. For each point x2 T₀, there exists a unique maximal u-calibrated curve

g_x:[ a(x);b(x)]!M such thatg(0)ˆx:This curve satisfies the relations

dxuˆ rvL(x;g__x(0)) or equivalently

g__x(0)ˆ rpH(x;dxu):

For each e>0, the differential x7!d_xu is locally Lipschitz on T_e; or equivalently the map x7!g__x(0)is locally Lipschitz onTe.

For a proof see [7].

DEFINITION 2.13. For x2M, we will denote by R_x the union of the transport rays containingx. We also denote

R^‡_x :ˆ fy2Mju(y) u(x)ˆc(x;y)g:

We observe thatRxˆg_x([ a(x);b(x)]) for allx2 T0.

In order to conclude this section, we recall two results of [5].

LEMMA2.14. We have

R^‡_x ˆ g_x([0;b(x)]) if x2 T₀; fxg if x2Mn T; (

whereg_xis given by Theorem2.12.

PROOF. Letxbe a pointofT₀. By the calibration property ofg_x, we have

u(g_x(t)) u(g_x(0))ˆc(g_x(0);g_x(t)) 8t2[0;b(x)];

that is

g_x(t)2R^‡_x 8t2[0;b(x)];

and so we haveg_x([0;b(x)])R^‡_x. Conversely, letus fix x2M, y2R^‡_x. Then we know that there exists a u-calibrated curve g:[0;T]!M

such that Z^T

0

L…g(t);g(t)_ †dtˆc(x;y)ˆu(y) u(x); g(0)ˆx; g(T)ˆy:

So, if x2 T0, by Theorem 2.12 gˆg_xj_[0;T] and hence yˆg(T)ˆ

ˆg_x(T)2g_x([0;b(x)]), while, ifx62 T, there is no non-trivial u-calibrated curve and then we must haveyˆxin the above discussion, that implies

R^‡_x ˆ fxg. p

PROPOSITION2.15. A transport plangis optimal for the costcif and only if it is supported on the closed set

[

x2M

fxg R^‡_x ˆ f(x; y)2MMjc(x; y)ˆu(y) u(x)g:

PROOF. By Theorem 2.6,gis optimal if and only if Z

MM

c(x;y)dg(x;y)ˆ Z

M

u(x)d(n m)(x)ˆ Z

MM

(u(y) u(x))dg(x;y):

The conclusion follows observing that c(x;y)u(y) u(x) for all

x;y2M. p

3. Proof of the main theorem.

The line of the proof is essentially the same as in [5], where the authors, using ideas of Lagrangian Dynamics, extend to a Riemannian setting the results obtained in the Euclidean case in [2].

As before, we fix aC¹functionUand ad>0 given by Proposition 2.2, and then fix a Kantorovich potentialugiven by Theorem 2.6 (see Sections 2.2, 2.3). We now define the second cost function

~c(x;y):ˆf(c(x;y) U(y)‡U(x));

withf(t):ˆ 

1‡t²

p . Consider the secondary variational problem ming2O

Z

MM

~c(x;y)dg(x;y);

…4†

whereOis the set of optimal transport plan, and select a minimizerg₀of this secondary variational problem. We now want to prove that it is supported on a graph.

The idea is the following: first one sees that the measure g₀ is con- centrated on a s-compactsetG S

x2Mfxg R^‡_x which is ~c-cyclically monotone in a weak sense that we will define later in the proof. Then one considers the set on which the transport plan is not a graph, that is

L:ˆ fx2Mj#(Gx)2g;

whereGx:ˆ fy2Mj(x;y)2Ggand#denotes the cardinality of the set.

In this way, intersecting Lwith a transport ray R, thanks to the mono- tonicity ofL\Rit is simple to see thatL\Ris atmostcountable. Finally Theorem 2.12 allows us to parameterize the transport rays in a locally Lipschitz way. This and the fact thatL\Rhas zeroH¹-measure for each transport rayR(whereH^kdenote thek-dimensional Hausdorff measure) imply thatLhas null volume measure, and som(L)ˆ0 as wanted.

We divide the proof in many steps, in order to make the overall strategy more clear.

