A finite dimensional linear programming approximation of Mather's variational problem

DOI:10.1051/cocv/2009039 www.esaim-cocv.org

A FINITE DIMENSIONAL LINEAR PROGRAMMING APPROXIMATION OF MATHER’S VARIATIONAL PROBLEM

Luca Granieri

Abstract. We provide an approximation of Mather variational problem by finite dimensional mini- mization problems in the framework of Γ-convergence. By a linear programming interpretation as done in [9] we state a duality theorem for the Mather problem, as well a finite dimensional approximation for the dual problem.

Mathematics Subject Classification.37J50, 49Q20, 49N60, 74P20, 65K10.

Received December 18, 2008.

Published online October 9, 2009.

1. Introduction

Mather variational principle, introduced by Mather in [18,19], is an important tool in Lagrangian dynamic.

In recent years, several authors have studied this topic in connections with various fields such as weak Kam Theory [5,8,10,11], Monge-Kantorovich mass transportation and geometric measure theory [3,4,6,13,14]. In this paper we discuss a finite dimensional approximation of Mather variational problem on the flat torusT^N, following a linear programming interpretation of Mather’s variational principle outlined in [9].

1.1. Mather’s problem

We will consider a compact Riemannian manifold (M, g) without boundary and a LagrangianL:T M →R which satisfies the following properties:

(1) Lis regular (at leastC²);

(2) L(x,·) is superlinear in the fiber uniformly onx∈M; (3) L(x,·) is strictly convex in the fiber for all x∈M.

The regularity requirement on L is crucial to consider regular solutions of the Hamilton-Jacobi equation (see Sect.3.1).

A measureμ∈ M(T M) is said to be closed if for all exact formsω it results

T Mω(x), vdμ= 0. (1.1)

Keywords and phrases. Mather problem, minimal measures, linear programming, Γ-convergence.

1 Dipartimento di Matematica Politecnico di Bari, via Orabona 4, 70125 Bari, Italy. l.granieri@poliba.it, granieriluca@libero.it

Article published by EDP Sciences c EDP Sciences, SMAI 2009

According to this definition we set

M^c={μ∈ M(T M)|μis closed, with compact support, μ∈ P(T M)},

where P(T M) stands for probability measures,i.e. μ≥0 and μ(T M) = 1. To each measure μ∈ M^c we can associate the homology class ofμwhich we will denote by [μ]∈H₁(M,R) (by duality withH¹). Indeed,μacts in a natural way on the set of the closed 1-forms onM by

ω→

T Mω(x), vdμ (1.2)

and thanks to condition (1.1) this action passes to the quotient by the exact forms.

Once we fix an homology class [h], Mather’s variational problem amounts to:

minM^c

T ML(x, v)dμ| [μ] = [h]

· (1.3)

We also define the action ofμby

A(μ) :=

T M

L(x, v)dμ.

A remarkable property of problem (1.3) is the following: we minimize an action functional which depends onLon measures which are merely closed. However, it turns out that the minimal measures are also invariant for the flow associated to the LagrangianL, see for example [3,5,11,17].

Consider theN-dimensional flat torus T^N. In this case it is known that the homologyH₁(T^N,R) and the cohomology H¹(T^N,R) are N-dimensional vector spaces. Moreover, the cohomology on T^N is generated by the classes [dx₁],[dx₂], . . . ,[dx_N]. In other words, for every closed 1-formsω on T^N there existsP ∈R^N and f ∈ C¹(T^N) such thatω=P+df. For details we refer for instance to [15]. FixedQ∈R^N, denoted byMc the space of measures having compact support, in this setting Mather’s problem can be formulated as follows:

Minimize

A(μ)|μ∈ M_c(T^N ×R^N) under the following constraints:

(1) μ∈ P(T^N ×R^N);

(2) ∀f ∈ C¹(T^N) :

T^N×R^Ndf, vdμ= 0;

(3)

T^N×R^Nv dμ=Q.

Observe that the second constraint in the above list correspond to the closeness of μ while the third one is equivalent to fix the homology class [μ]. Therefore, Mather’s problem corresponds to minimize the functionalP defined as

μ∈ Mc(T^N ×R^N)→P(μ) =A(μ) +I_K(μ), (1.4) whereI_K is the indicator function of the setK of measures which satisfy the above constraints.

