A continuous model of transportation revisited
aa r X i v : . [ m a t h . F A ] F e b A CONTINUOUS MODEL OF TRANSPORTATION REVISITED
LORENZO BRASCO AND MIRCEA PETRACHE
Abstract.
We review two models of optimal transport, where congestion effects duringthe transport can be possibly taken into account. The first model is Beckmann’s one,where the transport activities are modeled by vector fields with given divergence. Thesecond one is the model by Carlier et al. (SIAM J Control Optim 47: 1330–1350, 2008),which in turn is the continuous reformulation of Wardrop’s model on graphs. We dis-cuss the extensions of these models to their natural functional analytic setting and showthat they are indeed equivalent, by using Smirnov decomposition theorem for normal1 − currents. Contents
1. Introduction 11.1. Theoretical background 11.2. Goals of the paper 31.3. Plan of the paper 52. Well-posedness of Beckmann’s problem 53. Duality for Beckmann’s problem 104. A Lagrangian reformulation 13Appendix A. Decompositions of acyclic 1 − currents 16A.1. Definitions and links to vector fields 16A.2. Lipschitz curves as currents 19A.3. Smirnov decomposition theorem 20A.4. The case of flat currents 21References 241. Introduction
Theoretical background.
The present work is motivated by the study of transportproblems for distributions started in [6] and [12], which we desire to try and connect torelated works in the theory of currents presented in [1, 27, 36]. One of the important
Mathematics Subject Classification.
Key words and phrases.
Monge-Kantorovich problem, Beckmann problem, Smirnov Theorem, flat norm. motivations for this problem is the study of distributions of the form ∞ X i =1 ( δ P i − δ Q i ) , with ∞ X i =1 | P i − Q i | < ∞ , as in [33]. Such distributions arise as topological singularities in several geometric varia-tional problems as described for example in [11, 23, 32, 34, 35].To start with, we formally define three variational problems which can be settled (forsimplicity) on the closure of an open convex subset Ω ⊂ R N having smooth boundary. Forthe moment we are a little bit imprecise about the datum f but we will properly settle ourhypotheses later. The first problem is the minimization of the total variation of a Radonvector measure under a divergence constraint:( B ) min V (cid:26)Z Ω d | V | : − div V = f, V · ν Ω = 0 (cid:27) . The above problem can be connected by duality with the following one, called the Kan-torovich problem:( K ) max φ (cid:8) h f, φ i : k∇ φ k L ∞ (Ω) ≤ (cid:9) , where now the variable φ is a Lipschitz function and h· , ·i represents a suitable dualitypairing. Finally the third problem is the minimization of the total length( M ) min Q (cid:26)Z P ℓ ( γ ) dQ ( γ ) : ( e − e ) Q = f (cid:27) , where P is the space of Lipschitz continuous paths γ : [0 , → Ω, the length functional ℓ isdefined by ℓ ( γ ) = Z | γ ′ ( t ) | dt,e , e are the evaluation functions giving the starting and ending points of a path and thevariable Q is a measure concentrated on P .The classical setting for the above problems is when f is of the form f = f + − f − where f + and f − are positive measures on Ω having the same mass (for example conventionallyone can consider them to be probability measures). We point out that in this case a morefamiliar formulation of ( M ) is the so-called Monge-Kantorovich problem ( M ′ ) min η (cid:26)Z Ω × Ω | x − y | dη ( x, y ) : ( π x ) η = f + and ( π y ) η = f − (cid:27) , where π x , π y : Ω × Ω → Ω stand for the projections on the first and second variable,respectively. It is useful to recall that the link between ( M ) and ( M ′ ) is given by the factthat if η is optimal for Monge-Kantorovich problem then the measure which concentrateson transport rays i.e. Q = Z δ x y dη ( x, y ) , where x y stands for the segment connecting x and y, CONTINUOUS MODEL OF TRANSPORTATION REVISITED 3 is optimal in ( M ) and Z P ℓ ( γ ) dQ ( γ ) = Z Ω × Ω | x − y | dη ( x, y ) . When f has the above mentioned form f + − f − the equivalence of the three problems aboveis well understood. The equivalence of ( M ′ )=( M ) and ( K ) is the classical Kantorovichduality (see [24]), while that between ( B ) and ( K ) seems to have been first identified in[37].Recently the equivalence of the above three problems has been shown in [6] for f belong-ing to a wider class, i.e. when f is in the completion of the space of zero-average measureswith respect to the norm dual to the C (or flat) norm. This wider space was studied in[22] and characterized recently in [6, 7]. A different point of view is also available in [25],where the space of such f is called W − , .1.2. Goals of the paper.
Our starting observation is that problem ( B ) pertains to a wideclass of optimal transport problems introduced by Martin J. Beckmann in [2], which areof the form( B H ) min V (cid:26)Z Ω H ( V ) dx : − div V = f, V · ν Ω = 0 (cid:27) , where c | z | p ≤ H ( z ) ≤ c | z | p , for a suitable density-cost convex function H : R N → R + and p ≥
1. For a problem ofthis type the question of finding equivalent formulations of the form ( K ) and ( M ) hasalready been addressed in [10] (see also [8]) under some restrictive assumptions on f , likefor example f = f + − f − with f + , f − ∈ L p (Ω) and Z Ω f + = Z Ω f − = 0 . The goal of this paper is to complement and refine this analysis, first of all by studyingproblem ( B H ) in its natural functional analytic setting, i.e. when f belongs to a dualSobolev space W − ,p (whose elements are not measures, in general). By expanding theanalysis in [8, 10] we will also see that alternative formulations of the type ( K ) and ( M )are still possible for ( B H ) in this extended setting. These formulations are still well-posedon the dual space W − ,p and equivalence can be proved in this larger space. The problemcorresponding to ( K ) will now have the form (see Section 3 for more details)( K H ) max φ (cid:20) h f, φ i − Z Ω H ∗ ( ∇ φ ) dx (cid:21) , and the equivalence with ( B H ) will just follow by standard convex duality arguments (whichare later recalled, for the convenience of the reader). On the contrary, in the proof of theequivalence between ( B H ) and its Lagrangian formulation( M H ) min Q (cid:26)Z Ω H ( i Q ) dx : ( e − e ) Q = f (cid:27) , LORENZO BRASCO AND MIRCEA PETRACHE some care is needed and we will require to f be a finite measure belonging to W − ,p . Herethe measure i Q will be some sort of transport density generated by Q , which takes intoaccount the amount of work generated in each region by our distribution of curves Q (seeSection 4 for the precise definition). In particular the proof of this equivalence will pointout another not emphasized connection to Geometric Measure Theory.The main result of this paper can be formulated as follows (see Theorems 3.1 and 4.5for more precise statements): Main Theorem.
