Mass transport and geometric inequalities by Felipe Garcı́a Ramos Aguilar B. Sc. Mathematics, Universidad Nacional Autónoma de México, 2008 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Master of Science in THE FACULTY OF GRADUATE STUDIES (Mathematics) The University Of British Columbia (Vancouver) October 2010 c© Felipe Garcı́a Ramos Aguilar, 2010 Abstract In this thesis we will review some recent results of Optimal Mass Transportation emphasizing on the role of displacement interpolation and displacement convexity. We will show some of its recent applications, specially the ones by Bernard, and Agueh-Ghoussoub-Kang. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v Dedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Kantorovich problem . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.0.1 Brenier’s theorem . . . . . . . . . . . . . . . . . . . . . . 5 2.1 Kantorovich-Rubinstein space . . . . . . . . . . . . . . . . . . . 8 3 Time dependent mass transportation . . . . . . . . . . . . . . . . . . 10 3.1 Displacement interpolation . . . . . . . . . . . . . . . . . . . . . 10 3.2 Displacement convexity . . . . . . . . . . . . . . . . . . . . . . . 11 3.3 Benamou-Brenier formula . . . . . . . . . . . . . . . . . . . . . 13 4 Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.1 Young measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2 Transport measures . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.3 Generalized curves . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.4 Tonelli theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 5.1 General Sobolev inequality . . . . . . . . . . . . . . . . . . . . . 27 iii 5.1.1 Euclidian Log Sobolev inequalities . . . . . . . . . . . . 29 5.1.2 Sobolev and Gagliardo-Nirenberg inequalities . . . . . . . 31 5.2 General inequality . . . . . . . . . . . . . . . . . . . . . . . . . . 33 5.2.1 HWI inequalities . . . . . . . . . . . . . . . . . . . . . . 39 5.2.2 Gaussian inequalities . . . . . . . . . . . . . . . . . . . . 41 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 iv Acknowledgments I would like to thank Prof. Young-Heon Kim, for carefully reading this thesis and making very useful observations and explanations; Ramón Zárate and Craig Cowan, for the Functional Analysis, PDE and Mass Transport conversations that helped the completion of this thesis; Ignacio Rozada, who showed me several things, including how to compile this thesis; and Lee Yupitun for all her time and help at the Math Department. This thesis was done under the supervision of Prof. Nassif Ghoussoub, and I was economically supported by a CONACyT fellowship. v Al pollo que llegó, dejó su legado y se fué. ...y al Pollo también. vi Chapter 1 Introduction Mathematics is the art of giving the same name to different things. — J.H. Poincaré The most basic problem of modern mass transportation is the Monge-Kantorovich problem for quadratic cost function, that is, given the measures µ0 and µ1 of Rn find a measure γ0 of Rn×Rn, that satisfies inf γ∈Γ(µ0,µ1) ∫ Rn×Rn |x− y|2 dγ(x,y) = ∫ Rn×Rn |x− y|2 dγ0(x,y), where Γ(µ0,µ1) is the set of measures in Rn×Rn with marginals µ0 and µ1. This actually defines a distance dw(µ0,µ1) in the probability space P(Rn), which we shall denote by Wasserstein distance. In Chapter 1 we will review the basic results of Optimal Mass Transportation. A very good text book for this subject is the one written by Villani [13]. Brenier and Benamou [8] studied a different point of view of the transport prob- lem, one involving time, which is to find a one parameter family of pairs (µt ,Vt) , that minimize inf ∫ 1 0 ∫ |Vt |2 dµtdt, where µt is a measure and Vt a vector field that depends on t and satisfy the conti- nuity equation, or also known as the conservation of mass equation µt +div(Vµ) = 0. 1 In Chapter 2 we will show the proof of Brenier’s theorem, that under some circum- stances both problems have the same infimum or optimal cost. In this chapter we will also review the concepts of displacement convexity and displacement interpo- lation, which are related to the time dependent point of view. The main idea of chapters 3 and 4 is to explain some of the results given by Bernard [1] and Agueh, Ghoussoub and Kang [10] emphasizing on the role of time dependent mass transportation. In Chapter 3 we will explain some basic results in displacement interpolation and displacement convexity. Then we will relax Brenier’s problem by considering instead of classical flows Vt , generalized flows ηt,x(v), which are probability mea- sures that indicate the probability of having velocity v ∈ T M at the point x in time t. To generalize the continuity equation we will define a Transport measure, as a Young measure η that, for a set of test functions g, satisfies∫ I×T M [∂tg+∂xg · v]dηt,x(v)dµ(t,x) = ∫ I×T M [∂tg+∂xg · v]dη(v, t,x) = 0. (1.1) For certain functionals L we will show existence of minimizers of ∫ Ldη , first in the case where η is a generalized curve and, under some conditions, we will see that the infimum coincides with the one taken over the classical curves, hence showing a known result previously proved by Tonelli. Using a simple optimal transport argument and the geometric-arithmetic in- equality one can prove the isoperimetric inequality and Sobolev-Nirenberg in- equalities. With the use of the Monge-Ampere equation it is also possible to prove log-Sobolev inequalities, HWI, Brunn-Minkowski, and many others (see See [3],[2],[11]). An interesting result is that actually many of this inequalities belong or can be obtained from a more general inequality. Using displacement convexity we will first show a general Sobolev inequality, that involves a positive measur- able function ρ that represents density, force F , pressure PF , the internal energy functional H, the Young’s function c∗, and a constant Kc, HF+nPF (ρ)≤ ∫ Ω c∗(−∇F ′(ρ))ρdx+Kc. (1.2) 2 Afterwards we will show that if we choose different forms of the force, we can derive interesting inequalities from this one, like the Log-Sobolev, and the Gagliardo Nirenberg inequality. Also in this chapter we will prove the general Agueh-Ghoussoub-Kang in- equality shown in [10], that involves a Young’s function c and its dual c∗, energy functionals HF,WV , relative entropy production Ic∗(ρ0 | ρV ), Wasserstein distance d2w(ρ0,ρ1) and barycentre b(ρ0), namely HF,WV+c(ρ0 | ρ1)+ λ +ν 2 d2w(ρ0,ρ1)− ν 2 |b(ρ0)−b(ρ1)|2 ≤ H−nPF ,2x·∇Wc+∇V ·x (ρ0)+Ic∗(ρ0 | ρV ). We will show this inequality generalizes (1.2) and other inequalities including the HWI inequality. 