1.2. Linear programming problems

A finite dimensional linear programming problem is a minimizing problem of the following form

⎧⎨

⎩

minc, x x≥0 Ax=b,

(1.5)

forc∈R^N, b∈R^M andAa N×M matrix. Here we writex≥0 to meanx_i≥0 for everyi= 1, . . . , N. We can associate to problem (1.5) a dual problem in the following canonical way

maxb, y

A^ty≤c, (1.6)

whereA^tdenotes the transpose matrix ofA. Observe that in the dual problem (1.6) we do not ask for positive solutions. Denoting respectively byK,H the set of constraints of problem (1.5) and (1.6) one has

∀x∈K, ∀y∈H : b, y ≤ c, x·

Therefore, if we denote byP the minimum value of problem (1.5) and byDthe maximum value of problem (1.6) we have the following comparison

D ≤ P. (1.7)

We have the equality sign in the above inequality if the followingcomplementary slackness condition holds.

Lemma 1.1 (slackness condition). Letx₀∈K andy₀∈H such that

b, y0=c, x0· (1.8)

Then x₀ is a solution of problem(1.5), while y₀ is a solution of problem(1.6). Moreover it resultsP=D.

A remarkable fact in finite dimensional linear programming is the following duality theorem:

Theorem 1.1(strong duality). If one among the problems(1.5)and(1.6)admits a solution then also the other problem admits a solution. Moreover, it resultsP =D.

As references for linear programming problems we refer for instance to [2,21]. An infinite dimensional linear programming problem could be stated for real topological vector spacesX, Y. We assume X endowed with a partial ordering, compatible with the linear structure, which induces a partial ordering on the dual spaceX^∗. Givenc^∗∈X^∗,b∈Y andA:X→Y a continuous linear operator, denoting by,the pairing between a space and its dual, a linear programming problem could be stated as follows

⎧⎨

⎩

minc^∗, x x≥0 Ax=b.

(1.9)

Denoting by A^∗:Y^∗→X^∗ the adjoint operator ofA, the dual problem of (1.9) amounts to maxy^∗, b

A^∗y^∗≤c^∗. (1.10)

In general, in this setting a strong duality assertion like Theorem 1.1 does not hold. However, the slackness condition remains true. Denoting byK, H the set of constraints of problem (1.9) and (1.10) respectively, we have the following:

Theorem 1.2 (weak duality). For everyx∈K,y^∗∈H it results

y^∗, b ≤ c^∗, x· (1.11)

Moreover, if for somex₀∈K,y^∗₀∈H it results

y^∗₀, b=c^∗, x₀, (1.12)

thenx₀ is a solution of problem(1.9), while y₀^∗ is a solution of problem(1.10).

1.3. Description of the results

In the paper [9] the authors outline a linear programming formulation of Mather problem. In a large heuristic way they discuss some formal implications of linear programming duality to the Mather theory. Following these guide-lines, in this paper we provide an approximation of Mather problem by finite dimensional minimization problems. We perform this approximation in the framework of Γ-convergence. In particular, discretizing the phase-spaceT^N×R^N, we find a sequence of finite dimensional linear programming problemsPnwhich Γ-converge to the Mather problem. Therefore we can recover a Mather minimizing measure by solving finite dimensional linear programming problems (see Thm.2.2). As a consequence of recent developments of Hamilton-Jacobi and Mather theory [5,9–11,17,19] we state a duality result for Mather problem (see Sect.3.1). Moreover, the finite dimensional duality provides also an approximation result for the dual problem (see Thm.3.2).

2. Discrete approximation of Mather problem

Following Mather we restrict to consider measures with compact support. To pursue variational arguments, one need a notion of convergence which preserves the constraints. Therefore we get the following:

Definition 2.1. Let μ_n, μ ∈ M(T M). We say that μ_n μ if μ_n μ^∗ and

T M|v| dμ_n →

T M|v| dμ as n→+∞.

Given a continuous functionf : T M →Rwe say that f grows at most linearly if there exist two constant A, B ≥0 such that for every (x, v)∈T M it results|f(x, v)| ≤A+B|v|. For the reader convenience we state the following (see also [17]):

Lemma 2.1. Let μ_n, μ∈ Pc(T M)be probability measures having compact support. Ifμ_n μthen

n→+∞lim

T Mf dμ_n=

T Mf dμ for every continuous function which grows at most linearly.

Proof. Fix a large compact setKsuch that supp(μ)⊂ Kandμ(∂K) = 0. Given a continuous functionsf which grows at most linearly we setϕ(v) =A+B|v|. We evaluate

T Mf d(μ−μ_n) ≤

Kf d(μ−μ_n) +

K^cf d(μ−μ_n)

≤

Kf d(μ−μ_n) +

K^c|f|dμ_n ≤

Kf d(μ−μ_n) +

K^cϕ(v) dμ_n

=

Kf d(μ−μ_n) +

T Mϕ(v) dμ_n−

Kϕ(v) dμ_n−

T Mϕ(v) dμ−

Kϕ(v) dμ

. Observe that in the above computation we have used the condition supp(μ)⊂ Kto infer

K^cf d(μ−μ_n) =

K^cf dμ_n ;

T Mϕ(v) dμ−

Kϕ(v) dμ=

K^cϕ(v) dμ= 0 respectively in the second and the third line. We can now estimate

T Mf d(μ−μ_n) ≤

Kf d(μ−μ_n) +

Kϕd(μ−μ_n) +

T Mϕd(μ_n−μ). (2.1)

By standard properties of weak-star convergence (see for example Prop. 1.62 of [1]) we get

n→+∞lim

Kf d(μ−μ_n) = 0 = lim

n→+∞

Kϕd(μ−μ_n).