Let < p < ∞ . Suppose Ω ⊂ R N is the closure of a smooth boundedopen set, let f ∈ W − ,p (Ω) and let H be a strictly convex function having p − growth. Thenthe minimum in ( B H ) and the maximum in ( K H ) are achieved and coincide. Moreoverthe unique minimizer V of ( B H ) and any maximizer v of ( K H ) are linked by the relation ∇ v ∈ ∂ H ( V ) , as specified in Theorem 3.1.If in addition f is a Radon measure then we have the following relationship among theoptimizers of ( B H ) and ( M H ): (i) the unique minimizer of ( B H ) corresponds to a minimizer of ( M H ) in the sense ofProposition 4.3; (ii) each minimizer of ( M H ) corresponds to the unique minimizer of ( B H ) in the senseof Proposition 4.3. The connection of the above theorem to Geometric Measure Theory lies in the basictheory of normal − currents , whose basic steps are recalled in the (long) appendix at theend of the paper. Indeed, in order to show equivalence of ( B H ) and ( M H ) our cornerstoneis Smirnov decomposition theorem for 1 − currents.For the sake of completeness and in order to neatly motivate the studies performed inthis paper it is worth recalling that the proof of this equivalence in [10] was based on the Dacorogna-Moser construction to produce transport maps (see [13]), which has revealed tobe a powerful tool for optimal transport problems . In a nutshell, this method consists inassociating to the “static” vector field V which is optimal for ( B H ) the following dynamicalsystem ∂µ t + div (cid:18) V (1 − t ) f + + t f − µ t (cid:19) = 0 , µ = f + , i.e. a continuity equation with driving velocity field e V t given by V rescaled by the linearinterpolation between f + and f − . Assuming that one can give a sense (either deterministicor probabilistic) to the flow of e V t , the construction of the measure Q V concentrated on theflow lines of e V t paves the way to the equivalence between the Lagrangian model ( M H ) and( B H ) (see [8, 10] for more details). When H ( t ) = | t | problem ( M H ) is again the Monge-Kantorovich one and i Q for an optimal Q is nothingbut the usual concept of transport density, see [5, 15, 20]. It is worth remarking that the first proof of the existence of an optimal transport map for problem( M ′ ), more than 200 years after Monge stated it, was based on a clever use of this construction (see [18]). CONTINUOUS MODEL OF TRANSPORTATION REVISITED 5
Plan of the paper.
In Section 2 we describe the function space W − ,p (Ω) and weprove the existence of a minimizer for ( B H ). Section 3 treats the equivalence of ( B H )with ( K H ) by appealing to classical convex analysis results. The aim of Section 4 is tointroduce the Lagrangian counterpart of Beckmann’s model and to show that the twomodels are equivalent. A self-contained Appendix complements the paper. There weintroduce relevant concepts from Geometric Measure Theory and we translate Smirnov’sdecomposition theorem into the language of L vector fields.2. Well-posedness of Beckmann’s problem
Let Ω ⊂ R N be the closure of an open bounded connected set having smooth boundary.In what follows Ω will always be compact . Given 1 < q < ∞ we indicate with W ,q (Ω) theusual Sobolev space of L q (Ω) functions whose distributional gradient is in L q (Ω; R N ) aswell. We then define the quotient space˙ W ,q (Ω) = W ,q (Ω) ∼ , where ∼ is the equivalence relation defined by u ∼ v ⇐⇒ there exists c ∈ R such that u ( x ) − v ( x ) = c for a.e. x ∈ Ω . When needed the elements of ˙ W ,q (Ω) will be identified with functions in W ,q (Ω) havingzero mean. We endow the space ˙ W ,q (Ω) with the norm k u k ˙ W ,q (Ω) := (cid:18)Z Ω |∇ u | q dx (cid:19) q , ˙ u ∈ W ,q (Ω) , then we denote by ˙ W − ,p (Ω) its dual space, equipped with the dual norm. The latter isdefined as usual by k T k ˙ W − ,p (Ω) := sup n h T, ϕ i : ϕ ∈ ˙ W ,q (Ω) , k ϕ k ˙ W ,q = 1 o , where p = q/ ( q − Lemma 2.1.
Let T ∈ ˙ W − ,p (Ω) . Then k T k ˙ W − ,p (Ω) = p p " max ϕ ∈ ˙ W ,q (Ω) |h T, ϕ i| − q Z Ω |∇ ϕ | q dx p . Proof.
For every ϕ ∈ ˙ W ,q (Ω) we have |h T, ϕ i| − q Z Ω |∇ ϕ | q dx ≤ sup λ ≥ (cid:20) λ |h T, ϕ i| − λ q q Z Ω |∇ ϕ | q dx (cid:21) . On the other hand the supremum on the right is readily computed: this corresponds tothe choice λ = |h T, ϕ i| q − (cid:18)Z Ω |∇ ϕ | q dx (cid:19) − q − , LORENZO BRASCO AND MIRCEA PETRACHE which gives sup λ ≥ (cid:20) λ |h T, ϕ i| − λ q q Z Ω |∇ u | q dx (cid:21) = 1 p (cid:18) |h T, ϕ i|k ϕ k ˙ W ,q (cid:19) p . Passing to the supremum over ϕ ∈ ˙ W ,q (Ω) and using the definition of the dual norm weget the thesis. (cid:3) We also denote by E ′ (Ω) the space of distributions of order 1 with (compact) support inΩ. In what follows we tacitly identify this space with the dual of the space C (Ω), endowedwith the norm k ϕ k C (Ω) = sup x ∈ Ω | ϕ ( x ) | + sup x ∈ Ω |∇ ϕ ( x ) | . We denote by ν Ω the outer normal unit vector to ∂ Ω. We have the following characterizationfor the dual space ˙ W − ,p (Ω). Lemma 2.2.
Let p = q/ ( q − . We say that a vector field V ∈ L p (Ω; R N ) and T ∈ E ′ (Ω) satisfies (2.1) − div V = T in Ω , V · ν Ω = 0 on ∂ Ω , if Z Ω ∇ ϕ · V dx = h T, ϕ i , for every ϕ ∈ C (Ω) . If we set E ′ ,p (Ω) = { T ∈ E ′ (Ω) : there exists V ∈ L p (Ω; R N ) satisfying (2.1) } , we then have the identification ˙ W − ,p (Ω) = E ′ ,p (Ω) . Proof.
Let T ∈ ˙ W − ,p (Ω). We observe that then T ∈ E ′ (Ω) as well. Now consider thefollowing maximization problemsup v ∈ ˙ W ,q (Ω) h T, v i − q Z Ω |∇ v | q dx. By means of the Direct Methods, it is not difficult to see that there exists a (unique)maximizer u ∈ ˙ W ,q (Ω) for this problem. Moreover such a maximizer satisfies the relevantEuler-Lagrange equation given by Z Ω |∇ u | q − ∇ u · ∇ ϕ dx = h T, ϕ i , for every ϕ ∈ ˙ W ,q (Ω) . By taking V = |∇ u | q − ∇ u ∈ L p (Ω; R N ), the previous identity implies T ∈ E ′ ,p (Ω).Conversely let us take T ∈ E ′ ,p (Ω). Then for every ϕ ∈ C (Ω) equation (2.1) implies |h T, ϕ i| = (cid:12)(cid:12)(cid:12)(cid:12)Z Ω ∇ ϕ · V dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ k ϕ k ˙ W ,q k V k L p (Ω) . CONTINUOUS MODEL OF TRANSPORTATION REVISITED 7
Using the density of C (Ω) in ˙ W ,q (Ω) we obtain that T can be extended in a unique wayas an element (that we still denote T for simplicity) of ˙ W − ,q (Ω). This extension satisfies k T k ˙ W − ,p (Ω) ≤ k V k L p (Ω) , as can be seen by taking the supremum in the previous inequality. (cid:3) Remark 2.3.
We remark that the elements of E ′ ,p (Ω) have “zero average” i.e. h T, i = 0 , as follows by testing the weak formulation of (2.1) with ϕ ≡
1. This is coherent with theprevious identification ˙ W − ,p (Ω) = E ′ ,p (Ω) since by construction the space ˙ W ,q (Ω) doesnot contain any non trivial constant function. Example 2.4.