3 Chapter 2 Kantorovich problem The original transport problem was proposed by Monge around the 1780´s, the question was how to move given pile of soil into an excavation with the least amount of work. Kantorovich relaxed the problem in terms of probability mea- sures. In this chapter we explain some basic results in this direction, where a basic reference is [13]. Whenever T is map from a measure space (X ,µ) to arbitrary space Y , we can equip Y with the pushforward measure T#µ , where T#µ(B) = µ(T−1(B)), for every set B⊂ Y. We will denote the space of probability measures of X , asP(X) , andPAC(X)⊂ P(X) the space of absolutely continuous probability measures. Let M± ⊂ Rn be two compact sets , and µ± ∈P(M±). We denote the projection functions as pi+ : M+ ×M− 7→ M+, where pi+(x,y) = x, and pi− : M+ ×M− 7→ M−, where pi−(x,y) = y. So we define Γ(µ+,µ−) = { γ ∈P(M+×M−) | (pi+)#γ = µ+ and (pi−)#γ = µ− } . Let c : M+×M− 7→R, be a continuous cost function. The Kantorovich problem is to minimize the total cost defined as C (γ) = ∫ M+×M− c(x,y)dγ, where γ ∈ Γ(µ+,µ−) . As we shall see the existence of minimizers is not hard, as we have that C is linear 4 in γ and we can use the Banach-Alaouglu theorem. Theorem 2.1 There exists a minimizer for the Kantorovich problem. Proof. Since the space of probability measures is contained in the unit ball of the dual space, which is weak* compact. Since Γ(µ+,µ−) is closed, it is weak-* compact. Since C (γ) is continuous in the weak-* topology, it attains a minimum in the compact set. We will say γ is optimal if γ ∈ Γop(µ+,µ−) := { γ ∈ Γ(µ+,µ−) | C (γ) = inf γ∈Γ(µ+,µ−) ∫ M+×M− c(x,y)dγ } . 2.0.1 Brenier’s theorem First we state two definitions and their relationship. Definition 2.2 Given a set Γ ⊂ X ×Y, and a cost function c(x,y), we say that Γ is c-cyclically monotone if for any finite set of pairs {(xi,yi) | 1≤ i≤ N} ⊂ Γ. we have that N ∑ i c(xi,yi)≤ N ∑ i c(xi,yi+1). (with N+1=1) Definition 2.3 A function φ : X 7→ R∪{+∞} is said to be c-convex if it is not identical to {+∞} , and there exists a function g : Y 7→ R∪{+∞} , such that φ(x) = sup y∈Y (g(y)− c(x,y)) ∀x ∈ X . It´s c-transform is the function φ c(y) := inf x∈X (φ(x)+ c(x,y)), and it’s c-subdifferential is the c-cyclically monotone set ∂ cφ := {(x,y) ∈ X×Y | φ c(y)−φ(x) = c(x,y)} . 5 Remark 2.4 The fact that a set Γ is c-cyclically monotone if and only if Γ = ∂ cφ for a c-convex function φ is called Rockafellers’s theorem, a proof can be found in [13]. Remark 2.5 If we take X =Y =Rn and c(x,y) =−x ·y, the the c-transform is the usual Legendre transform or dual, and c-convexity is just convexity. Lemma 2.6 The support of an optimal mapping γ is c-cyclically monotone. Proof. If supp γ is not c-cyclically monotone then we have that, this means there exists a set of pairs {(xi,yi) | 1≤ i≤ N} ⊂ Γ such that N ∑ i c(xi,yi)− N ∑ i c(xi,yi+1)> 0. Even more, since c is continuous we can find a set of open neighbourhood Ui×Vi, of (xi,yi) such that N ∑ i c(ui,vi)− N ∑ i c(ui,vi+1)> 0 ∀ (ui,vi) ∈Ui×Vi. We define the measures γi(E) = γ(E∩(Ui×Vi))γ(Ui×Vi) ; let η = Πγi a measure of (X ×Y )N and Hi = ( fi,gi ) the projections such that γi = Hi#η . We define γ̃ = γ+ λ n N ∑ i ( fi+1×gi)#η− ( fi×gi)#η , where λ = infγ(Ui×Vi)> 0.We check that (pi+)#γ̃ = µ++ λn ∑Ni ( fi+1)#η−( fi)#η = µ+, similarly (pi−)#γ̃ = µ−, and γ̃(X×Y ) = 1+ λn ∑Ni γ((Ui+1×Vi))γ(Ui+1×Vi) − γ((Ui×Vi)) γ(Ui×Vi) = 1. Finally we compute C (γ)−C (γ̃) = ∫ M+×M−∑c( fi+1,gi)− c( fi,gi)dη > 0 Now we proceed to prove a theorem by Brenier, that applies to quadratic cost functions, i.e. c(x,y) = 12 |x− y|2 . 6 Theorem 2.7 Let M± ⊂ Rn be open and bounded sets, and µ± ∈ PAC(M±). Hence there exists an optimal mapping T#µ+ = µ−, and ϕ convex function such that T = ∇ϕ almost everywhere. Proof. By the previous results, we know there exists an optimal measure γop ∈ Γop(µ+,µ−), let S be a maximal c-cyclical monotone set, hence suppγop ⊂ S . By Rockafellar theorem, there exists a c-convex function φ , such that S = ∂ cφ(x), by definition of c-convexity we have that φ(x) = sup y∈M− {−c(x,y)−φ c(y)}= sup y∈M− { −1 2 |x− y|2−φ c(y) } = sup y∈M− { −1 2 |x|2− 1 2 |y|2+ xy−φ c(y) } . Now we can define a function ϕ, ϕ(x) := φ(x)+ 1 2 |x|2 = sup y∈M− { −1 2 |y|2+ xy−φ c(y) } . This function is convex and bounded and moreover, ∂ϕ(x) = ∂ cφ . This means ϕ is Lipschitz and hence differentiable almost everywhere by Rademacher’s theorem So almost everywhere ∂ϕ(x) = ∇ϕ(x). Monge-Ampere equation Let M± ⊂ Rn be open and bounded sets, and µ± ∈PAC(M±), that is µ+ = f dx and µ− = gdy, from the last theorem there exists an optimal mapping T#µ+ = µ−, and ϕ convex function such that T = ∇ϕ, in other words we have that∫ ψ(y)g(y)dy = ∫ ψ(∇ϕ(x)) f (x)dx for all test functions ψ. 7 Alexandrov’s theorem says that D2ϕ exists almost everywhere if ϕ is convex, so if we do a change of variables y = ∇ ϕ(x) we get that∫ ψ(y)g(y)dy = ∫ ψ(∇ϕ(x))g(∇ϕ(x))detD2ϕdx. From this we get the Monge-Ampere’s equation g(∇ϕ)detD2ϕ = f (x). 