Moreover, since we are dealing with probability measures,μ_n μ⇒

T Mϕ(v) dμ_n→

T Mϕ(v) dμasn→+∞.

Therefore, by (2.1) the statement follows.

To realize a discrete approximation, we will allow to a measureμ to have an atom of negative mass at the origin. If the measure μ is not positive we consider the total variation |μ| given by |μ| = μ⁺ +μ⁻, where μ⁺ andμ⁻ denote the positive and negative parts corresponding to the Hahn decomposition ofμ. Moreover, since Mather problem deals with probability measures, by adding a constant to the LagrangianL(x, v) we may assume L(0,0) = 0 without changing the minimizing problem. Therefore, throughout the rest of this paper M =T^N and L(0,0) = 0.

2.1. Formulation of the discrete functional P

Letn≥1 be the discretization step. We consider the following discretization ofT^N I=

(i₁, ..., i_N)|i_h∈

0,1

n, . . . ,n−2 n ,n−1

n

·

The number of elements of I(which isn^N) will be denoted by|I|. The discretization ofR^N is given by J =

(j₁, ..., j_N)|j_h∈

−n, . . . ,−1 n,0,1

n, . . . , n

· We define the setC as the set of atomic measuresμ=

i,jμ_i,jδ_i,j ∈ Mc(T M) which satisfy the constraints:

⎧⎪

⎪⎨

⎪⎪

⎩

μ_i,j≥0, (i, j)= (0,0),

i,jμ_i,j = 1,

i,jjμ_i,j =Q,

∀i∈I:

j

_N

h=1j_h(μ_i−eh/n,j−μ_i,j) = 0.

(2.2)

We define the functionalP_n onM_c(T M) as P_n(μ) =

T ML(x, v) dμ+ Φ_n(μ) +I_C(μ), (2.3) whereI_C(μ) is the indicator function of the setC, while

Φ_n(μ) =O(n)μ⁻_0,0+

i,|j|=n

|j|μi,j−

i,|j|=n

L(i, j)μ_i,j, (2.4)

where|j|denotes the norm, for instance the maximum norm, of the vectorj,O(n) is a strictly positive function such that lim_n→+∞O(n) = +∞ and lim_n→+∞^O(n)_n = 0, while μ⁻_i,j = max(−μ_i,j,0). The introduction of the penalization term Φ_n(μ) is motivated by the construction of discrete measures which approximate a given Mather’s measure μ performed in Lemma 2.4. Indeed, to realize this approximation, a crucial step is to consider discrete measures which satisfy all the constraints (2.2). If such discrete measures do not satisfy all the constraints, we get a modified sequence of measures by adding suitable terms at the origin and on the points of the discretization such that|j|=n. The term Φ_n, see formula (2.27) and (2.28), ensures that these changes do not affect the approximation process.

We observe that the constraint

∀i∈I:

j

h

j_h(μ_i−eh/n,j−μ_i,j) = 0, (2.5)

where (e_h)_h=1,...,N denotes the canonical basis ofR^N, corresponds to the closeness condition, while

i,j

jμ_i,j=Q (2.6)

is the discrete version of the momentum constraint of Mather’s measure, which is equivalent to fix the homology class of the measureμ.

Of course, we haveC=∅. Indeed, setting for every (i, j)= (0,0)

⎧⎪

⎨

⎪⎩

μ_i,eh =_|I|¹ max(Q_h,0), μ_i,−eh = _|I|¹ max(−Q_h,0), μ_i,j= 0 forj=±eh, andμ_0,0= 1−_N

h=1|Qh|, it resultsμ=

i,jμ_i,jδ_i,j∈C.

Lemma 2.2. For every n≥1Pn admits minimizers.

Proof. Letμ_k be a minimizing sequence. SinceC=∅, we havePn(μ_k)<+∞and we may assume thatμ_k are atomic measures. It follows that the sequence|μ_k|is uniformly bounded. Therefore, by weak-star compactness of measures, there exists a convergent subsequence to a measureμhaving compact support. SinceCis a closed convex subset of a finite dimensional simplex, it resultsμ∈C. Since the point of the discretization are fixed, it turns out thatP_n is continuous alongμ_k. Therefore,μis a minimum point forP_n.