Step 1: the construction ofG.

Firstwe observe that~cis integrable with respect tomn. Indeed, since fhas linear growth, it suffices to prove that

c(x;y):ˆc(x;y) U(y)‡U(x)

ismn-integrable. The uniform boundedness in the fiber ofL(x;v) implies that the Hamiltonian H(x;p) is uniformly superlinear. By this and the inequality H(x;d_xU)0, we get that the gradient of U is uniformly bounded, which implies thatUis Lipschitz, that is

jU(y) U(x)j Cd(x;y) 8x;y2M:

So we have

0c(x;y)(C‡A(1))d(x;y);

and thenc(x;y) is integrable with respect to mn, since so is d(x;y) by assumption.

Let us consider the lower semicontinuous function z:MM!

![0;‡ 1] given by

z(x;y)ˆ ~c(x;y) ifu(y) u(x)ˆc(x;y);

‡ 1 otherwise:

(

Note that, asg₀(f(x;y)2MMju(y) u(x)ˆc(x;y)g)ˆ1, Z

MM

z(x;y)dg₀(x;y)ˆ Z

MM

~c(x;y)dg₀(x;y)<‡ 1

and, thanks to Proposition 2.15, we have that g₀ is a minimizer for the Kantorovich problem

g2P(m;n)min Z

X

z(x;y)dg(x;y) 8<

:

9=

;:

It is then a standard result thatg₀is concentrated on a setG~ that isz-cy- clically monotone, that is if (x_i;y_i)

1ilis a finite family of points ofG~and s(i) is a permutation, we have

X^l

iˆ1

z(x_i;y_s(i))X^l

iˆ1

z(x_i;y_i)

(for a proof see, for example, [2, Theorem 3.2]). By the definition ofzthis implies the following monotonicity property ofG:~

if (xi;yi)

1ilis a finite family of points ofG~ ands(i) is a permutation such that (x_i;y_s(i))

1il is still contained inG, then~ X^l

iˆ1

~c(x_i;y_s(i))X^l

iˆ1

~c(x_i;y_i):

By inner regularity of the Borel measure g₀, there exists a s-compact subsetG G~ on whichg₀is concentrated. ObviouslyGis still monotone in the sense defined above.

Step 2:Lis a Borel set andL T.

Now that we have constructedG, we define L:ˆ fx2Mj#Gx2g;

where G_x:ˆ fy2Mj(x;y)2Gg. LetKⁱ be an countable increasing se- quence of compactsetsuch thatGˆS

i Kⁱ(we recall thatGiss-compact).

For eachx2M, we consider the compact setKⁱ_x:ˆ fy2Mj(x;y)2Kⁱg and we define the upper semicontinuous function

d_i(x):ˆdiam(K_xⁱ);

where diam denotes the diameter of the set. Then d(x):ˆsup

i di(x)ˆ

ˆdiam(Gx) is a Borel function and so

Lˆ fx2Mjd(x)>0g is a Borel subsetofM.

Letus now show thatL T. Ifx62 T, then, by Lemma 2.14,R^‡_x ˆ fxg.

Hence, asGxR^‡_x (see Proposition 2.15),Gx fxgandx62L.

Step 3:L\Ris at most countable for each transport rayR.

We fix a transport ray, that is the image of a non-trivial maximal u- calibrated curve g:[a;b]!M, and we consider the strictly increasing function f :[a;b]!R defined by f ˆ(u U)g (see Lemma 2.9). We observe that

~c(g(s);g(t))ˆf(f(t) f(s)) 8st; s;t2[a;b]:

In view of the monotonicity ofG we have

f(f(t) f(s))‡f(f(t⁰) f(s⁰))f(f(t⁰) f(s))‡f(f(t) f(s⁰)) whenever (g(s);g(t))2G, (g(s⁰);g(t⁰))2G,st⁰,s⁰t. Now, following [2], we show the implication

(g(s);g(t))2G; (g(s⁰);g(t⁰))2G; s<s⁰ ) tt⁰: …5†

Assume by contradiction that t>t⁰. Since st and s⁰t⁰, we have s<s⁰t⁰<t. In this case, setting aˆf(s⁰) f(s), bˆf(t⁰) f(s⁰), cˆf(t) f(t⁰), we have

f(a‡b‡c)‡f(b)f(a‡b)‡f(b‡c):

On the other hand, sincec>0, the strictly convexity offgives f(a‡b‡c) f(b‡c)>f(a‡b) f(b);

and therefore we have a contradiction. By (5), we obtain that the vertical sectionsG_xofGare ordered along a transport ray, i.e.