2.1 Kantorovich-Rubinstein space Now that we know about the existence of solutions of the Kantorovich problem, we can define a very useful metric. Let Pn(X) be the space of Borel probability measures on µ ∈ X with finite nth moment, which means∫ |d(x0,x)|n dµ(x)< ∞ for some x0 ∈ X , and hence for any x0 ∈ X . Definition 2.8 We can define a distance in the spacePn(X) is defined as follows, dn(µ,η) = ∣∣∣∣minλ ∫ X×X |d(x,y)|n dλ (x,y) ∣∣∣∣1/n . Where λ runs along de probability measures on X×X, whose marginals are µ and η . Particularly for n = 2, we shall call it the Wasserstein distance and we will denote it as dw. We can characterize the topological space P1(X), let´s consider C1(X), the space of functions f : X 7→ R, such that sup x∈X | f (x)| 1+d(x0,x) < ∞ for one (and hence for all) x0 ∈ X . Then we have that d1(µn,µ) 7→ 0 if and only if∫ f dµn 7→ ∫ f dµ 8 for every f ∈C1(X). We also define a weaker topology , called the narrow topology Definition 2.9 We say µn converges narrowly to µ if∫ f dµn 7→ ∫ f dµ for each bounded continuous function f . 9 Chapter 3 Time dependent mass transportation 3.1 Displacement interpolation So far we only minded the starting and ending point of the mass transportation problem, without giving any information of what could happen in the middle. This view point is related to fluid dynamics and has been studied principally by Brenier [8] Definition 3.1 Let µ+,µ− ∈P2(Rn) and γ0 ∈ Γ(µ+,µ−) be the solution to the Kantorovich problem with quadratic cost . For every s ∈ [0,1] we define pis : Rn× Rn→ Rn, as pis(x,y) := (1− s)x+ sy, and we will call µs = (pis)#γ0 the displacement interpolation between µ+ and µ−. As we shall see the solutions of the time dependent minimization problems can be represents as displacement interpolation of two measures. We now prove a result that shows that the displacement interpolation of two measures is a constant speed geodesic. Theorem 3.2 Let µs be the displacement interpolation between µ0 and µ1 then 10 ∀s, t ∈ [0,1] , we have that dw(µt ,µs) = |t− s|dw(µ0,µ1) . Proof. First we take pist = (pis×pit)# γ0 = ((1− s)x+ sy),(1− t)x+ ty)#γ0 ∈ Γ(µs,µt). So that d2w(µt ,µs)≤ ∫ |x− y|2 dpist = ∫ |(1− s)x+ sy− ((1− t)x+ ty)|2 dγ0 = (t− s)2 ∫ |x− y|2 dγ0 = (t− s)2d2w(µ0,µ1). To get the equality use the triangle inequality dw(µ0,µ1)≤ dw(µ0,µs)+dw(µs,µt)+dw(µt ,µ1) ≤ sdw(µ0,µ1)+dw(µs,µt)+(1− t)dw(µ0,µ1). So we conclude that dw(µt ,µs) = |t− s|dw(µ0,µ1) . 3.2 Displacement convexity In this chapter we explain an important concept called displacement convexity, originally due to R. McCann, which inspired a lot of development in Optimal Transportation theory. Definition 3.3 We will say H : dom(H)⊂P2→ R is displacement convex if H(ρs)≤ (1− s)H(ρ0)+ sH(ρ1), for all ρs displacement interpolation of ρ0 and ρ1 ∈ dom(H). 11 Lemma 3.4 Suppose h:(0,∞)→R∪{∞} is convex and non increasing, and g:[0,1]→(0,∞) is concave. Then h◦g will be convex. Proof. Let s, t0, t1 ∈ [0,1] , then h◦g((1− s)t0+ st1)≤ h((1− s)g(t0)+ sg(t1)) ≤ (1− s)h◦g(t0)+ sh◦g(t1). Definition 3.5 Let F : [0,∞)→Rn differentiable, then we can define the associated Internal Energy Functional as HF(ρ) := ∫ Ω F(ρ(x))dx . Proposition 3.6 Let HF be the internal energy functional. If we suppose F : [0,∞)→ Rn is differentiable with F(0) = 0, and x 7→ xnF( rxn ) is convex and non increasing for all r > 0, then HF is displacement convex. Proof. Let ρ0 and ρ1 ∈P2,ac, , ∇ψ be the optimal mapping, and the displacement interpolation of ρ0 and ρ1. Since Supp(ρs) = Supp(ρ0)((1− s)I+ s∇ψ) we have that HF(ρs) = ∫ Ω F(ρs(x))dx = ∫ Ω∩Supp(ρs) F(ρs(x))dx = ∫ Ω∩Supp(ρ0) F(ρs(1− s)x+ s∇ψ(x))det((1− s)I+ s∇ψ)dx. Using the Monge-Ampere formula, and defining λ (s) = det((1− s)I+ s∇ψ) 1n we can conclude that HF(ρs) = ∫ Ω∩Supp(ρ0) F( ρ0(x) det((1− s)I+ s∇ψ))det((1− s)I+ s∇ψ)dx = ∫ Ω∩Supp(ρ0) F( ρ0(x) λ n )λ ndx 12 Using the fact that λ (s) is concave, the remark and the lemma, we see that s 7→ F(ρ0(x)λ n )λ n is convex, this means HF(ρs)≤ (1− s)HF(ρ0)+ sHF(ρ1). 3.3 Benamou-Brenier formula In physics if we have a density µt and a vector field V ,and we assume the mass is conserved, then the density must satisfy the continuity equation. Inspired on this we have the following definition. Definition 3.7 We will call (µt ,Vt), an admissible pair if ·t→ µt is weak* continuous ·t→ ∫ |x|dµt is continuous ·∫ ‖V (t,x)‖2 dµtdt < ∞. ·∂tµ+∇ · (µV ) = 0 in a weak sense. Theorem 3.8 Let X be a complete smooth manifold, let µ0 be a probability mea- sure on X. If v is an integrable field, that is, there exists a locally Lipschitz family of diffeomorphisms (Tt)0≤t≤T , such that dTt dt (x) =Vt(T (x)), then (µt ,Vt) is an admissible pairing, where µt = Tt#µ is the unique solution to the continuity equation . Proof. Let ϕ be a test function and t ∈ (0, t), by definition of push-forward we have ∫ ϕdµt = ∫ (ϕ ◦Tt)dµ so for h > 0 we can write 1 h ( ∫ ϕdµt+h− ∫ ϕdµt) = ∫ ϕ ◦Tt+x(x)−ϕ ◦Tt h dµ. 13 Since T−1t is continuous, then ϕ ◦ Tt is Lipschitz and compactly supported uni- formly for t ∈ [0, t] , so the right hand of the equation is uniformly bounded for t ∈ [0, t−h] and for almost all t,x converges point-wise to ∂ ∂ t (ϕ ◦Tt) = (∇ϕ ◦Tt) · ∂∂ t Tt = (∇ϕ ◦Tt) · (vt ◦Tt). By Lebesgue´s dominated convergence theorem we deduce that for almost all t we have d dt ∫ ϕdµt = ∫ (∇ϕ ◦Tt) · (vt ◦Tt) = ∫ ∇ϕ · vtdµt . To prove uniqueness we will prove that if µt satisfies the continuity equation then for any T ∈ [0, t] , if µ0 = 0 then µT = 0. We first assume we can find a Lipschitz compactly supported function ϕ(t,x) that satisfies ∂ϕ ∂ t + vt ·∇ϕ = 0 ϕ |t=T= ϕT . Where ϕT ∈D(X), the space of distribution, so we can compute for almost all t d dt ∫ ϕtdµt = ∫ ∂ϕt ∂ t dµt + ∫ ϕtd( ∂µt ∂ t ) =− ∫ vt ·∇ϕ+ ∫ ϕtd(∇ · vtµt) = 0. Since µ0 = 0, then ∫ ϕT dµT = 0 =⇒ µT = 0. Finally we can check that ϕt = ϕT ◦TT ◦T−1t Lipschitz with compact support, and is a solution of d dt ϕt(Ttx) = ∂ϕ ∂ t + v ·∇ϕ = 0. We will need the following lemma to prove the Benamou-Brenier theorem. 