2.2. Approximation

The main result of this section is Theorem2.2which allow to recover a Mather minimizing measure by solving finite dimensional minimization problems. We state the following approximation result

Theorem 2.1. P_n →^Γ Pwith respect to the convergence.

We refer to [16] for an introduction to the Γ-convergence. To simplify reading, we divide the proof of the above theorem in two lemmas.

Lemma 2.3. Let μ∈ M_c(T M). Then, for every sequenceμ_n∈ M_c(T M)we have μ_n μ⇒P(μ)≤lim inf

n→+∞Pn(μ_n).

Proof. Suppose that lim inf_n→+∞Pn(μ_n)<+∞. Hence, we may assume that μ_n ∈ C. To simplify notation we denote μⁿ_i,j := (μ_n)_i,j. If there are infinitely manyμⁿ_0,0 <0, the penalization term Φ_n forces (μⁿ_0,0)⁻ → 0 as n→+∞. Therefore, we get|μn| μ. This implies^∗ μ≥0. Since (μⁿ_0,0)⁻ →0 as n→+∞, it follows that actuallyμis a probability measure. Indeed, by weak-star convergenceμ(T M)≤1. On the other hand, fixed a large compact setKsuch that supp(μ)⊂ K, μ(∂K) = 0 and|v| ≥1 for every (x, v)∈ K^c we get

0≤μ_n(K^c)≤

K^c|v|dμ_n=

K^c|v|d(μ_n−μ) =

T M|v|d(μ_n−μ)−

K|v|d(μ_n−μ).

Sinceμ_n μ it resultsμ_n(K^c)→0 asn→+∞. Therefore μ(T M) =μ(K)≥lim sup

n→+∞|μ_n|(K) = lim sup

n→+∞

1−μ_n(K^c) + 2(μⁿ_0,0)⁻

= 1.

Since μ_n μ and (μⁿ_0,0)⁻ → 0, Lemma 2.1 still holds and then we deduce that the measure μ verifies the momentum constraint. It remains to test the closeness constraint. If we denoteDⁿ_hφ(x) = ^φ(x+eh_1/n^/n)−φ(x), by changing variables, the constraint (2.5) forμ_n is equivalent to

∀φ∈ C¹(M) :n

i

φ(i) N h=1

j_h(μⁿ_i−eh/n,j−μⁿ_i,j) =

i,j

N h=1

j_hD_hⁿφ(i)μⁿ_i,j =

T Mv, Dⁿφ(x)dμ_n= 0, (2.7) whereDⁿφ(x) = (D₁ⁿφ(x), . . . , Dⁿ_Nφ(x)). We evaluate

0 =

T Mv, Dⁿφ(x)dμ_n =

T Mv, Dφ(x) dμ_n−

T Mv, Dφ(x)−Dⁿφ(x)dμ_n

≤

T Mv, Dφ(x) dμ_n+Dφ−Dⁿφ_∞

T M|v|dμ_n. Passing to the limit asn→+∞, because of Lemma2.1, we obtain

T Mv, Dφdμ≥0.

By the arbitrariness ofφ it follows that μis a closed measure. Finally, since Lis superlinear, without loss of generality we may assume thatLis positive onK^c. Recalling thatL(0,0) = 0, by the weak-star convergence of measures we obtain

P(μ) =

T ML(x, v)dμ=

KL(x, v)dμ= lim

n→+∞

KL(x, v) dμ_n

= lim

n→+∞ T ML(x, v)dμ_n−

K^cL(x, v)dμ_n

+

i,|j|=n

|j|μⁿ_i,j−

i,|j|=n

|j|μⁿ_i,j+O(n)(μⁿ_0,0)⁻−O(n)(μⁿ_0,0)⁻

= lim

n→+∞

⎛

⎝Pn(μ_n)−

K^c\|j|=n

L(x, v)dμ_n−O(n)(μⁿ_0,0)⁻−

i,|j|=n

|j|μⁿ_i,j

⎞

⎠≤lim inf

n→+∞Pn(μ_n).

Lemma 2.4. For every μ∈ Mc(T M)there exists a sequenceμ_n∈ Mc(T M)such thatμ_n μ and satisfying P(μ) = lim_n→+∞Pn(μ_n).