8y12Gx1; 8y22Gx2; y1y2 wheneverx1ˆg(s1); x2ˆg(s2); s1<s2: As a consequence, the set of allx2Rsuch thatG_xis nota singleton is at mostcountable, since, if for suchxwe considerIxthe smallest open interval such thatg Ix

Gx, we obtain a family of pairwise disjoints open intervals ofR.

Step 4: covering the setL.

AsL T, we will coverT with a countable family of, so called, trans- portbeams.

DEFINITION3.1. We calltransport beama couple (B;x) whereBRⁿis a Borel subsetandx:B!Mis a locally Lipschitz map such that:

± there exist a bounded Borel set VR^{n 1} and two Borel func- tionsa<b:V!Rsuch that

Bˆ f(v;s)2VRja(v)sb(v)g RⁿˆR^{n 1}R;

± for each v2V, the curve x_v:[a(v);b(v)]!M given by x_v(s)ˆx(v;s) isu-calibrated.

We remark that we do not assume thatxis injective.

We now want to prove that there exists a countable family (Bj;k;x_j;k)_j;k2Nof transport beams such that the imagesx_j;k(Bj;k) cover the setT.

So lettakeDR^{n 1}the closed unit ball and letc_j :D!M,j2N, be a family of smooth embeddings such that, for each maximal u-calibrated curveg:[a;b]!R, the embedded arcg((a;b)) intersect transversally the image ofc_jfor somej2N. Indeed, in order to construct such embeddings, it suffices to take a countable atlas (U_i;u_i)_i2Nsuch thatu_i(U_i)ˆB2(0)Rⁿ (where Br(0) denote the n-dimensional ball of radius r centered at the origin) and that satisfies S

i2N u_i¹(B₁(0))ˆM, and to consider the image by u_i¹of the countable family (D_l;q)_{1ln;q2Q\[ 1;1]}of (n 1)-dimensional balls of radius 1 defined by

Dl;q :ˆn

(x1;. . .;xn)jxlˆq; X

m6ˆl

jxmj² 1o :

For each (j;k)2N² letus consider the setVj;k ˆD\c_j¹(T_k¹). Letus define

a_j;k(v):ˆ ac_j:V_j;k!R;

b_j;k(v):ˆbc_j:V_j;k!R;

whereaandbwere defined in Section 2.3. We observe that, by Lemma 2.10, a_j;kandb_j;k are Borel functions. We can now define the Borel sets

B_j;k :ˆ f(v;s)2V_j;kRja_j;k(v)sb_j;k(v)g:

Finally, we define onBj;kthe map

x_j;k(v;s):ˆg_cj(v)(s):

We now observe thatx_j;k is locally Lipschitz. In fact, we can write an ex- tremal using the Euler-Lagrange flowf_s:TM!TM, which is complete because of the energy conservation. Thanks to the hypotheses made onL, the map

(s;x;v)7!fs(x;v) is of classC¹. As we have

x_j;k(v;s):ˆp_Mf_s(c_j(v);g__cj(v)(0));

wherepM:TM!Mis the canonical projection, in view of Theorem 2.12 we deduce that this map is locally Lipschitz. It is clear that, for each transport rayR, there existj;k2Nsuch thatRis contained inx_j;k(B_j;k).

Step 5:m(L)ˆ0.

In order to conclude thatm(L)ˆ0, it suffices to prove that, if (B;x) is a transport beam, then the set L\x(B) is negligible with respect to the volume measure.