Consider the measure T = δ a − δ b for two points a = b ∈ R N . We claimthat T = δ a − δ b ∈ ˙ W − ,p (Ω) if and only if 1 ≤ p < N/ ( N − , where Ω is a sufficiently large ball containing a, b in its interior. We prove this by usingthe characterization of Lemma 2.2.Suppose indeed that there exists some V ∈ L p (Ω) such that − div V = T . We pick a ball B r ( a ) centered at a and having radius r such that 2 r < | a − b | . Then for each ε < r weconsider a C ( B r ( a )) function η ε such that η ε ≡ B r − ε ( a ) and k∇ η ε k L ∞ ≤ C ε − . Thanks to our assumption we have1 = h T, η ε i = Z B r ( a ) V · ∇ η ε dx, so that Z B r ( a ) \ B r − ε ( a ) | V | dx ≥ εC . By H¨older inequality this easily implies a lower bound on the L p norm of V , namely Z B r ( a ) | V | p dx ≥ ε p | B r ( a ) \ B r − ε ( a ) | − p = C N,p ε p r N (1 − p ) (cid:20) − (cid:16) − εr (cid:17) N (cid:21) − p . We now make the choice ε = r/
2, so that from the previous we can infer Z B r ( a ) | V | p dx ≥ e C N,p r p + N (1 − p ) = e C N,p r N − p ( N − . The previous estimate clearly contradicts the assumption V ∈ L p (Ω) if the exponent N − p ( N −
1) is not strictly positive. Therefore we see by Lemma 2.2 that p < N/ ( N −
1) is anecessary condition for T ∈ ˙ W − ,p (Ω).This condition on p is also sufficient for T to belong to ˙ W − ,p (Ω), as we now proceed toshow. Set 2 τ = | a − b | and for simplicity assume that a = ( − τ, , . . . ,
0) and b = ( τ, , . . . , LORENZO BRASCO AND MIRCEA PETRACHE
We use the notation x = ( x , x ′ ) for a generic point in R N , where x ′ ∈ R N − . We definethe following vector field V a,b ( x ) = ( x + τ, x ′ )( x + τ ) N , if | x ′ | ≤ τ and | x ′ | − τ ≤ x ≤ , ( x − τ, x ′ )( x − τ ) N , if | x ′ | ≤ τ and τ − | x ′ | ≥ x ≥ , (0 , . . . , , otherwise . It is easily seen that div V a,b = δ a − δ b and that V a,b is supported on the set D a,b = (cid:26) ( x , x ′ ) ∈ R N : | a − b | ≥ | x ′ | + | x | (cid:27) , which is just the the union of two cones centered at a and b having opening 1 and height τ = | a − b | /
2. By construction we have Z Q a,b | V a,b | p dx = 2 Z − τ Z { x ′ : | x ′ | = x + τ } (cid:16)p ( x + τ ) + | x ′ | (cid:17) p ( x + τ ) Np dx ′ dx = 2 p +22 N ω N Z − τ ( x + τ ) − Np + p + N − dx so that finally k V a,b k pL p ≤ C N,p | a − b | N − p ( N − , thanks to our assumption p < N/ ( N − Proposition 2.5.
Let < p < ∞ . Let H : Ω × R N be a Carath´eodory function such that z
7→ H ( x, z ) is convex on R N for every x ∈ Ω . We further suppose that H satisfies thegrowth conditions (2.2) λ ( | z | p − ≤ H ( x, z ) ≤ λ ( | z | p + 1) , ( x, z ) ∈ Ω × R N for some < λ ≤ . Then the following problem (2.3) min V ∈ L p (Ω; R N ) (cid:26)Z Ω H ( x, V ) dx : − div V = T, V · ν Ω = 0 (cid:27) admits a minimizer with finite energy if and only if T ∈ ˙ W − ,p (Ω) .Proof. Let T ∈ ˙ W − ,p (Ω). Thanks to Lemma 2.2 there exists at least one admissible vectorfield V with finite energy, thus the infimum (2.3) is finite. If { V n } n ∈ N ⊂ L p (Ω; R N ) is aminimizing sequence then the hypothesis (2.2) on H guarantees that this sequence is weakly CONTINUOUS MODEL OF TRANSPORTATION REVISITED 9 convergent to some e V ∈ L p (Ω; R N ). Thanks to the convexity of H the functional is weaklylower semicontinuous, i.e. Z Ω H ( x, e V ) dx ≤ lim inf n →∞ Z Ω H ( x, V n ) dx = min V ∈ L p (Ω; R N ) (cid:26)Z Ω H ( x, V ) dx : − div V = T,V · ν Ω = 0 (cid:27) . Moreover the vector field e V is still admissible since Z Ω ∇ ϕ · e V dx = lim n →∞ Z Ω ∇ ϕ · V n dx = h T, ϕ i , for every ϕ ∈ C (Ω) , by weak convergence. Therefore e V realizes the minimum.On the other hand suppose that T ˙ W − ,p (Ω). Again thanks to Lemma 2.2 we have thatthe set of admissible vector fields is empty so the problem is not well-posed. (cid:3) We need the following definition.
Definition 2.6.
We say that a vector field V ∈ L (Ω; R N ) is acyclic if whenever we canwrite V = V + V with | V | = | V | + | V | and div V = 0 with homogeneous Neumannboundary condition, namely Z Ω V · ∇ ϕ dx = 0 , for every ϕ ∈ C (Ω) , there must result V ≡ isotropic case i.e.when H depends on the variable z only through its modulus. This becomes crucial in orderto equivalently reformulate (2.3) as a Lagrangian problem, where the transport is describedby measures on paths. Proposition 2.7.
Assume that H satisfies the hypotheses of Proposition 2.5. In additionassume that z
7→ H ( x, z ) is a strictly convex increasing function of | z | for every x. Then there exists a unique minimizer V for (2.3) and V is acyclic.Proof. The uniqueness of V follows by strict convexity. We now prove that V is acyclic.Suppose that we can write V = V + V for some vector fields V , V ∈ L (Ω; R N ) such that | V | = | V | + | V | and div V = 0 . It follows that div V = div V and | V | ≥ | V | . Thus V is a competitor for problem (2.3)with energy not larger than that of V thanks to the monotonicity of H . Since V is theunique minimizer, it must have energy equal to that of V . Thus | V | = | V | and | V | = 0almost everywhere. This shows that V is acyclic, concluding the proof. (cid:3) Duality for Beckmann’s problem
We need the following general convex duality result (for the proof the reader is referredto [17, Proposition 5, page 89]). The statement has been slightly simplified in order to bedirectly adapted to our setting.
Convex duality.
Let F : Y → R be a convex lower semicontinuous functional on thereflexive Banach space Y . Let X be another reflexive Banach space and A : X → Y abounded linear operator, with adjoint operator A ∗ : Y ∗ → X ∗ . Then we have (3.1) sup x ∈ X h x ∗ , x i − F ( A x ) = inf y ∗ ∈ Y ∗ {F ∗ ( y ∗ ) : A ∗ y ∗ = x ∗ } , x ∗ ∈ X ∗ , where F ∗ : Y ∗ → R ∪{ + ∞} denotes the Legendre-Fenchel transform of F . If the supremumin (3.1) is attained at some x ∈ X then the infimum in (3.1) is attained as well by a y ∗ ∈ Y ∗ such that y ∗ ∈ ∂ F ( A x ) . Thanks to the above result we obtain that Beckmann’s problem admits a dual formula-tion which is a classical elliptic problem in Calculus of Variations.