Proof. We may assume that P(μ) < +∞, namely that μ satisfy the constraints of Mather problem. For every (i, j) ∈ I×J consider disjoint squares Q_i,j with center in the points (i, j) and size 1/n, in such a way, for large n, they cover the supp(μ). We set μ_i,j = μ(Q_i,j) and μ_n =

i,jμ_i,jδ_i,j. We claim that

T Mf(x, v)dμ_n →

T Mf(x, v)dμasn→+∞for every uniformly continuous functionf. Indeed

T Mf(x, v)dμ−

T Mf(x, v)dμ_n =

i,j

Qi,j

f(x, v)dμ−

i,j

f(i, j)μ_i,j

=

i,j

Qi,j

(f(x, v)−f(i, j)) dμ ≤

i,j

Qi,j

|f(x, v)−f(i, j)|dμ→0

as n→+∞sincef is uniformly continuous andμis a probability measure. Therefore, it also resultsμ_n μ.

Although μ_n are probability measures, in general μ_n does not belong to the constraints setC. Therefore, we have to modify this construction. With respect to the closeness constraint, we look for coefficientsα_i,j ≥0 such that the discrete measureμ_n defined by (μ_n)_i,j =μ_i,j+α_i,j satisfies the following conditions

∀i∈I:

j

_N

h=1j_h(μ_i−eh/n,j−μ_i,j) = 0 (2.8)

|μ_n−μ_n| →0. (2.9)

For everyi∈I, condition (2.8) is equivalent to

j

N h=1

j_h(α_i−eh/n,j−α_i,j) =c_i:=

j

N h=1

j_h(μ_i,j−μ_i−eh/n,j), (2.10)

while condition (2.9) is equivalent to

i,j

α_i,j→0. (2.11)

Setting c= (c_i)_i∈I,z=μ_i,j,α=α_i,j, condition (2.10) can be written as

Bα=c, (2.12)

for a suitable matrix B. We search solutionsαof (2.12) supported on supp(μ),i.e. α_i,j= 0 whenever|j|> R for a suitable largeR >0. By definition of the coefficientsc_i,−z is a such solution of (2.12). Therefore, all the solutions of (2.12) are of the formβ −z with β ∈ker(B). Since the vectors whose components are all equal belong to ker(B), we can find positive solutions of (2.12). Denoting byIthe vector whose components are all equal to 1, we takeαas a solution of the following minimization problem

⎧⎨

⎩

minα,I α≥0 Bα=c.

(2.13)

By duality we also have a solutiony= (y_i)_i∈I of the problem maxc, y

B^ty≤I. (2.14)

We use duality to computeB^t as follows B^ty, x=y, Bx=

i

y_i

⎛

⎝

j

N h=1

j_h(x_i−eh/n,j−x_i,j)

⎞

⎠=

i,j

N h=1

j_h(y_i+eh/n−y_i)x_i,j.

Therefore, the (i, j)th component of B^ty is N h=1

j_h(y_i+eh/n−y_i).

Hence, the constraint of problem (2.14) yields

∀(i, j)∈I×J : N h=1

j_h(y_i+eh/n−y_i)≤1.

Choosingj=±e_h in the above formula we obtain

∀i∈I, ∀h= 1, . . . , N: |y_i+eh/n−y_i| ≤1. (2.15) By the slackness condition we have

i,j

α_i,j=α,I = c, y=

i

y_ic_i =

i

y_i

⎛

⎝

j

N h=1

j_h(μ_i,j−μ_i−eh/n,j)

⎞

⎠

=

i,j

N h=1

j_h(y_i−y_i+eh/n)μ_i,j. (2.16)

In particular, the above computation shows that

ic_i = 0 (choose y_i = constant). Therefore, by adding a constant, we may assume that y_i = 0 for some i ∈I. We have to prove that actually

iy_ic_i →0. First, because of (2.15), it results that the functionφ_n(i) =y_iis a Lipschitz function on the discrete gridI. Therefore, by standard extension theorems for Lipschitz functions, we obtain a Lipschitz function φ_n on the whole flat torus T^N whose Lipschitz constant grows as the discrete step n, i.e. Lip(φ_n) ≈n. Therefore, the sequence of functions ϕ_n = _n¹φ_n is equi-Lipschitz and equi-bounded. By the Ascoli-Arzel`a theorem, by passing to a subsequence, we find a Lipschitz functionϕsuch thatϕn−ϕ∞→0. On the other hand, sinceμis a closed measure andμ_n μ, with the same notation of the proof of Lemma2.3, for everyφ∈ C¹we get

−n

i

φ(i)c_i=

i,j

N h=1

nφ(i)j_h(μ_i−eh/n,j−μ_i,j) =

i,j

N h=1

j_hDⁿ_hφ(i)μⁿ_i,j=

T Mv, Dⁿφ(x) dμ_n

≤ Dφ−Dⁿφ_∞

T M|v|dμ_n+

T MDφ(x), vdμ_n →0 (2.17) as n → +∞. Consider the linear operator on the space (Lip(M), · _∞) of Lipschitz functions defined by L_n(f) :=n

if(i)c_i. By the uniform boundedness principle on the space ofC¹functions, taking into account the computation (2.17), for everyf ∈ C¹it results

|Ln(f)| ≤K(f∞+ Lip(f)).