14 Lemma 3.9 Let σ be a measure inRn, f ∈L2(σ), and T a map such that T#( fσ)= hT#(σ) . Then ‖h‖L2(T#σ) ≤ ‖ f‖L2(σ) Proof. Let g ∈ L2(T#σ), computing 〈T#( fσ),g〉= 〈g◦T, fσ〉 ≤ ‖ f‖L2(σ) ‖g◦T‖L2(σ) = ‖ f‖L2(σ) ‖g‖L2(T#σ) . Using Riesz representation theorem we know the continuous linear functional F such that F(g) = 〈T#( fσ),g〉= 〈hT#(σ),g〉 , has norm ‖h‖L2(T#σ) . This means ‖h‖L2(T#σ) ≤ ‖ f‖L2(σ) . The following result has an interesting physical interpretation, as the Wasser- stein distance between two measures can be seen as the infimum of the energy needed to translate one density to the other. Theorem 3.10 (Benamou-Brenier) If µ0, µ1 ∈P2,AC, then we have the equality d2w(µ0,µ1) = infVt ,µt admissible ∫ 1 0 ∫ |Vt |2 dµtdt. Proof. Since we are assuming absolute continuity for µ0 and µ1, we know there is a convex funtion ψ such that ∇ψ#µ0 = µ1 a.e. Let µt be the displacement inter- polation function between µ0 and µ1. For 0≤ t ≤ 1, let µt = (Tt)#µ0 where Tt = (1− t)Id+ t∇ψ. So we define Vt(x) := d dt Tt(x) = ∇ψ(x)− x 15 We claim that Vt(Tt)#µ0 =Vtµt = (Tt)#((∇ψ− Id)µ0). So using lemma 3.9 we have that ‖Vt‖L2(Tt#µ) ≤ ‖∇ψ− Id‖L2(µ) , this means ∫ |Vt |2 dµt ≤ ∫ |x−ψ(x)|2 dµ0 = d2w(µ0,µ1). For the other inequality we take an admissible pairing, and first we suppose Vt is sufficiently regular so there exists a flow map T such that dTt dt (x) =Vt(T (x)). T0(x) = x We know the unique solution of the continuity equation is a displacement interpo- lation so µt = (Tt)#µ0. We can compute ∫ 1 0 ∫ |Vt |2 dµtdt = ∫ 1 0 ∫ |Vt(Tt(x))|2 dµ0dt ≥ ∫ |T1(x)− x|2 dµ0 ≥ dw(µ0,µ1). 16 Chapter 4 Relaxation 4.1 Young measures Young measures are an important tool in the Calculus of Variations and Optimal Control Theory. It gives a description of limits of minimizing sequences; most of the basic results can be found in L.C. Young’s book [14]. In this chapter we will explain some work of Bernard [1], showing how he used the concept of Young measures to generalize Brenier’s theory and prove some interesting results. So far we have been working with measures that depend on time, instead of this we will define Young measures in (I×X), where I = [a,b] with λ the normalized Lebesgue measure, and (X ,d) is a complete and separable metric space. Definition 4.1 A Young Measure in (I×X), is a positive measure η on (I×X), such that for any measurable set A ⊂ I,η(A×X) = λ (A). We denote the set of Young measures as Y1(I,X) ⊂P1(I,X), and we endow the metric d1 (see defini- tion 2.8). Note that Y1(I,X) is closed inP1(I,X). There is another way to express a Young Measure by using the disintegration theorem [9], as there exist is a family of measures {ηt}t∈I in X , such that∫ I×X f (t,x)dη = ∫ I ∫ X f (t,x)dηtdλ . (4.1) 17 Now we would like to study some properties of the map η 7→ ∫ I×X f (t,x)dη . (4.2) This map is continuous if | f (t,x)|/(1+ d(x0,x)) is bounded for some x0 and f is continuous, but we can generalize this result. For this we need to define Caratheodory integrands , and remind the reader of some results. Definition 4.2 A Caratheodory integrand is a Borel-measurable function f (t,x) : I×X 7→ R, which is continuous in the second variable. A normal integrand is a Borel function f (t,x) : I×X 7→ (−∞,∞] , which is lower semi-continuous in the second variable. Definition 4.3 We say Y ⊂P(X) has uniformly integrable first moment if for every ε > 0 there exists a ball B ⊂ X such that∫ X−B d(x0,x)dµ ≤ ε ∀µ ∈ Y, for one and hence for all x0 ∈ X .We will use the following result in the proposition. Definition 4.4 A set Y ⊂P(X) is called tight if for every ε > 0 ∃ Kε compact such that µ(X−Kε)≤ ε ∀µ ∈ Y. Theorem 4.5 The function g(t,x) : I×X 7→ R is a normal integrand if and only if g = supn∈N gn(t,x), where gn is a sequence of Caratheodory integrands. Proof. See Berliocchi, Lasry [7]. Theorem 4.6 (Prokhorov) Let K ⊂P(X), K is tight if and only if it is relatively compact. Proof. See Ambrosio, Gigli, Savare [9]. Theorem 4.7 The following properties are equivalent · The family Y is tight with uniformly integrable first moment. 18 ·There exists a function f ;X 7→ [0,∞] whose sub-levels are compact, a constant C and a point x0 such that∫ X (1+d(x0,x) f (x)dµ ≤C ∀µ ∈ Y. Proof. See Ambrosio, Gigli, Savaré [9] Proposition 4.8 The map (4.2) is continuous on Y1(I×X) if f is a Caratheodory integrand such that | f (t,x)|/(1+d(x0,x)) is bounded for some x0 ∈ X . It is lower semi-continuous if f is a normal integrand such that | f (t,x)|/(1+ d(x0,x)) is bounded from below for some x0 ∈ X . Proof. Using the Scorza-Dragoni Theorem [12], we know there exists a sequence of compact sets Jn ⊂ I, such that f is continuous on Jn×X , and λ (Jn) 7→ 1 as n 7→ ∞. For every set Jn we can extend the function f continuously to a function fn with a bounded norm, so | fn(t,x)|/(1+ d(x0,x)) is bounded for every n. This means η 7→ ∫I×X fn(t,x)dη is continuous, and converges uniformly to (4.2) , and therefore is continuous. For the second part we define g= f (t,x)/(1+d(x0,x)), then g is a normal inte- grand which is bounded from below. Using Theorem 4.5, we see g= supn∈N gn(t,x) , where gn have to be bounded Caratheodory integrands. So now we can see the map (4.2) as the increasing limit of the continuous maps η 7→ ∫ (1+d(x0,x))gn(t,x)dη . Hence it is lower semi-continuous. From this proposition we can conclude our first important result. Theorem 4.9 Let f (t,x) be a normal integrand such that f (t,x)≥ l(x)(1+d(x0,x))+ g(t), where g : I 7→R, is an integrable function, and l : X 7→ [0,∞) is a proper func- tion. Then for each C ∈ R, the set{ η ∈ Y1(I,X) | ∫ f dη ≤C } is compact. 19 Proof. The set{ η ∈ Y1(I,X) | ∫ l(x)(1+d(x0,x))dη ≤C } ⊃ { η ∈ Y1(I,X) | ∫ f dη ≤C } is closed and 1-tight by the equivalence results, hence it is compact. Since the map (4.2) is lower semi continuous the set {η ∈ Y1(I,X) | ∫ f dη ≤C} is closed. 