Theorem 3.1 (Duality) . Let < p < ∞ and q = p/ ( p − . Let H be a function satisfyingthe hypotheses of Proposition 2.5 and T ∈ ˙ W − ,p (Ω) . Then min V ∈ L p (Ω; R N ) (cid:26)Z Ω H ( x, V ) dx : − div V = T,V · ν Ω = 0 (cid:27) = max v ∈ ˙ W ,q (Ω) (cid:26) h T, v i − Z Ω H ∗ ( x, ∇ v ) dx (cid:27) , (3.2) where H ∗ is the partial Legendre-Fenchel transform of H , i.e. H ∗ ( x, ξ ) = sup z ∈ R N ξ · z − H ( x, z ) , x ∈ Ω , ξ ∈ R N . Moreover if V ∈ L p (Ω) and v ∈ ˙ W ,q (Ω) are two optimizers for the problems in (3.2) then we have the following primal-dual optimality condition (3.3) V ∈ ∂ H ∗ ( x, ∇ v ) in Ω , where ∂ H ∗ denotes the subgradient with respect to the ξ variable, i.e. ∂ H ∗ ( x, ξ ) = { z ∈ R N : H ∗ ( x, ξ ) + H ( x, z ) = ξ · z } , x ∈ Ω . Proof.
To prove (3.2) it is sufficient to apply the previous result with the choices Y = L q (Ω; R N ) , X = ˙ W ,q (Ω) , F ( φ ) = Z Ω H ∗ ( x, φ ( x )) dx and A ( ϕ ) = ∇ ϕ. The operator A is bounded since k A ( ϕ ) k Y = k∇ ϕ k L q (Ω) = k ϕ k X , for every ϕ ∈ X, CONTINUOUS MODEL OF TRANSPORTATION REVISITED 11 and F ∗ ( ξ ) = Z Ω H ∗∗ ( x, ξ ( x )) dx = Z Ω H ( x, ξ ( x )) dx, since ξ
7→ H ( x, ξ ) is convex and lower semicontinuous, for every x ∈ Ω. We only needto compute the adjoint operator A ∗ : L p (Ω; R N ) → ˙ W − ,p (Ω). Let us define the mapΨ : L p (Ω; R N ) → E ′ ,p (Ω) byΨ( V ) ∈ E ′ (Ω) such that h Ψ( V ) , ϕ i = Z Ω ∇ ϕ · V dx, for every ϕ ∈ C (Ω) . Observe that Ψ is a linear operator whose image is contained in E ′ ,p (Ω) = ˙ W − ,p (Ω) byconstruction and by the definition of E ′ ,p (Ω). Moreover for ϕ ∈ C (Ω) and V ∈ L p (Ω; R N )we have h A ϕ, V i = Z Ω ∇ ϕ · V dx = h ϕ, Ψ( V ) i . By density of C (Ω) in W ,q (Ω) we obtain that Ψ = A ∗ , thus (3.2) follows from (3.1).The primal-dual optimality condition (3.3) is a direct consequence of the second part ofthe convex duality result as well. It is sufficient to observe that the maximum in (3.2) isattained at some v ∈ ˙ W ,p (Ω) by the Direct Methods. Thus, by the above convex dualitytheorem, a minimizer V of Beckmann’s problem has to satisfy V ∈ ∂ F ( ∇ v ) , which implies directly (3.3). (cid:3) A significant instance of the previous result corresponds to H ( x, z ) = | z | p . Thanks toLemma 2.1 we have the following result. Corollary 3.2.
For every T ∈ ˙ W − ,p (Ω) we have k T k ˙ W − ,p (Ω) = min V ∈ L p (Ω; R N ) n k V k L p (Ω) : − div V = T, V · ν Ω = 0 o . Proof.
It is sufficient to use (3.2) and Lemma 2.1 and to observe thatmax ϕ ∈ ˙ W ,q (Ω) |h T, ϕ i| − q Z Ω |∇ ϕ | q dx = max ϕ ∈ ˙ W ,q (Ω) h T, ϕ i − q Z Ω |∇ ϕ | q dx. This establishes the thesis. (cid:3)
Corollary 3.3.
Under the hypotheses of Theorem 3.1 we have that the functional F H : ˙ W − ,p (Ω) → R + T minimal value (2.3) is convex and weakly lower semicontinuous. Proof.
It is sufficient to observe that thanks to Theorem 3.1 the value (2.3) can be writtenas a supremum of the affine continuous functionals L ϕ defined by L ϕ ( T ) = h T, ϕ i − Z Ω H ∗ ( x, ∇ ϕ ) dx, ϕ ∈ ˙ W ,q (Ω) . Then the thesis follows. (cid:3)
Some comments are in order about the duality result of Theorem 3.1.
Remark 3.4 (Economic interpretation) . By the so-called
Legendre reciprocity formula inConvex Analysis the primal-dual optimality condition (3.3) can be equivalently written as(3.4) ∇ v ∈ ∂ H ( x, V ) , in Ω , so this result is the rigorous justification of the necessary optimality conditions derivedin [2, Lemma 2]. Such v is called a Beckmann potential and its economic interpretationis that of an efficiency price, i.e. it represents a system price for moving commodities inthe most efficient regime for a transport company. It can be seen as a generalization of aKantorovich potential to a situation where the cost to move some unit of mass from x to y is not fixed. Indeed it depends on the quantity of traffic generated by the transport V itself. Heuristically observe that in this case the minimal cost is given by the “congestedmetric” d V ( x , x ) = min γ : γ ( i )= x i Z (cid:12)(cid:12) ∇H ( · , V ) ◦ γ | | γ ′ ( t ) (cid:12)(cid:12) dt. In other words each mass particle is charged for the marginal cost it produces, the latterbeing the derivative of the function H (we suppose for simplicity that H possesses a truegradient and not just a subgradient). Then v acts as a Kantorovich potential for theOptimal Transport problemmin (cid:26)Z Ω × Ω d V ( x, y ) dη ( x, y ) : ( π x ) η = T + and ( π y ) η = T − (cid:27) , where we assume for simplicity that T = T + − T − , with T + and T − positive measureshaving equal total masses. It should be remarked that ∇ ϕ does not give the direction ofoptimal transportation in Beckmann’s problem since ∇ ϕ and V are only linked throughthe relation (3.4) and they are not parallel in general. They are guaranteed to be parallelonly when the cost function H is isotropic , i.e. when it just depends on | V | for everyadmissible vector field V . This is the case studied by Beckmann in his original paper [2]. Remark 3.5 (Regularity of optimal vector fields) . We point out that if z
7→ H ( x, z ) isstrictly convex then ξ
7→ H ∗ ( x, ξ ) is C . In this case the optimal V is unique and we have V = ∇H ∗ ( x, ∇ v ) . Then the regularity of the optimal vector field V can be recovered from the regularity ofa Beckmann potential, which solves the following elliptic boundary value problem(3.5) (cid:26) − div ∇H ∗ ( x, ∇ u ) = T, in Ω ∇H ∗ ( x, ∇ u ) · ν Ω = 0 , on ∂ Ω . CONTINUOUS MODEL OF TRANSPORTATION REVISITED 13
For instance if H ∗ in uniformly convex “at infinity”, meaning that there exist C , C , M > C (1 + | z | ) q − ≤ min | ξ | =1 h D H ∗ ( x, z ) ξ, ξ i , for every | z | ≥ M, x ∈ Ω , and such that | D H ∗ ( x, z ) | ≤ C (1 + | z | ) q − , ( x, z ) ∈ Ω × R N , then V is bounded provided that T ∈ L N + ε (Ω), with ε >