We recall that, for each v2V, the curve x_v is a locally bilipschitz homeomorphism onto its image, and so, as we know that the set L\x(fvg [a(v);b(v)]) is countable, the set x ¹(L)\Bintersects each vertical linevRalong a countable set, and so in particular has zeroH¹- measure. Then, by Fubini's theorem,x ¹(L)\Bhas zeroHⁿ-measure in Rⁿ, and so, since locally Lipschitz maps sendHⁿ-null sets into Hⁿ-null sets, we get

Hⁿ(L\x(B))Hⁿ(x(x ¹(L)\B))ˆ0;

that implies thatL\x(B) is negligible with respect to the volume measure.

Step 6: uniqueness.

We now prove that the transport plan selected with the secondary variational problem is unique.

Letg₀,g₁be two optimal transport plans, which are optimal also for the secondary variational problem. By what we proved above, we know that

they are induced by two transport mapst0:M!Mandt1:M!M, re- spectively. Let us now considerg:ˆg₀‡g₁

2 . By the linear structure of the two variational problems (2) and (4),gis still optimal for both, and so it is induced by a transport map t. This implies that both g₀ andg₁ are con- centrated on the graph oft, and sot0ˆt1m-a.e.

REMARK3.2. We observe that exactly this argument shows also that the setEof ray ends is negligible with respect to the volume measure.

Acknowledgments. It's a pleasure to thank Albert Fathi for his sug- gestions and his support, and to gratefully acknowledge the hospitality of the EÂcole Normale SupeÂrieure of Lyon, where this paper was written.

REFERENCES

[1] L. AMBROSIO- N. GIGLI- G. SAVAREÂ,Gradient flows in metric spaces and in the Wasserstein space of probability measures. Lectures in Mathematics, ETH Zurich, BirkhaÈuser, 2005.

[2] L. AMBROSIO- A. PRATELLI,Existence and stability results in the L¹theory of optimal transportation. Lectures notes in Mathematics, Springer Verlag, 1813(2003), pp. 123-160.

[3] L. AMBROSIO, B. KIRCHHEIM- A. PRATELLI, Existence of optimal transport maps for crystalline norms.Duke Math. J.,125, no. 2 (2004), pp. 207-241.

[4] P. BERNARD- B. BUFFONI,Optimal mass transportation and Mather theory.

Journal of the European Mathematical Society,9, no. 1 (2007), pp. 85-121.

[5] P. BERNARD - B. BUFFONI, The Monge problem for supercritical ManÄe potential on compact manifolds.Adv. Math.,207, no. 2 (2006), pp. 691-706.

[6] G. CONTRERAS- J. DELGADO- R. ITURRIAGA,Lagrangians flows: the dynamics of globally minimizing orbits. II.Bol. Soc. Brasil. Mat. (N.S.),28, no. 2 (1997), pp. 155-196.

[7] A. FATHI,Weak KAM theorems in Lagrangian Dynamics.Book to appear.

[8] A. FATHI- A. FIGALLI,Optimal transportation on manifolds.Preprint, 2006.

[9] A. FATHI - E. MADERNA, Weak KAM theorem on noncompact manifolds.

Nonlin. Diff. Eq. and Appl., to appear.

[10] M. FELDMAN- R. J. MCCANN,Monge's transport problem on a Riemannian manifold.Trans. Amer. Math. Soc.,354(2002), pp. 1667-1697.

[11] L. V. KANTOROVICH,On the transfer of masses.Dokl. Akad. Nauk. SSSR,37 (1942), pp. 227-229.

[12] L. V. KANTOROVICH,On a problem of Monge.Uspekhi Mat.Nauk.,3(1948), pp.

225-226.

[13] R. MANOÄ,Lagrangians flows: the dynamics of globally minimizing orbits.

Bol. Soc. Brasil. Mat. (N.S.),28, no. 2 (1997), pp. 141-153.

[14] S. T. RACHEV - L. RUSCHENDORF, Mass transportation problems. Vol. I:

Theory, Vol. II: Applications. Probability and ita applications, Springer, 1998.

[15] C. VILLANI,Topics in optimal transportation.Graduate Studies in Mathe- matics,58(2004), American Mathematical Society, Providence, RI.

[16] C. VILLANI,Optimal transport, old and new.Lecture notes, 2005 Saint-Flour summer school.

Manoscritto pervenuto in redazione il 17 febbraio 2006