4.2 Transport measures In this section we will consider Young measures acting on I×T M where M is a complete Riemannian manifold without boundary, and d is a distance on T M such that the quotient 1+d((x0,0),(x,v)) 1+‖v‖x , and its inverse are bounded for any point x0 ∈ X . If η ∈ Y1(I,T M) is a Young measure, the image of η of the projection I× T M 7→ I×M will be denoted as µ. We can think of µ as a density in M. Using the disintegration theorem [9] with respect to this projection, we obtain the measurable family ηt,x of probability measures on TxM such that η = µ⊗ηt,x. We define the vector field V (t,x) : I×M 7→ T M by the expression V (t,x) = ∫ TxM vdηt,x(v). We note that V (t,x) is a Borel vector field, that satisfies the integrability condition ∫ ‖V (t,x)‖x dµ(t,x)< ∞. We would like to know wheter µ satisfies the continuity equation, ∂tµ+div(Vµ) = 0, (4.3) in the sense of distributions. We have the following characterization result. 20 Lemma 4.10 The measure µ satisfies equation (4.3) if and only if∫ I×T M [∂tg+∂xg · v]dη(t,x,v) = 0 (4.4) for all smooth compactly supported test functions g ∈C∞c ((a,b)×M). Proof. If we disintegrate η , we have that ∫ I×T M [∂tg+∂xg · v]dη(t,x,v) = ∫ I×T M [∂tg+∂xg · vd]ηt,x(v)dµ(t,x), for each test function. Considering the definition of V , we have the equality∫ T M ∂xg · vdηt,x(v) = ∂xg ·V (t,x). This means η satisfies equation (4.4) if and only if∫ I×M ∂tg+∂xg ·V (t,x)dµ(t,x) = 0. Which is equivalent to say µ satisfies equation (4.3). Any η ∈ Y1(I,T M), that satisfies equation (4.4) will be called transport mea- sure, and we will denote the space of transport measures as T (I,M). For the boundary conditions we do the following, given two probability mea- sures µi and µ f on M, we say η is a transport measure between µi and µ f , if in addition we have that∫ I×T M [∂tg+∂xg · v]dη(t,x,v) = ∫ M gb(x)dµ f − ∫ M ga(x)dµi, for all g : [a,b]×M 7→ R, smooth compactly supported function. We denote by T µ f µi (I,M), the set of transport measures between µi and µ f . 4.3 Generalized curves A particular case of transport measures are generalized curves as studied by L.C. Young. The way he defined boundary points is equivalent as the way defined above for this particular case. 21 Definition 4.11 A transport measure η is called a generalized curve if for each t ∈ I we have that µt = δγ(t), for a continuous curve γ(t) : I 7→ M. We say η is a generalized curve over γ, and we denote them as G (I,M). The following result shows us some regularity we can obtain from our new continuity equation. Lemma 4.12 Let Γ ∈ T (I,M) be a generalized curve over γ, then γ is absolutely continuous. Proof. By the disintegration theorem, the measure Γ can be written in the from dΓ = dt⊗ δγ(t)⊗ dΓt , with some measurable family {Γt} of probability measure in Tγ(t)M. In other words ∫ I×T M f (t,x,v)dΓ(t,x,v) = ∫ 1 0 ∫ Tγ(t)M f (t,γ(t),v)dΓt(v)dt ∀ f ∈ L1(Γ). Now, for each f ∈ C∞c (a,b) and ϕ ∈ C∞c (M), let´s apply the equation (4.4) to the function g(t,x) = f (t)ϕ(x), to get 0 = ∫ I×T M [ f ′(t)ϕ(x)+ f (t)dϕx · v ] dη(t,x,v) = ∫ 1 0 f (́t)ϕ(γ(t))dt+ ∫ 1 0 f (t) ∫ Tγ(t)M dϕγ(t) · vdΓt(v)dt. This means, in the sense of distributions, that (ϕ ◦ γ )́(t) = ∫ Tγ(t)M dϕγ(t) · vdΓt(v) ∀ϕ ∈C∞c (M). Hence γ is absolutely continuous and∫ Tγ(t)M vdΓt(v) = . γ(t). (4.5) Theorem 4.13 The set G (I,M) is closed in Y1(I,T M), and the map Γ 7→ γ (4.6) 22 is continuous. Proof. Let Γn be a sequence of generalized curves converging to η inP1(I,T M). The set {Γn}∪η is compact hence, it has uniformly integrable first moment, so if the Γńs are generalized curves over γn , then the sequence γn is absolutely equi- continuous. Hence there exists a subsequence γnm and a curve γ0 absolutely con- tinuous, such that γnm → γ0. 4.4 Tonelli theorem In this section we will prove the existence of minimizers of normal integrands L, by finding conditions for which sets of the type {Γ | ∫ LdΓ≤C} are compact. We will consider the space ACx fxi of absolutely continuous curves γ : I 7→M, such that γ(a) = xi and γ(b) = x f , and the set G x f xi =T µ f µi (I,M)∩G (I,M), of generalized curves above elements of ACx fxi . We will notice convexity of L is not needed for the result in G x fxi , but it is for AC x f xi , which is one of the advantages of working with generalized curves. For the following results, we suppose L : [a,b]×T M 7→ R∪{+∞} is a normal integrand. Definition 4.14 We say L is fiber-wise convex if, the function v 7→ L(t,x,v), is con- vex on TxM, for every t ∈ [a,b], and x ∈M. Definition 4.15 We say L is uniformly super-linear over a compact K, if there exists a function l :R+ 7→R, such that limr 7→∞ l(r)/r =∞ and such that L(t,x,v)≥ l(‖v‖x) for every (t,x,v) ∈ [a,b]×TkM. Lemma 4.16 Let L be a fiber-wise convex normal integrand. If Γ is a generalized curve above γ, then ∫ 1 0 L(t,γ(t), γ(t))dt ≤ ∫ LdΓ. 23 Proof. Using equation (4.5) and Jensen´s inequality we have L(t,γ(t), γ(t)) = L(t,γ(t), ∫ Tγ(t)M vdΓt(v))≤ ∫ Tγ(t)M L(t,γ(t),v)dΓt(v). Hence ∫ 1 0 L(t,γ(t), γ(t))dt ≤ ∫ 1 0 ∫ Tγ(t)M L(t,γ(t),v)dΓt(v)dt = ∫ LdΓ. Theorem 4.17 Let L be a normal integrand such that the quotient L(t,x,v) 1+‖v‖x (4.7) is bounded from below. Conclusion: for each C ∈ R, the set A gC := { Γ ∈ G x fxi | ∫ LdΓ≤C } is compact in G x fxi . If L is fiber-wise convex, the set AC := { γ ∈ ACx fxi | ∫ b a L(t,γ(t), γ(t))dt ≤C } is compact in ACx fxi for the uniform topology. Proof. The compactness of A gC follows from theorem 4.9. If L is fiber-wise convex, using lemma 4.16 we know the image of A gC with the continuous map (4.6) is AC, hence it is compact. A more general result is due originally to Tonelli. Theorem 4.18 (Tonelli) Let L be a normal integrand such that ·L is uniformly super-linear over each compact subset of M. ·There exists a positive constant such that L(t,x,v)≥ c‖v‖x−1 . Then we have the same conclusion as in the last theorem. 