0. Indeed, in this case solutionsto (3.5) are Lipschitz. These assumptions are verified for example by (see [10]) H ∗ ( z ) = 1 q ( | z | − δ ) q + , z ∈ R N , where ( · ) + stands for the positive part and where we assume δ ≥
0, but they are violatedby anisotropic functions of the type H ∗ ( z ) = N X i =1 q ( | z i | − δ i ) q + , z = ( z , . . . , z N ) ∈ R N , considered for example in [8, 9].4. A Lagrangian reformulation
The aim of this section is to introduce a Lagrangian counterpart of Beckmann’s modeland to show how the two models turn out to be equivalent. The model we are going topresent is a continuous version of a classical discrete model on networks by Wardrop (see[40]). This continuous model has already been addressed in [12] and the equivalence hasbeen discussed in [10]. We prove well-posedness of the Lagrangian problem and equivalenceof the models by imposing in addition that the datum T is a finite measure belonging to˙ W − ,p (Ω). The proofs use Smirnov’s decomposition theorem for 1 − currents (see TheoremA.14).Given two Lipschitz curves γ , γ : [0 , → Ω we say that they are equivalent if there existsa continuous surjective nondecreasing function t : [0 , → [0 ,
1] such that γ ( t ) = γ ( t ( t )) , for every t ∈ [0 , . We call L (Ω) the set of all equivalence classes of Lipschitz paths in Ω. We introduce atopology on this set by defining the following distance d ( γ , γ ) := max {| ˆ γ ( t ) − ˆ γ ( t ) | : t ∈ [0 , , ˆ γ i equivalent to γ i } . Observe that convergence in this metric is nothing but the usual uniform convergence, upto reparameterizations.We denote the class of finite positive Borel (with respect to the above topology) measureson L (Ω) by M + ( L (Ω)). For Q ∈ M + ( L (Ω)). We define the corresponding traffic intensity by h i Q , ϕ i := Z L (Ω) (cid:18)Z ϕ ( γ ( t )) | γ ′ ( t ) | dt (cid:19) dQ ( γ ) , ϕ ∈ C (Ω) , provided that the outer integral converges, in which case we say that “the traffic intensity i Q exists”. If this is the case then the following integral also converges: h i Q , ϕ i = Z L (Ω) (cid:18)Z ϕ ( γ ( t )) · γ ′ ( t ) dt (cid:19) dQ ( γ ) , ϕ ∈ C (Ω; R N ) . These definitions do not depend on the particular representative of the equivalence classwe chosen, since the integrals in brackets are invariant under time reparameterization.
Remark 4.1.
Observe that i Q counts in a scalar way the traffic generated by Q while i Q computes it in a vectorial way. This means that in principle i Q and | i Q | could be verydifferent: in i Q two huge amounts of mass going in opposite direction give rise to a lotof cancellations, as the orientation of curves is taken into account. As a simple examplesuppose to have two distinct points x = x and consider the measure Q = 12 δ γ + 12 δ γ , with γ ( t ) = (1 − t ) x + t x and γ ( t ) = (1 − t ) x + t x . By computing the traffic intensitywe obtain i Q = H x x x , which takes into account the intuitive fact that on the segment x x globally there is a nonnegligible amount of transiting mass. On the other hand it is easily seen that i Q ≡ . Given a Radon measure T on Ω we define the following space Q p ( T ) := n Q ∈ M + ( L (Ω)) : i Q ∈ L p (Ω) and ( e − e ) Q = T o , where e i : L (Ω) → Ω is defined by e i ( γ ) = γ ( i ), for i = 0 ,
1. Now consider a Carath´eodoryfunction H : Ω × R + → R + such that(4.1) λ ( t p − ≤ H ( x, t ) ≤ λ ( t p + 1) , x ∈ Ω , t ∈ R + for some 0 < λ ≤ t
7→ H ( x, t ) is convex, for every x ∈ Ω . If Q p ( T ) = ∅ then we define the following minimization problem:(4.2) inf Q ∈Q p ( T ) Z Ω H ( x, i Q ( x )) dx. Remark 4.2.
Similar Lagrangian formulations have been studied in connection with trans-port problems involving concave costs , e.g. problems where to move a mass m of a length ℓ costs m α ℓ (0 < α < CONTINUOUS MODEL OF TRANSPORTATION REVISITED 15
Proposition 4.3.
Let ≤ p ≤ ∞ . Assume that V ∈ L p (Ω , R N ) and that it is acyclic. Let T = − div V be a Radon measure on Ω . It is then possible to find Q ∈ M + ( L (Ω)) such that ( e ) Q = T − and ( e ) Q = T + . Moreover we have i Q = V and i Q = | V | . In particular Q ∈ Q p ( T ) . Thanks to the above result we can prove the following.
Proposition 4.4.
Let T be a Radon measure on Ω . The set Q p ( T ) is not empty if andonly if T ∈ ˙ W − ,p (Ω) .Proof. Let us suppose that T ˙ W − ,p (Ω) and assume by contradiction that there exists Q ∈ Q p ( T ). In particular(4.3) Z Ω | i Q | p dx < + ∞ . The vector measure i Q satisfies (2.1), since Z Ω ∇ ϕ · d i Q = Z L (Ω) (cid:18)Z ∇ ϕ ( γ ( t )) · γ ′ ( t ) dt (cid:19) dQ ( γ )= Z L (Ω) h ϕ ( γ (1)) − ϕ ( γ (0)) i dQ ( γ ) = h T, ϕ i , for every ϕ ∈ C (Ω). Thanks to the fact that | i Q | ≤ i Q and to (4.3) we have that i Q ∈ L p (Ω; R N ). This contradicts the fact that T ˙ W − ,p (Ω), as desired.Now take T ∈ ˙ W − ,p (Ω). Then there exists a minimizer V of problem (2.3) with H ( x, z ) = | z | p . Thanks to Proposition 2.7 we know that V is acyclic. Since T is a Radon measurewe can apply Proposition 4.3 and we infer the existence of Q ∈ Q p ( T ). This gives directlythe thesis. (cid:3) We now prove our equivalence statement, which is the main result of this section. Ob-serve that we prove at the same time existence of a minimizer for (4.2).
Theorem 4.5.
Let H : Ω × R + → R + be a Carath´eodory function satisfying (4.1) andsuch that t
7→ H ( x, t ) is strictly convex and increasing , x ∈ Ω . If T is a Radon measure belonging to ˙ W − ,p (Ω) then we have (4.4) inf Q ∈Q p ( T ) Z Ω H ( x, i Q ) dx = min V ∈ L p (Ω; R N ) (cid:26)Z Ω H ( x, | V | ) dx : − div V = T,V · ν Ω = 0 (cid:27) and the infimum on the left-hand side is achieved.Moreover, if Q ∈ Q p ( T ) is optimal then i Q ∈ L p (Ω; R N ) is a minimizer of Beckmann’sproblem. Conversely, if V is optimal then there exists Q V ∈ Q p ( T ) such that i Q V = | i Q V | minimizes the Lagrangian problem. Proof.