Iff is a Lipschitz function, by standard approximation results (using for instance convolution kernels) we find f_n∈ C¹such that f−f_n∞≤1/nand Lip(f_n)≤Lip(f). Since μ∈ Pc(T M), it follows

i|ci| ≤M. Hence we compute

|L_n(f)| ≤ |L_n(f−f_n)|+|L_n(f_n)| ≤nf−f_n∞

i

|c_i|+K(f_∞+ Lip(f))≤C(f).

Applying again the Banach-Steinhaus theorem we find

n

i

|c_i|=L_n|| ≤C. (2.18)

Iff ∈Lip(M), for everyφ∈ C¹ we have the estimate

|n

i

f(i)c_i| ≤n

i

|ci|f −φ∞+|n

i

φ(i)c_i| ≤Cf−φ∞+|n

i

φ(i)c_i|.

Using the boundedness condition (2.18), by (2.17) and by density ofC¹functions in Lip(M), for every Lipschitz functionf it results

n

i

f(i)c_i→0

as n→+∞. Now we come back to the estimation of the coefficient α_i,j. By (2.16) we have

i,j

α_i,j=

i

y_ic_i =n

i

ϕ_n(i)c_i≤nϕ−ϕ_n∞

i

|ci|+n

i

ϕ(i)c_i ≤Cϕ−ϕ_n∞+n

i

ϕ(i)c_i→0

as n →+∞. Therefore, condition (2.11) holds. Without loss of generality, normalizing we may assume that μ_n(T M) = 1. At this point we need another modification to satisfy the momentum constraint. To this aim we look for coefficients s_i,j≥0 such that the discrete measure ¯μ_n defined by (¯μ_n)_i,j=μ_i,j+s_i,j has the right momentum. Forh= 1, . . . , N, denoting by

R_h=

i,j

j_hμ_i,j (2.19)

R_h=

i,j

j_hμ_i,j (2.20)

s_h= Q_h−R_h

n|I| , (2.21)

for every (i, j)∈I×J, we set

⎧⎨

⎩

s_i,neh= max(s_h,0) s_i,−neh = max(−s_h,0) s_i,j= 0 forj=±neh.

(2.22)

Therefore, the momentum of ¯μ_n is given by

i,j

j_h(¯μ_n)_i,j =R_h+

i,j

j_hs_i,j=R_h+n|I|sh=R_h+Q_h−R_h=Q_h.

Observe that the choice ofs_i,j does not modify the closeness constraint. Indeed, for everyi∈Iwe have

j

N h=1

j_h

(¯μ_n)_i−eh/n,j−(¯μ_n)_i,j

=±n

j=±neh

N h=1

μ_i−eh/n,j+s_h−μ_i,j−s_h

+

j=±neh

N h=1

j_h

μ_i−eh/n,j−μ_i,j

=

j

N h=1

j_h

μ_i−eh/n,j−μ_i,j

= 0.

Since

|Qh−R_h|=

i,j

Qi,j

(v_h−j_h) dμ ≤K

n, (2.23)

for a constantK >0 which depends only on the dimensionN of the space, we can evaluate

i,j

s_i,j=|I|^N

h=1

|s_h|= 1 n

N h=1

|Q_h−R_h| ≤ 1 n

N h=1

|Q_h−R_h|+ 1 n

i,j

|j|α_i,j

≤ N K n² + 1

n

i,j

|j|α_i,j→0, (2.24)

i,j

|j|si,j=n

i,j

s_i,j ≤N K

n +

i,j

|j|αi,j→0, (2.25)

as n→+∞because of (2.11) and since αhas compact support. Moreover, we estimate

|μ¯_n−μ_n|=

i,j

|(¯μ_n)_i,j−μ_i,j| ≤

i,j

α_i,j+

i,j

s_i,j.

By (2.11) and (2.24) we infer

n→+∞lim |μ_n−μ¯_n|= 0. (2.26)

It remains to satisfy the probability constraint. This can be done by subtracting the mass needed at the origin, without changing neither the momentum or the closeness constraint. More precisely, since

i,j

(¯μ_n)_i,j=

i,j

μ_i,j+

i,j

s_i,j = 1 +

i,j

s_i,j,

if we put (¯μ_n)_0,0 =μ_0,0−

i,js_i,j we also have

i,j(¯μ_n)_i,j = 1. Condition (2.26) still holds since by (2.24) we have added infinitesimal quantities. At this point it is easy to verify the weak-star convergence. Indeed, if f ∈ C_c(T M) we have

T Mf(x, v)dμ−

T Mf(x, v)d¯μ_n ≤

T Mf(x, v)dμ−

T Mf(x, v)dμ_n +

T Mf(x, v)dμ_n−

T Mf(x, v)d¯μ_n

≤

T Mf(x, v)dμ−

T Mf(x, v)dμ_n

+f_∞|μ¯_n−μ_n| →0.