24 Proof. If Γ is a generalized curve over γ such that ∫ LdΓ ≤ C, using ‖v‖x ≤ (L(t,x,v)+1)/c ∫ b a ∥∥∥ γ(t)∥∥∥ γ(t) dt ≤ C+b−a c . This means the curve γ lies in the ball B(C+b−ac ,xi) , which is compact since M has finite dimension and d is complete. So if we define the convex integrand LB(t,x,v) = { L(t,x,v) if x ∈ B(C+b−ac ,xi) ∞ if x /∈ B(C+b−ac ,xi) , we have that Γ satisfies ∫ LdΓ ≤C if and only if ∫ LBdΓ ≤C. Using the fact L is uniformly super-linear on B(C+b−ac ,xi), we see that the quotient LB(t,x,v) 1+‖v‖x (4.8) is bounded below. So we can use the previous theorem. 25 Chapter 5 Inequalities Mass transport has already shown it is a powerful tool to prove known inequalities in sometimes remarkably simpler ways, for example one of the most simple in- equalities one can prove using mass transportation techniques is the isoperimetric inequality. Using only the arithmetic-geometric inequality in the following sense n(detD2ϕ) 1 n ≤ tr(D2ϕ) = ∆ϕ, we give a sketch, ignoring subtle analytic issues, of the original proof due to M. Gromov (see [13]) Theorem 5.1 Let Ω be an open set, such that |Ω| = 1, then we have that |∂Ω| ≥ |∂B|= n, where B is the ball with area one. Sketch. We take the unitary functions in Ω and B , 1Ω and 1B. Both are probability functions so we can take the optimal transport ∇ϕ, from Ω onto B. Hence this function satisfies the Monge-Ampere equation detD2ϕ = 1. Since |∇ϕ| ≤ 1, using Gauss theorem we can compute |∂Ω|= ∫ ∂Ω 1ds≥ ∫ ∂Ω ∇ϕ ·−→n ds = ∫ Ω ∆ϕdx≥ ∫ Ω n(detD2ϕ) 1 n = n |Ω|= n. 26 It has been known that there is a relationship between the isoperimetric inequal- ity and the Sobolev inequality. In fact the Sobolev inequality, can be proven using optimal transport in a similar spirit. There are several other applications like Brunn- Monkowski, HWI, Log-sobolev, and Gagliardo-Nirenberg. See [3],[2],[11]. Re- cently Agueh-Ghoussoub-Kang [10] showed that many of this inequalities actually belong to the same family of inequalities, in other words they are particular cases of the same general inequality. It is the purpose of this chapter to explain this re- sult emphasizing on displacement convexity by proving first a general Sobolev in- equality that can be used to obtain Log-Sobolev, Sobolev and Gagliardo-Nirenberg inequalities. Afterwards we will prove the Agueh-Ghoussoub-Kang’s general in- equality , and show that it generalizes the general Sobolev inequality as well as other general inequalities like the HWI and Gaussian inequalities . 5.1 General Sobolev inequality In this section we will use the energy functional HF (see definition 3.5) In this chapter, T represents the optimal map from ρ0 to ρ1, and ρt := ((1− t)I+T )#ρ0. Lemma 5.2 Suppose F : [0,∞)→ Rn is differentiable with F(0) = 0, and x 7→ xnF( rxn ) is convex and non increasing for all r > 0, then we have that HF(ρ1)−HF(ρ0)≥ ∫ Ω ρo(T −1) ·∇F ′(ρ0)dx, for all ρ0,ρ1 ∈P2,AC. Proof. Since HF(ρt) is convex then we obtain HF(ρ1)−HF(ρ0) 1 ≥ [ d dt HF(ρt) ] t=0 = [ d dt ∫ Ω F(((1− t)I+ tT )#ρ0)dx ] t=0 =− ∫ Ω F ′(ρ0)div(ρ0(T − I))dx = ∫ Ω ρo(T −1) ·∇F ′(ρ0)dx. 27 Definition 5.3 We will call a Young function, any strictly convex super-linear C1- function c :Rn→R, such that c(0) = 0, and we will denote by c∗ its Legendre dual, as defined in remark 2.5. Theorem 5.4 (General Sobolev inequality) Under the hypotheses of the previous lemma, let Ω be any open bounded convex set, then for any ρ ∈P2,AC, satisfying suppρ ⊂Ω and PF(x) := xF ′(x)−F(x) ∈W 1,∞(Ω) , we have that HF+nPF (ρ)≤ ∫ Ω c∗(−∇F ′(ρ))ρdx+Kc. (5.1) Proof. Using the previous lemma for ρ0 = ρ, and ρ1 = ρc, where ρc ∈P2,AC is a solution of ∇(F ′(ρc)+ c) = 0, we get that HF(ρc)−HF(ρ)≥ ∫ Ω ρ(T x− x) ·∇F ′(ρ). We note that since ρ∇(F ′(ρ)) = ∇(PF(ρ)) we have that∫ Ω ρ∇(F ′(ρ)) · x = ∫ Ω −nPF(ρ) = H−nPF (ρ). We obtain HF(ρ)−HF(ρc)≤ ∫ ρ(x−T x) ·∇(F ′(ρ)) ≤ H−nPF (ρ)− ∫ Ω ρ∇(F ′(ρ)) ·T xdx. For the last term we can use the generalized Young’s inequality to obtain that −∇(F ′(ρ)) ·T x≤ c(T x)+ c∗(−ρ∇(F ′(ρ)). 28 Integrating this to the inequality we have HF(ρ)−HF(ρc) ≤ H−nPF (ρ)+ ∫ Ω c(T x)ρdx+ ∫ Ω c∗(−∇F ′(ρ))ρdx = H−nPF (ρ)+ ∫ Ω c(x)ρcdx+ ∫ Ω c∗(−∇F ′(ρ))ρdx. Finally we get HF+nPF (ρ)≤ ∫ Ω c∗(−∇F ′(ρ))ρdx+ ∫ Ω (c(x)+F ′(ρc))ρcdx−HPF (ρc) We name the constant c(x)+F ′(ρc) = Kc, and we note that HPF (ρc) ≥ 0 to conclude the proof. In the following pages we will see that using different F’s this inequality gen- eralizes Log-Sobolev inequalities and Sobolev-Nirenberg-Gagliardo inequalities. 5.1.1 Euclidian Log Sobolev inequalities The Log-Sobolev inequality was first introduced by L. Gross, see [6], here we prove it as a corollary of the previous inequality. Corollary 5.5 Let Ω ⊂ Rn be an open bounded and convex set, and let c be a Young functional , such that c∗ is p-homogeneous, for p > 1, we have that for all probability densities ρ, with supp(ρ)⊂Ω, and ρ ∈W 1,∞(Rn) ∫ Rn ρ logρdx≤ n p log( p nep−1σ p/nc ∫ Rn ρc∗(−∇ρ ρ )dx), where σc = ∫ Rn e −c(x)dx. Proof. Let F(x)= x log(x), and F(0)= 0.We check that x 7→ xnF(x−n)=−n log(x) is convex and non increasing. Considering that in this case PF(x) = x, we get that for any probability measure ρ , HPF = ∫ ρ = 1. We take inequality (5.1), HF+nPF (ρ)≤ ∫ Ω c∗(−∇F ′(ρ))ρdx+ ∫ (F ′(ρc)+ c)ρcdx, (5.2) 29 where ρc is a solution of the equation ∇(logρc+ c) = 0, which we take ρc(x) = e−c(x)/σc, so we get ∫ ρ logρ+n≤ ∫ c∗(−∇ρ ρ )ρdx+ ∫ ( loge−c(x)− log( ∫ Rn e−c(x)dx)+ c ) ρcdx (5.3) = ∫ c∗(−∇ρ ρ )ρdx− log( ∫ Rn e−c(x)dx). (5.4) Let cλ (x) := c(λx), hence c∗λ (y) = c ∗( yλ ). If we apply the inequality to this Young function we get∫ ρ logρ+n≤ ∫ c∗(−∇ρ λρ )ρdx− log( ∫ Rn e−c(λx)dx) = ∫ c∗(−∇ρ λρ )ρdx− log( ∫ Rn e−c(x)dx)+n logλ . Considering that c∗( yλ ) = 1 λ p c ∗(y), the infimum over λ is attained when λ po = p n ∫ c∗(−∇ρ ρ )ρdx. So we get the inequality for all probability densities ρ, with supp(ρ) ⊂ Ω, and ρ ∈W 1,∞(Rn) ∫ ρ logρ ≤ n p ∫ c∗(−∇ρλρ )ρdx ∫ c∗(−∇ρ ρ )ρdx− log(σc)+ np log ( p n ∫ c∗(−∇ρ λρ )ρdx ) −n ≤ n p log ( p n ∫ c∗(−∇ρ λρ )ρdx ) − log(σc)−n+ np = n p log( p nep−1σ p/nc ∫ Rn ρc∗(−∇ρ ρ )dx). 