By the previous result the set Q p ( T ) is not empty. For every admissible Q we have | i Q | ≤ i Q , therefore i Q is admissible for Beckmann’s problem. Using the monotonicity of H ( x, · ) we then obtainmin V ∈ L p (Ω; R N ) (cid:26)Z Ω H ( x, | V | ) dx : − div V = T,V · ν Ω = 0 (cid:27) ≤ inf Q ∈Q p ( T ) Z Ω H ( x, i Q ( x )) dx < + ∞ . Now let V ∈ L p (Ω; R N ) be a minimizer for Beckmann’s problem. By Proposition 2.7 V is acyclic. Thus by Proposition 4.3 there exists Q ∈ Q p ( T ) such that | V | = i Q , i.e.min V ∈ L p (Ω; R N ) (cid:26)Z Ω H ( x, | V | ) dx : − div V = T,V · ν Ω = 0 (cid:27) = Z Ω H ( x, i Q ( x )) dx. This shows that (4.4) holds true and that the infimum in the left-hand side is indeed aminimum.The relation between minimizers of the two problems is an easy consequence of theprevious constructions. (cid:3)
Appendix A. Decompositions of acyclic − currents A.1.
Definitions and links to vector fields.
The classical references which we use forcurrents are [19, 21]. We translate however all results in the language of the previoussections. In what follows, by Ω ⊂ R N we still denote the closure of an open boundedconnected set having smooth boundary. Definition A.1.
A 0 − current on Ω is a distribution on Ω in the usual sense. A 1 − current on Ω is a vector valued distribution on Ω. The relevant duality is the one with 1-forms ω ( x ) = P Ni =1 ω i ( x ) dx i having smooth coefficients, i.e. ω i ∈ C ∞ (Ω). We denote by C ∞ (Ω , ∧ R N ) the space of such forms. More generally a k − current is an element in thedual of smooth k − forms C ∞ (Ω , ∧ k R N ).The above definition automatically gives the space of currents a natural weak topology,defined via the duality with smooth forms. For any current there is a natural definition ofboundary. Definition A.2. If I is a k − current on Ω then we can define its boundary ∂I to be the( k − h ∂I, ϕ i = −h I, dϕ i , for all ϕ ∈ C ∞ (Ω , ∧ k − R N ) , where d is the exterior derivative. For example if k = 1 we must take ϕ ∈ C ∞ (Ω) and dϕ is the 1-form P i ∂ x i ϕ dx i . Definition A.3.
Let I be a k − current. The mass of I is defined as M ( I ) = sup (cid:26) |h I, ω i| : ω ∈ C ∞ (Ω , ∧ k R N ) , sup x ∈ Ω k ω ( x ) k ≤ (cid:27) , where the norm k ω k for an alternating k -tensor ω is defined as k ω k = sup {h ω, e i : e unit simple k − vector } . CONTINUOUS MODEL OF TRANSPORTATION REVISITED 17
For k = 1 this coincides with the usual norm k ω k = q ω + . . . + ω N .We use just 1 − currents which are distributions of order 0 (i.e. vector valued Radonmeasures) and in their boundaries (which are scalar distributions of order 1). On thispoint a comment is in order. Remark A.4.
Finite mass 1 − currents can be identified with vector-valued Radon mea-sures as follows. To every smooth 1 − form ω we may associate naturally a vector field X ω := ( ω , . . . , ω N ). We can then write for a 1 − current I Z X ω dI := h I, ω i . Since C ∞ is dense in C the resulting linear functional on smooth vector fields can beidentified via Hahn-Banach theorem to a unique linear functional on C vector fields. Thelatter is indeed a vector-valued Radon measure by Riesz representation theorem. Since k ω ( x ) k = k X ω ( x ) k by the above definition, we automatically obtain that M ( I ) = Z Ω d | I | , i.e. the mass equals the total variation of I regarded as a Radon measure. The samereasoning can be applied to 0 − currents of finite mass, by identifying them with scalarRadon measures . Definition A.5 (variation of a current) . Let A be a k − current with k ∈ { , } . Then wemay define the variation measure µ A of A in the usual sense by identifying A with a Radonmeasure as in Remark A.4. Thus for a Borel set E we define µ A ( E ) := sup ( k X i =1 (cid:12)(cid:12)(cid:12)(cid:12)Z E i dA (cid:12)(cid:12)(cid:12)(cid:12) : E i form a Borel partition of E ) . An equivalent way of defining µ A would be as the infimum of all measures µ such that h A, ω i ≤ R Ω k ω k dµ for all smooth 1-forms ω .We recall that a k − current T is said to be normal if M ( T ) + M ( ∂T ) < + ∞ . We now define flat currents , a class useful for its closure properties.
Definition A.6.
We define the flat norm of a k − current A as follows F ( A ) = inf { M ( A − ∂I ) + M ( I ) : I is a ( k + 1) − current with M ( I ) < ∞} . Then the space of flat k − currents is defined as the completion of normal k − currents inthe flat norm. See also “
Distributions representable by integration ” in [19, 4.1.7] For k = 0 we define ∂A = 0 and thus the condition on ∂A can be omitted. Flat currents of finite mass have the following characterization, which will be exploitedin the sequel.
Lemma A.7.
Let T be a k − current of finite mass. Then T is flat if and only if thereexists a sequence of normal k − currents { T n } n ∈ N such that lim n →∞ M ( T n − T ) = 0 . Proof.
This is a standard fact but we provide a proof for the sake of completeness. Bydefinition of flat convergence there exists a sequence { I n } n ∈ N of normal k − currents and asequence { Y n } n ∈ N of ( k + 1) − currents such thatlim n →∞ (cid:20) M ( T − I n − ∂Y n ) + M ( Y n ) − n (cid:21) ≤ lim n →∞ F ( T − I n ) = 0 . Then we set T n = I n + ∂Y n , which by construction is a k − current andlim n →∞ M ( T − T n ) = 0 , thanks to the previous estimate. To prove that T n is a normal current we first write M ( T n ) ≤ M ( I n ) + M ( ∂Y n ) ≤ M ( I n ) + M ( T − I n − ∂Y n ) + M ( T ) < + ∞ . Observe that we used that T has finite mass and the triangular inequality. Secondly wenote that M ( ∂T n ) = M ( ∂I n ) < + ∞ , since ∂ ( ∂Y n ) = 0.The converse implication is simpler: by definition of flat norm we have F ( T − T n ) ≤ M ( T − T n ) . This concludes the proof. (cid:3)
A significant instance of flat 1 − currents with finite mass is given by L vector fields. Lemma A.8.