In order to verify the convergence ¯μ_n μ, we need to check the convergence of momentum. By (2.9) and (2.25) we get

T M|v|dμ−

T M|v|d¯μ_n ≤

T M|v|dμ−

T M|v|dμ_n +

i,j

|j|si,j→0.

Finally, recalling thatL(0,0) = 0, we evaluate

|P(μ)−P_n(¯μ_n)|=

T ML(x, v)dμ−

T ML(x, v) d¯μ_n−Φ_n(¯μ_n)

≤

T ML(x, v)dμ−

i,j

L(i, j)μ_i,j +

i,|j|=n

|j|si,j+O(n)((¯μ_n)_0,0)⁻. (2.27)

Using (2.9), (2.24) and (2.25) it follows

|P(μ)−Pn(¯μ_n)| ≤

T ML(x, v)dμ−

i,j

L(i, j)μ_i,j +

i,|j|=n

|j|si,j+O(n)((¯μ_n)_0,0)⁻

=

T ML(x, v)dμ−

i,j

L(i, j)μ_i,j +

i,|j|=n

|j|s_i,j+O(n)

i,j

s_i,j→0. (2.28)

Theorem 2.2. Let μ_n be minimum points ofPn. Then there exists a sequence of discrete measuresμ¯_n and a Mather minimizing measureμ such thatμ¯_n μ. Moreover, by passing to a subsequence, it results

P(μ) = lim

n→+∞P_n(μ_n).

Proof. Let K = max{L(x, v) : (x, v) ∈ T M, v = 1}. Without loss of generality we may assume that 0≤K≤1/√

N. By superlinearity, there exists k >1 such that for everyx∈M it resultsL(x, v)>√ N K|v| whenever|v|> k. For every (i, j)∈I×J we set

⎧⎪

⎪⎨

⎪⎪

⎩

(¯μ_n)_i,eh = (μ_n)_i,eh +

|j|>kmax(j_h(μ_n)_i,j,0) (¯μ_n)_i,−eh = (μ_n)_i,−eh+

|j|>kmax(−jh(μ_n)_i,j,0) (¯μ_n)_i,j = (μ_n)_i,j forj=±e_h, 0<|j| ≤k (¯μ_n)_i,j = 0 for|j|> kor|j|= 0.

(2.29)

By the above modification it results that the new discrete measures ¯μ_nis positive in T M and, since we may assume μ_n ∈ C, they still satisfy the momentum and closeness constraints (2.6) and (2.5). We need at most to check the probability constraint. Let M =

i,j(¯μ_n)_i,j. If M < 1, we add the needed mass at the origin.

If M > 1 we normalize by considering the measure ¯μ_n/M. This normalized measure is still closed but the momentum amounts to _M^Q. Following the construction performed in the proof of Lemma2.4, we can obtain the right momentum by adding the quantities

s_h= Q_h−Q_h/M n|I|

as done in (2.22). It remains to subtract the mass in excess, which is an infinitesimal quantity, at the origin.

Settingε_n=O(n)

i,js_i,j, because of (2.24) it resultsε_n→0 asn→+∞. In any case, for everyν∈ M_c(T M) we get

Pn(¯μ_n)≤

i,|j|≤k

L(i, j)(μ_n)_i,j+

i,|j|>k

N h=1

L(i,±e_h)|j_h|(μ_n)_i,j+ε_n

≤

i,|j|≤k

L(i, j)(μ_n)_i,j+√

N K

i,|j|>k

|j|(μn)_i,j+ε_n<Pn(μ_n) +ε_n ≤Pn(ν) +ε_n. (2.30)

Therefore, ¯μ_n is anε_n-minimizer forPn. Furthermore, since|¯μ_n|is equibounded and the supports are contained in a same large compact set, by passing to a convergent subsequence we have ¯μ_n μ. Moreover, for every ν ∈ Mc(T M), by Lemma 2.4 we find a sequence ν_n ∈ Mc(T M) such that Pn(ν_n) → P(ν) as n → +∞. By (2.30) we getPn(¯μ_n)≤Pn(ν_n) +ε_n. Letting n→+∞, by Lemma2.3we infer

P(μ)≤lim inf

n→+∞P_n(¯μ_n)≤ lim

n→+∞P_n(ν_n) =P(ν).