30 5.1.2 Sobolev and Gagliardo-Nirenberg inequalities We will now derive the Gagliardo-Nirenberg inequality from the general Sobolev inequality. A classical proof can be found in [4], and a proof using mass-transport approach can be found in [2]. Corollary 5.6 (Gagliardo-Nirenberg) Let 1 < p < n and r ∈ (0, npn−p) such that r 6= p. We define γ := 1r + 1q , where 1p + 1q = 1. For any f ∈W 1,p(Rn) we have that there exists θ such that ‖ f‖r ≤C(p,r)‖∇ f‖θp ‖ f‖1−θrγ . Proof. We will use inequality (5.1) with F(x) = x γ γ−1 . Since r 6= p we have that γ 6= 1 and since r ∈ (0, npn−p ), we have that 1 > γ > 1− 1n . To use the inequality we check that F(0) = 0, and x 7→ xnF(x−n) = xn−nγγ−1 is convex and non increasing since n− nγ < 1 and γ − 1 < 0. Let c(x) = rγq |x|q , so c∗(x) = 1p(rγ)p−1 |x|p . Using inequality (5.1) we get ∫ F(ρ)+nρF ′(ρ)−nF(ρ)dx≤ ∫ Ω ρ 1 p(rγ)p−1 (−∇F ′(ρ))pρdx+Kc. (5.5) Making the substitution of F(x) = x γ γ−1 we get 1 γ−1 ∫ ργ −nργ +nγργdx≤ ∫ Ω ρ 1 p(rγ)p−1 (−γργ−2∇ρ)pρdx+Kc, and rearranging the equation we get ( 1 γ−1 +n) ∫ ργdx≤ ∫ Ω ρ rγ p(r)p (∇ρ)pρdx+Kc. If we suppose that ‖ f‖r = 1, we take ρ = | f |r to get ( 1 γ−1 +n) ∫ | f |rγ dx≤ ∫ Ω rγ p |∇ f |pρdx+Kc, and for general f we get this inequality 31 rγ p ‖∇ f‖pp ‖ f‖pr − ( 1 γ−1 +n ) ‖ f‖rγrγ ‖ f‖r ≥−Kc. If we have the function fλ (x) = f (λx), with a change of variables we get the following equalities ‖ fλ‖pr = λ−np/r ‖ f‖pr , ‖ fλ‖rγrγ = λ−n ‖ f‖rγrγ , ‖ fλ‖r = λ−n/r ‖ f‖r , ‖∇ fλ‖pp = ‖λ∇ f (xλ )‖pp = λ p−n ‖∇ f‖pp . So the inequality becomes λ p−n+np/r rγ p ‖∇ f‖pp ‖ f‖pr −λ−n+n/r ( 1 γ−1 +n ) ‖ f‖rγrγ ‖ f‖r ≥−Kc. We take λ = ‖∇ f‖ap ‖ f‖br ‖ f‖crγ , and we pick a, b, and c, so that the powers of the norms are the same in both terms, that is a = pr pr+np−n , b = (p−1)r pr+np−n , c = r pr+np−n . So we obtain 1 Kc ( −rγ p + 1 γ−1 −n ) ‖∇ f‖a′p ‖ f‖c ′ rγ ≥ ‖ f‖b ′ r , 32 where a′ = −npr+np pr+np−n b′ = (p−1)r(−n+n/r) pr+np−n −1 c′ = rp−nr+np pr+np−n Finally we note that if we take the limit as r→ p∗ = npn−p , we have that c′→ 0, and a′,b′→ np(n−np−p)(n−p)(pr+np−n) so we get the Sobolev inequality ‖ f‖p∗ ≤C(p,n)‖∇ f‖p . 5.2 General inequality In this section we will generalize the previous result by showing an inequality that contains even more information, like HWI inequalities (see [10]). For this, inspired by the physics of interacting gases, we will define more energy functionals inP2,ac, and we will use the concept of semi-convexity. Definition 5.7 Let F : [0,∞)→ Rn differentiable, and V,W : R→ [0,∞) , twice differentiable, then we can define the associated Free Energy Functional as HF,WV (ρ) := H F(ρ)+HV (ρ)+HW (ρ). Where we have ·Internal energy HF(ρ) := ∫ Ω F(ρ(x))dx, . ·Potential energy HV (ρ) =: ∫ Ω ρ(x)V (x)dx. ·Interaction energy HW (ρ) =: 1 2 ∫ Ω ρ(W ∗ρ), 33 where ∗ denotes the convolution product. Furthermore we define the relative energy of ρ0 with respect to ρ1 as HF,WV (ρ0 | ρ1) := HF,WV (ρ1)−HF,WV (ρ0), and the relative entropy production of ρ with respect to ρV as I2(ρ | ρV ) := ∫ Ω ∣∣∇(F ′(ρ)+V +W ∗ρ)∣∣2ρdx. So if ρV is a probability density that satisfies ∇(F ′(ρV )+V +W ∗ρ) = 0, then I2(ρ | ρV ) := ∫ Ω ∣∣∇(F ′(ρ)−F ′(ρV )+W ∗ (ρ−ρV ))∣∣2ρdx. We will also work with non-quadratic versions of entropy, so we define the generalized relative entropy production-type function of ρ with respect to ρV mea- sured against c∗ as Ic∗(ρ0 | ρV ) := ∫ Ω c∗ (−∇(F ′(ρ0)+V +W ∗ρ0))ρ0dx, where c∗ is the Legendre conjugate of c. Lemma 5.8 Assume V:Rn→ R satisfies that D2V ≥ λ I, for some λ ∈ R, then we have that HV (ρ1)−HV (ρ0)≥ ∫ Ω ρo(T −1) ·∇V dx+ λ2 d 2 w(ρ0,ρ1), for all ρ0,ρ1 ∈P2,AC. Proof. Expanding and using D2V ≥ λ I, we obtain V (b)−V (a)≥ ∇V (a) · (b−a)+ λ 2 |a−b|2 . 34 This means that V (T x)−V (x)≥ ∇V (x) · (T x− x)+ λ 2 |x−T x|2 . Hence integrating we obtain HV (ρ1)−HV (ρ0)≥ ∫ V (T x)ρ0−V (x)ρ0dx ≥ ∫ Ω ∇V (x) · (T x− x)+ λ 2 |x−T x|2ρ0dx = ∫ Ω ρo(T −1) ·∇V dx+ λ2 d 2 w(ρ0,ρ1). Lemma 5.9 Assume W:Rn → R is even and satisfies that D2W ≥ νI, for some ν ∈ R, then we have that HW (ρ1)−HW (ρ0)≥ ∫ Ω ρo(T−1)·∇(W ∗ρ0)dx+ ν2 (d 2 w(ρ0,ρ1)−|b(ρ0)−b(ρ1)|2), for all ρ0,ρ1 ∈P2,AC, and where b represents the centre of mass denoted by b(ρ)=∫ xρ(x)dx. Proof. First we note that we can write the interaction energy as follows HW (ρ1) = 1 2 ∫ Ω×Ω W (x− y)ρ1(x)ρ1(y)dxdy = 1 2 ∫ Ω×Ω W (T x−Ty)ρ0(x)ρ0(y)dxdy = 1 2 ∫ Ω×Ω W (x− y+(T − I)(x)− (T − I)(y))ρ0(x)ρ0(y)dxdy 35 Since D2W ≥ νI we obtain HW (ρ1)≥ 12 ∫ Ω×Ω [W (x− y)+∇W (x− y) · ((T − I)(x)− (T − I)(y))]ρ0(x)ρ0(y)dxdy + ν 4 ∫ Ω×Ω |(T − I)(x)− (T − I)(y)|2ρ0(x)ρ0(y)dxdy = HW (ρ0)+ 1 2 ∫ Ω×Ω ∇W (x− y) · ((T − I)(x)− (T − I)(y))ρ0(x)ρ0(y)dxdy + ν 4 ∫ Ω×Ω |(T − I)(x)− (T − I)(y)|2ρ0(x)ρ0(y)dxdy. Now we note the following equalities, for the last term∫ Ω×Ω |(T − I)(x)− (T − I)(y)|2ρ0(x)ρ0(y)dxdy = 2 ∫ Ω×Ω |(T − I)(x)|2ρ0(x)dx−2 ∣∣∣∣∫Ω×Ω(T − I)(x)ρ0(x)dx ∣∣∣∣2 = 2 [∫ Ω×Ω |(T − I)(x)|2ρ0(x)dx−|b(ρ1)−b(ρ0)|2 ] . For the second term we consider that ∇W is odd∫ Ω×Ω ∇W (x− y) · ((T − I)(x)− (T − I)(y))ρ0(x)ρ0(y)dxdy = 2 ∫ Ω×Ω ∇W (x− y) · ((T − I)(x))ρ0(x)ρ0(y)dydx = 2 ∫ Ω×Ω (∇W ∗ρ0) · (T − I)(x))ρ0(x)dx. Using these two equalities we get HW (ρ1)−HW (ρ0)≥ ∫ Ω ρo(T−1)·∇(W ∗ρ0)dx+ ν2 (d 2 w(ρ0,ρ1)−|b(ρ0)−b(ρ1)|2). Theorem 5.10 (Basic inequality) Under the hypotheses of the three previous lem- mas, let Ω be any open bounded convex set, then for any ρ0,ρ1 ∈P2,AC, satisfying 36 suppρ0 ⊂Ω and PF(x) := xF ′(x)−F(x) ∈W 1,∞(Ω) , we have that HF,WV+c(ρ0 | ρ1)+ λ +ν 2 d2w(ρ0,ρ1)− ν 2 |b(ρ0−b(ρ1)|2 ≤ H−nPF ,2x·∇Wc+∇V ·x (ρ0)+Ic∗(ρ0 | ρV ). Proof. First we note that since ρ0∇(F ′(ρ0)) = ∇(PF(ρ0)) we have that∫ Ω ρ0∇(F ′(ρ0)+V +W ∗ρ0) · x = ∫ Ω −nPF(ρ0)+ρ0 [∇(V +∇W ∗ρ0)] · x = ∫ Ω −nPF(ρ0)+ρ0∇V · x+ 12ρ0(2x ·∇W ∗ρ0)dx = H−nPF ,2x·∇W∇V ·x (ρ0). If we add the inequalities from the previous lemmas we get HF,WV (ρ1)−HF,WV (ρ0) ≥ ∫ Ω ρ0(T x− x) ·∇(F ′(ρ0)+V +W ∗ρ0)dx+ λ2 d 2 w(ρ0,ρ1) + ν 2 (d2w(ρ0,ρ1)−|b(ρ0)−b(ρ1)|2). Rearranging and using the first inequality we have HF,WV (ρ0)−HF,WV (ρ1)+ λ +ν 2 d2w(ρ0,ρ1)−|b(ρ0)−b(ρ1)|2) ≤ ∫ Ω ρ0(x−T x) ·∇(F ′(ρ0)+V +W ∗ρ0) ≤ H−nPF ,2x·∇W∇V ·x (ρ0)− ∫ Ω ρ0∇(F ′(ρ0)+V +W ∗ρ0) ·T xdx. For the last term we can use the generalized Young’s inequality to obtain that −∇(F ′(ρ0)+V +W ∗ρ0) ·T x ≤ c(T x)+ c∗(−∇(F ′(ρ0)+V +W ∗ρ0). 