Given V ∈ L (Ω; R N ) we naturally associate to it the − current I V of finitemass defined by h I V , ω i = N X i =1 Z Ω ω i V i dx := Z Ω ω ( V ) for every ω ∈ C ∞ (Ω , ∧ R N ) . This current has compact support contained in Ω and M ( I V ) = k V k L . Moreover I V is aflat current.Proof. We just prove that I V is a flat current, the first statement being straightforward.To this aim we use the characterization of Lemma A.7 and we construct the approximatingcurrents by convolution. For every ε ≪ ε = { x ∈ Ω : dist( x, ∂ Ω) > ε } . CONTINUOUS MODEL OF TRANSPORTATION REVISITED 19
Then we take a standard convolution kernel ̺ ∈ C ∞ supported on the ball { x : | x | ≤ } and we define ̺ ε ( x ) = ε − N ̺ (cid:16) xε (cid:17) , x ∈ R N . We also set V ε := ( V · Ω ε ) ∗ ̺ ε , where 1 E stands for the characteristic function of a set E . Define now I ε := I V ε and observethat V ε (and thus I ε ) has compact support contained in Ω. From the mass estimate andby H¨older inequality we obtain that masses are equi-bounded, since M ( I ε ) ≤ k V ε k L ≤ C k V k L . The boundedness of ∂I ε follows via a similar strategy. As V ε has compact support (strictlycontained) in Ω, we have |h ∂I ε , ϕ i| = |h I ε , d ϕ i| = (cid:12)(cid:12)(cid:12)(cid:12)Z Ω V ε · ∇ ϕ dx (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12)Z Ω div V ε ϕ dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ k div V ε k L k ϕ k L ∞ . Setting C ε = k div V ε k L and passing to the supremum on ϕ we obtain M ( ∂I ε ) ≤ C ε < + ∞ . This implies that { I ε } ε> is a sequence of normal currents. The mass convergence M ( I ε − I )easily follows from the convergence of V ε to V in L (Ω; R N ). (cid:3) Remark A.9.
It is easily seen that the boundary ∂I V corresponds to the distributionaldivergence of V i.e. h ∂I V , ϕ i = − Z Ω ∇ ϕ · V dx for every ϕ ∈ C ∞ (Ω) . This distribution has compact support in Ω as well.
Definition A.10.
A 1 − current I is called acyclic if whenever we can write I = I + I ,with M ( I ) = M ( I ) + M ( I ) and ∂I = 0, there must result I = 0.For I = I V with V ∈ L we have the correspondence with Definition 2.6, i.e.(A.1) V is acyclic ⇐⇒ I V is acyclic . A.2.
Lipschitz curves as currents.
We recall that L (Ω) is the space of equivalenceclasses of Lipschitz curves γ : [0 , → Ω (see the beginning of Section 4), with the topologyof uniform convergence.Here we remark that to each γ ∈ L (Ω) we may associate a vector valued distribution i.e.a 1 − current, denoted by [ γ ] and defined by requiring h [ γ ] , ω i := Z γ ω = Z ω ( γ ( t )) [ γ ′ ( t )] dt = N X i =1 Z ω i ( γ ( t )) γ ′ i ( t ) dt, for all ω ∈ C ∞ (Ω , ∧ R N ). Note that this expression is well-defined on L (Ω) since theintegral on the right is invariant under reparameterization.For γ ∈ L (Ω) we have M ([ γ ]) ≤ ℓ ( γ ) := R | γ ′ ( t ) | dt with equality exactly when γ has arepresentative which is injective for H -almost every time. We are interested in a strongerrequirement, namely that the curve does not even intersect itself. In this case an injectiverepresentative exists (such curves are called “arcs” in [27]). We fix a notation for suchclasses of curves. Definition A.11 (Arcs) . We define e L (Ω) the subset of L (Ω) made of those classes ofcurves γ : [0 , → Ω which have an injective representative.
Remark A.12.
One could think of the above-defined arcs as “acyclic curves”, where a“cycle” can mean two things: • we can have a cycle in the parameterization, where a cycle would be represented bya curve satisfying (up to reparameterization) γ ( t ) = γ (1 − t ) and “inserting a cycleof type γ ” in another curve γ such that γ ( t ) = γ (0) would result into the curve: e γ ( s ) = γ (2 s ) if s ∈ [0 , t / γ (2 s − t ) if s ∈ [ t / , ( t + 1) / γ (2 s −
1) if s ∈ [( t + 1) / , . • a curve which intersects itself (i.e. which has no injective parameterization) givesinstead rise to a 1 − current which is not acyclic, since it has a reparameterizationcontaining an injectively parameterized loop.The fact that in the decomposition of acyclic currents one restricts to using just arcs (forwhich neither type of cycle occurs) is then another natural consequence of the robustnessof the acyclicity requirement.On 1 − currents we consider the topology of distributions. The following result links thetwo topologies: Lemma A.13.
The map L (Ω) ∋ γ [ γ ] as defined above is continuous on the sublevelsof the length functional ℓ : L (Ω) → R .Proof. Assuming that γ i → γ and ℓ ( γ i ) ≤ C we then obtain that γ i converge uniformly. Inparticular they converge as distributions. (cid:3) A.3.
Smirnov decomposition theorem.
We can now state the theorem on the decom-position of 1 − currents due to Smirnov [36] and recently extended by Paolini and Stepanovin [27, 28] to metric spaces. Theorem A.14.
Suppose that I is a normal acyclic − current on Ω . Then there exists Q ∈ M + ( L (Ω)) concentrated on e L (Ω) and such that the following decompositions of I arevalid in the sense of distributions: (A.2) I = Z L (Ω) [ γ ] dQ ( γ ) and µ I = Z L (Ω) µ [ γ ] dQ ( γ ) , CONTINUOUS MODEL OF TRANSPORTATION REVISITED 21 (A.3) ∂I = Z L (Ω) ∂ [ γ ] dQ ( γ ) and µ ∂I = Z L (Ω) µ ∂ [ γ ] dQ ( γ ) . We now note down some reformulations of the items present in the above theorem interms of measures and vector fields: • the total variation is the mass norm, i.e. µ I (Ω) = M ( I ) and µ ∂I (Ω) = M ( ∂I ); • if V is a L loc vector field then I V has variation measure µ I V = | V | · L N ; • if ρ = ρ + − ρ − is the decomposition of a signed Radon measure into positive andnegative part then µ ρ = ρ + + ρ − ; • in particular for γ ∈ L (Ω) we have µ ∂ [ γ ] = δ γ (1) + δ γ (0) . Since this measure hastotal variation 2 for all γ we can quantify the total mass of the above Q by meansof the mass norm of the boundary of I . Namely, we have Q ( L (Ω)) = 12 µ ∂I (Ω) = M ( ∂I )2 ; • for γ ∈ e L (Ω) there holds µ [ γ ] = H x Im( γ ), i.e. this is the arclength measure of γ ; • by expanding the definitions and comparing to Section 4 we see that for I = I V with V ∈ L (Ω) and for Q as in Theorem A.14, there holds | V | · L N = µ I V = i Q · L N , V · L N = I = i Q · L N and div V = ∂I V = ( e − e ) Q. All these reformulations allow to translate Theorem A.14 in the case of I = I V , with V ∈ L p (Ω) for p ≥
1. This is the content of the next result.
Corollary A.15 (Reformulation of Theorem A.14) . Suppose that V ∈ L (Ω) is an acyclicvector field such that div V is a Radon measure. Then there exists Q ∈ M + ( L (Ω)) concen-trated on e L (Ω) and such that the following decompositions of V are valid in the sense ofdistributions: (A.4) i Q = V and i Q = | V | , (A.5) − div V = ( e − e ) Q and ( − div V ) + + ( − div V ) − = ( e + e ) Q. A.4.
The case of flat currents.