Thereforeμis a Mather minimizing measure. Moreover, again by Lemma2.4, we find a sequenceμ_n such that Pn(μ_n)→P(μ) as n→+∞. Sinceμ_n are minimum points ofPn we have

lim sup

n→+∞Pn(μ_n)≤ lim

n→+∞Pn(μ_n) =P(μ).

On the other hand, by (2.30) we get

P(μ)≤lim inf

n→+∞Pn(¯μ_n)≤lim inf

n→+∞Pn(μ_n).

3. Linear programming interpretation

The discrete problems of minimizingP_n can be reformulate into a linear programming framework. Indeed, splittingμ_0,0=μ⁺_0,0−μ⁻_0,0, therefore increasing the dimension of the problem, we set

λ_i,j =L(i, j) |j| =n λ⁺_0,0= 0, λ⁻_0,0=O(n)

λ_i,j =|j| for every|j|=n.

Therefore, minimize the functionalP_n is equivalent to

minimize

⎛

⎝

i,j

λ_i,jμ_i,j

⎞

⎠

subject to the following constraints

μ_i,j, μ⁺_0,0, μ⁻_0,0≥0

∀i∈I:_N

h=1

jj_h(μ_i−eh/n,j−μ_i,j) = 0

i,jμ_i,j+μ⁺_0,0−μ⁻_0,0= 1

i,jjμ_i,j=Q.

In this problem we haveh=|I|+N+ 1 equality constraints fork=|I||J|+ 1 unknowns. Denoting respectively byμandλthe vectors ofkunknown components μ_i,j andλ_i,j, and the vector

b= (0, . . . , 0,

|I|zeros

1, Q₁, . . . , Q_N), (3.1)

the above problem can be written in the form

⎧⎨

⎩

minimizeλ, μ μ≥0

Aμ=b

(3.2)

for a suitableh×kmatrixA. As standard in linear programming, problem (3.2) admits the dual formulation maximizey, b

A^ty≤λ (3.3)

whereA^tis the transpose matrix ofA.

3.1. The infinite dimensional dual problem

In [9] it is proposed a linear programming interpretation of Mather’s problem, while a linear programming interpretation is a well known fact for the Monge-Kantorovich problem (see for example [7]). In particular, for μ∈ Mc(T M) define:

φ∈ C¹(M)→L_μ(φ) =

T MDφ(x), vdμ, and consider the continuous linear operator

A:Mc(T M)→ C¹(M)^∗×R×R^N defined by

Aμ=

L_μ, μ(T M),

T Mvdμ

. If we denote by c^∗, the linear functional associated to L by μ →

T ML(x, v)dμ, fixed Q ∈ R^N and setting (0,1, Q) = b, Mather’s problem can be reformulated as the following infinite dimensional linear programming

problem ⎧

⎨

⎩

minc^∗, μ μ≥0 Aμ=b.

(3.4) The dual formulation of problem (3.4) amounts to

maxy^∗, b

A^∗y^∗≤c^∗. (3.5)

However, for infinite dimensional linear programming problems, the dual problem could not admit solutions despite the existence of solutions of the primal one. Moreover, since the space C¹(M) is not reflexive, to a solution y^∗ of the dual problem does not correspond in general a C¹ function. Actually, in [9] it is shown that there is no duality gap between the two problem and that if we have a solution y^∗ of (3.5) such that y^∗= (u, ω₀, P)∈ C¹(M)×R×R^N thenusolve the Hamilton-Jacobi equation

H(x, P+d_xu) = ¯H(P) on supp(π_x(μ)),

where H is the Hamiltonian associated to the LagrangianL(x, v), while ¯H(P) =−ω0,μ is a solution of (3.4) andπ_x(μ) is the push-forward ofμ through the projection ofT M onM. Hence the dual of Mather’s problem predict the weak KAM equation (see [5,8,10,11]). Although in this abstract framework only a weak duality result holds, we can justify a strong duality relation for the Mather’s problem by the following considerations.

First we recall that Mather’s problem can be formulated in terms of merely closed measures. Precisely we have the following (see [17,19]):

Theorem 3.1. For every Mather minimizing measure μwith [μ] =Qthere existsP ∈R^N such that A(μ) +P·Q=−c(L+P),

where−c(L+P) = min{

(L+P)dν |ν ∈ M^c} is the Ma˜n´e critical value (see also[5,10,13]) corresponding to the Lagrangian (L+P)(x, v) =L(x, v) +P, v.

We state the following approximation theorem for the dual problem (3.5):

Theorem 3.2. Let y_n be solutions of the finite dimensional linear programming problems (3.3). Then there exists a solution y^∗ of problem(3.5)such that, by passing to a subsequence,

n→+∞lim y_n, b=y^∗, b·