37 Integrating this to the inequality we have HF,WV (ρ0)−HF,WV (ρ1)+ λ +ν 2 d2w(ρ0,ρ1)−|b(ρ0)−b(ρ1)|2) ≤ H−nPF ,2x·∇W∇V ·x (ρ0)+ ∫ Ω c(T x)ρ0dx+ ∫ Ω c∗(−∇(F ′(ρ0)+V +W ∗ρ0)ρ0dx = H−nPF ,2x·∇W∇V ·x (ρ0)+ ∫ Ω c(x)ρ1dx+ ∫ Ω c∗(−∇(F ′(ρ0)+V +W ∗ρ0)ρ0dx. This proves the inequality. A simpler inequality is the one obtained when V and W are strictly convex hence ν ,λ ≥ 0. Lemma 5.11 Under the same hypothesis as theorem 5.10, assume that V and W are also convex. Then for any Young function c : Rn→ R, we have HF+nPF ,W−2x·∇WV−∇V ·x (ρ)≤−HPF ,W (ρV+c)+Ic∗(ρ | ρV )+KV+c. (5.6) Furthermore if we set W =V = 0, since HPF (ρc)≥ 0, we obtain HF+nPF (ρ)≤−HPF (ρc)+Ic∗(ρ | ρV )+KV+c (5.7) ≤ ∫ Ω c∗(−∇F ′(ρ))ρdx+KV+c. (5.8) Hence recovering inequality (5.1). Proof. Let´s consider the inequality we just proved HF,WV+c(ρ0 | ρ1)+ λ +ν 2 d2w(ρ0,ρ1)− ν 2 |b(ρ0−b(ρ1)|2 (5.9) ≤ H−nPF ,2x·∇Wc+∇V ·x (ρ0)+Ic∗(ρ0 | ρV ). (5.10) In particular if we take ρ0 = ρ and ρ1 = ρV+c , where ρV+c is a solution of ∇(F ′(ρV+c)+V + c+W ∗ρV+c) = 0. 38 Hence we have that for any ρ ∈Pc(Ω),with supp ρ ⊂Ω, and PF(ρ)∈W 1,∞(Ω) we have that HF+nPF ,W−2x·∇WV−∇V ·x (ρ)+ λ +ν 2 d2w(ρ,ρV+c)− ν 2 |b(ρ−b(ρ1)|2 (5.11) ≤−HPF ,W (ρV+c)+Ic∗(ρ | ρV )+ ∫ ( F ′(ρV+c)+V + c+W ∗ρV+c ) ρV+c. (5.12) Where we can define the constant F ′(ρV + c)+V + c+W ∗ρ := KV+c . Since ν ,λ ≥ 0 we get that λ +ν 2 d2w(ρ,ρV+c)− ν 2 |b(ρ)−b(ρV+c)|2 = λ +ν 2 ∫ |T x− x|2ρ0(x)dx− ν2 ∣∣∣∣∫ (T x− x)ρ0(x)dx∣∣∣∣2 ≥ 0. So we can remove the terms involving ν and λ in the inequality to get the wanted inequality. 5.2.1 HWI inequalities Now we proceed to get some corollaries when we apply a quadratic Young func- tion. Corollary 5.12 Under the same hypothesis as theorem 5.10, let µ ∈ R, and U :Rn→ R be a C2 function such that D2U ≥ µI, then for any σ > 0 we have that HFU (ρ0 | ρ1)+ 1 2 (µ− 1 σ )W 22 (ρ0,ρ1)≤ σ 2 ∫ Ω ρ ∣∣∇(F ′ ◦ρ0+U∣∣2 dx. Proof. If we take the basic inequality with c(x) = 12σ |x|2 , W = 0, and we set, V = U − c. Hence we have that c∗(p) = 12σ |σ p|2 = σ2 |p|2 , so using the general inequality we get HFU (ρ0 | ρ1)+ (µ−σ−1) 2 d2w(ρ0,ρ1)≤H−nPFc+∇(U−c)·x(ρ0)+ σ 2 ∫ Ω ρ0∇(F ′◦ρ0+U−c)dx. 39 We can compute σ 2 ∫ Ω ρ0 ∣∣∇(F ′ ◦ρ0+U− c)∣∣2 dx = σ 2 ∫ Ω ρ ∣∣∇(F ′ ◦ρ0+U∣∣2 dx+ 12σ ∫ Ω ρ0 |x|2 dx− ∫ Ω xρ0 ·∇(F ′ ◦ρ0+U)dx, and HnPFc+∇(U−c)·x(ρ0) = H nPF (ρ0)− ∫ Ω ρx ·∇Udx+ 1 2σ ∫ Ω |x|2ρ0dx. By combining the two and using integration by parts we get that H−nPFc+∇(U−c)·x(ρ0)+ σ 2 ∫ Ω ρ0 ∣∣∇(F ′ ◦ρ0+U− c)∣∣2 dx = σ 2 ∫ Ω ρ ∣∣∇(F ′ ◦ρ0+U∣∣2 dx−∫ Ω xρ0 ·∇(F ′ ◦ρ0)dx−HnPF (ρ0) = σ 2 ∫ Ω ρ ∣∣∇(F ′ ◦ρ0+U∣∣2 dx+∫ Ω div(xρ0) · (F ′ ◦ρ0)dx−HnPF (ρ0) = σ 2 ∫ Ω ρ ∣∣∇(F ′ ◦ρ0+U∣∣2 dx+∫ Ω nρ0 · (F ′ ◦ρ0)dx+ ∫ Ω x ·∇F(ρ0)dx−HnPF (ρ0) = σ 2 ∫ Ω ρ ∣∣∇(F ′ ◦ρ0+U∣∣2 dx+∫ Ω x ·∇F(ρ0)dx+ ∫ Ω n(F ◦ρ0)dx = σ 2 ∫ Ω ρ ∣∣∇(F ′ ◦ρ0+U∣∣2 dx. Returning to the first inequality we get that HFU (ρ0 | ρ1)+ (µ−σ−1) 2 d2w(ρ0,ρ1)≤ σ 2 ∫ Ω ρ ∣∣∇(F ′ ◦ρ0+U)∣∣2 dx. Corollary 5.13 Furthermore if we take µ > 0, that is, take U is uniformly convex, take σ = 1µ , we can get the Generalized Log-Sobolev inequality: HFU (ρ0 | ρ1)≤ 1 2µ ∫ Ω ρ ∣∣∇(F ′ ◦ρ0+U)∣∣2 dx = 12µI2(ρ0 | ρU). Corollary 5.14 (HWI ) Finally we can obtain the generalized HWI-inequality, 40 which is originally due to Otto and Villani (see [5]). HFU (ρ0 | ρ1)+ µ 2 d2w(ρ0,ρ1)≤ √ I2(ρ0 | ρU)dw(ρ0,ρ1). Proof. If we write the inequality of the last corollary as HFU (ρ0 | ρ1)+ µ 2 d2w(ρ0,ρ1)≤ σ 2 I2(ρ0 | ρU)+ 12σ d 2 w(ρ0,ρ1) and minimize over σ , we obtain the minimum when σ = dw(ρ0,ρ1)√ I2(ρ0|ρU ) , we can write the inequality as HFU (ρ0 | ρ1)+ µ 2 d2w(ρ0,ρ1)≤ √ I2(ρ0 | ρU)dw(ρ0,ρ1). 5.2.2 Gaussian inequalities By taking a particular F we can prove Otto-Villani’s HWI inequality. Corollary 5.15 Let µ ∈R, and U :Rn→R be a C2 function such that D2U ≥ µI, then for any σ > 0 , and any non-negative function f such that fρU ∈W 1,∞(Rn) and∫ fρU = 1, we have that ∫ f log( f )ρU + 1 2 (µ− 1 σ )W 22 (ρ0,ρ1)≤ σ 2 ∫ Ω ρU |∇ f |2 f dx. Where ρU = e−U/ ∫ e−U dx. Proof. The proof follows from corollary 5.12, taking ρ0 = ρU , ρ1 = fρU , and F(x) = x logx. So we compute HFU (ρU) = ∫ ρU logρU +UρU dx = ∫ [( e−U/ ∫ e−U dx ) log ( e−U/ ∫ e−U dx ) +U ( e−U/ ∫ e−U dx )] = 1∫ e−U dx ∫ e−U(− log ∫ e−U) =− log ∫ e−U , and 41 HFU ( fρU) = ∫ fρU log fρU +U fρU = 1∫ e−U dx ∫ e−U f (log f − log ∫ e−U) = ∫ f log( f )ρU − ( log ∫ e−U )∫ fρU = ∫ f log( f )ρU − log ∫ e−U Hence HFU (ρ0 | ρ1) = ∫ f log( f )ρU . Furthermore if U is uniformly convex , we can consider µ > 0, so we can simplify the inequality to get the original Log-Sobolev inequality of Gross ∫ f log( f )ρU ≤ 1µ ∫ Ω ρU |∇ f |2 f dx. 42 Bibliography [1] P. Bernard. Young measures, supersposition and transport. Indiana University Mathematics Journal, 57(1):247–246, 2008. → pages 2, 17 [2] C. V. D. Cordero-Erausquin, B. Nazaret. A mass-transportation approach to sharp sobolev and gagliardo-nirenberg inequalities. preprint, 2002. → pages 2, 27, 31 [3] W. G. D. Cordero-Erausquin and C. Houdre. Inequalities for generalized entropy and optimal transport. 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