In Section 4 we required T ∈ ˙ W − ,p (Ω) to be a Radonmeasure. As already mentioned, this further hypothesis permits to identify optimal vectorfields for Beckmann’s problems with acyclic normal currents. Then well-posedness andequivalence of the problems can be obtained by means of Smirnov’s Theorem. However inthe setting of Beckmann’s problem and of its dual it would be natural to allow T to be ageneric element of ˙ W − ,p (Ω). If one whishes to extend the analysis of the Lagrangian for-mulation to this larger space then one is naturally lead to consider a possible extension of Smirnov’s result to L vector fields having divergence which is not a Radon measure. Ob-serve that such vector fields correspond to flat currents (see Lemma A.8). In this subsectionwe investigate the possibility to have Smirnov’s Theorem for such a class of currents.We start by observing that the measure Q which decomposes I may not be finite in general. Example A.16.
For 1 ≤ p < NN − we consider an infinite sequence of small dipoles { ( a i , b i ) } i such that ∞ X i =1 | a i − b i | N − p ( N − < + ∞ , and D a i ,b i are disjoint , where the sets D a i ,b i are defined as in Example 2.4. If we consider the vector fields V a i ,b i as in Example 2.4 then the new vector field defined by V = P ∞ i =1 V a i ,b i verifies k V k pL p = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X i =1 V a i ,b i (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) pL p ≤ C N ∞ X i =1 | a i − b i | N − p ( N − < + ∞ , which implies T := P i ( δ a i − δ b i ) ∈ ˙ W − ,p (Ω). By observing that ∞ = M ( T ) = R L (Ω) dQ for any decomposing measure we see that no finite measure Q can be found. On the otherhand a σ − finite measure Q can be found, since each V a i ,b i can be separately decomposedwith a measure Q i of mass 2 and the Q i have disjoint supports. Example A.17.
We present now another version of Example A.16, which exploits theSobolev embedding theorem. Let us take again 1 ≤ p < N/ ( N − q = p/ ( p − >N . Then ˙ W ,q (Ω) can be identified with a space of functions which are H¨older continuousof exponent α = 1 − N/q . We consider the following two curves γ ( t ) = 1 t /α (cos t, sin t ) and γ ( t ) = g ( t ) t /α (cos t, sin t ) , t ≥ , where g : [1 , ∞ ) → R + is a continuous function such that1 > g ( t ) > (cid:18) tt + 2 π (cid:19) /α and t g ( t ) t /α is decreasing . We define the distribution h T, ϕ i = Z ∞ h ϕ ( γ ( t )) − ϕ ( γ ( t )) i dt, ϕ ∈ C ∞ (Ω) . This is an element of ˙ W − ,p (Ω) since by Sobolev embedding [ ϕ ] C ,α ≤ C Ω k ϕ k ˙ W ,q . Thus |h T, ϕ i| ≤ Z ∞ | ϕ ( γ ( t )) − ϕ ( γ ( t )) | dt ≤ C k ϕ k W ,q (Ω) Z ∞ | γ ( t ) − γ ( t ) | α dt = C k ϕ k W ,q (Ω) Z ∞ | − g ( t ) | α t dt ≤ α C k ϕ k W ,q (Ω) , ϕ ∈ ˙ W ,q (Ω) . CONTINUOUS MODEL OF TRANSPORTATION REVISITED 23
We then introduce the measure on paths Q T defined by Q T = Z ∞ δ γ ( t ) γ ( t ) dt, where for t ≥ γ ( t ) γ ( t ) the straight segment going from γ ( t ) to γ ( t ).Observe that for every ϕ we have Z L (Ω) [ ϕ ( γ (0)) − ϕ ( γ (1))] dQ T ( γ ) = Z ∞ h ϕ ( γ ( t )) − ϕ ( γ ( t )) i dt = h T, ϕ i and Z L (Ω) ℓ ( γ ) dQ T ( γ ) = Z ∞ | γ ( t ) − γ ( t ) | dt = Z ∞ | − g ( t ) | t /α dt ≤ αα − t − α (cid:12)(cid:12)(cid:12) ∞ = 2 α − α < ∞ , while Q T is not finite, but just σ -finite.The previous examples clarify that we cannot hope to give a distributional meaning tothe positive and negative parts of the divergence of V . The good definition of div V asa distribution relies in general on some sort of “almost–cancellation”. Therefore the lastno-cancellation requirement ( − div V ) + + ( − div V ) − = ( e + e ) Q of Smirnov’s TheoremA.14 should be relaxed when we try to extend it to a larger class of V ’s.Actually we can say more. For general flat currents even the existence of a (possibly σ -finite) decomposition Q satisfying just (A.2) is not granted, as shown in the next example. Example A.18.
Let Ω = [0 , N ⊂ R N and let us consider a totally disconnected closedset E ⊂ Ω such that L N ( E ) >
0. We then pick a vector ϑ ∈ R N \ { } and set V ( x ) = ϑ · E ( x ) , x ∈ Ω . Of course this is an L ∞ (Ω) vector field and thanks to Lemma A.8 we have that the asso-ciated 1 − current I V is flat. We claim that I V does not admit a Smirnov decomposition .Indeed, assume by contradiction that a decomposition Q ∈ M + ( L (Ω)) satisfying the fol-lowing condition (equivalent to (A.2)) exists:(A.6) µ I V = Z L (Ω) µ [ γ ] dQ ( γ ) . We see that spt( µ I V ) = spt( I V ) and from this we can infer that Q − a.e. curve γ hassupport included in spt( I V ) = E (see also [36, Remark 5]). Indeed, if the latter were nottrue then the supports of the two sides of (A.6) would differ, as follows by observing thatthe left-hand side is a superposition of the positive measures µ [ γ ] .By knowing now that Q − a.e. curve γ has support in the totally disconnected set E andthat the curves γ are connected, we deduce that Q − a.e. curve γ is constant. This impliesthat for Q − a.e. curve γ there holds [ γ ] = 0 . Therefore from (A.6) it follows µ I V = 0, which contradicts the fact that µ I V (Ω) = M ( I V ) = Z Ω | V | dx > . Since the existence of Q satisfying (A.6) leads to a contradiction, we conclude that noSmirnov decomposition of I V exists. Note that whether Q is assumed to be finite or only σ − finite is immaterial for this contradiction.We point out that by defining the distribution T as h T, ϕ i = Z Ω V ( x ) · ∇ ϕ ( x ) dx = Z E ϑ · ∇ ϕ ( x ) dx, for every ϕ ∈ C (Ω) , this can be considered as an element of ˙ W − ,p (Ω) for any 1 < p < ∞ , thanks to Lemma2.2.We also claim that V (and thus I V ) is acyclic . Suppose that we can write V = V + V with | V | = | V | + | V | and div V = 0. This implies that V = λ ϑ · E , where λ ∈ L (Ω)and it takes values in [0 , h− div V , ϕ i = Z E λ ( x ) ϑ · ∇ ϕ ( x ) dx, for every ϕ ∈ C (Ω) . By taking ϕ ( x ) = ϑ · x we observe that the integral is nonzero unless λ ≡ E , inwhich case V = 0. By appealing to Definition 2.6 we eventually prove that V is acyclic. Acknowledgements.
This work started during the conference “Monge-Kantorovich op-timal transportation problem, transport metrics and their applications” held in St. Pe-tersburg in June 2012. The authors wish to thank the organizers for the kind invitation.Guillaume Carlier and Eugene Stepanov are gratefully acknowledged for some commentson a preliminary version of the paper. L.B. has been partially supported by the
AgenceNational de la Recherche through the project ANR-12-BS01-0014-01
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E-mail address : [email protected] M. P. , ETH, Departement Mathematik, R¨amistrasse 101, 8092 Z¨urich, Switzerland
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