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Mass transport and geometric inequalities Garcia Ramos Aguilar, Felipe 2010

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Mass transport and geometric inequalities  by Felipe Garc´ıa Ramos Aguilar B. Sc. Mathematics, Universidad Nacional Aut´onoma de M´exico, 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 Garc´ıa 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  Kantorovich-Rubinstein space . . . . . . . . . . . . . . . . . . .  8  Time dependent mass transportation . . . . . . . . . . . . . . . . . .  10  3.1  Displacement interpolation . . . . . . . . . . . . . . . . . . . . .  10  3.2  Displacement convexity . . . . . . . . . . . . . . . . . . . . . . .  11  3.3  Benamou-Brenier formula . . . . . . . . . . . . . . . . . . . . .  13  Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  17  4.1  Young measures . . . . . . . . . . . . . . . . . . . . . . . . . . .  17  4.2  Transport measures . . . . . . . . . . . . . . . . . . . . . . . . .  20  4.3  Generalized curves . . . . . . . . . . . . . . . . . . . . . . . . .  21  4.4  Tonelli theorem . . . . . . . . . . . . . . . . . . . . . . . . . . .  23  Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  26  5.1  27  2.1 3  4  5  General Sobolev inequality . . . . . . . . . . . . . . . . . . . . .  iii  5.1.1  Euclidian Log Sobolev inequalities . . . . . . . . . . . .  29  5.1.2  Sobolev and Gagliardo-Nirenberg inequalities . . . . . . .  31  General inequality . . . . . . . . . . . . . . . . . . . . . . . . . .  33  5.2.1  HWI inequalities . . . . . . . . . . . . . . . . . . . . . .  39  5.2.2  Gaussian inequalities . . . . . . . . . . . . . . . . . . . .  41  Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  43  5.2  iv  Acknowledgments I would like to thank Prof. Young-Heon Kim, for carefully reading this thesis and making very useful observations and explanations; Ram´on Z´arate 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 lleg´o, dej´o su legado y se fu´e.  ...y al Pollo tambi´en.  vi  Chapter 1  Introduction Mathematics is the art of giving the same name to different things. — J.H. Poincar´e 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  |x − y|2 dγ(x, y) =  γ∈Γ(µ0 ,µ1 ) Rn ×Rn  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 problem, one involving time, which is to find a one parameter family of pairs (µt ,Vt ) , that minimize  1  inf 0  |Vt |2 dµt dt,  where µt is a measure and Vt a vector field that depends on t and satisfy the continuity equation, or also known as the conservation of mass equation µt + div(V µ) = 0. 1  In Chapter 2 we will show the proof of Brenier’s theorem, that under some circumstances both problems have the same infimum or optimal cost. In this chapter we will also review the concepts of displacement convexity and displacement interpolation, 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 measures 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 [∂t g + ∂x g · v] dηt,x (v)dµ(t, x) =  [∂t g + ∂x g · v] dη(v,t, x) = 0. (1.1)  I×T M  I×T M  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 inequality one can prove the isoperimetric inequality and Sobolev-Nirenberg inequalities. 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 measurable function ρ that represents density, force F, pressure PF , the internal energy functional H, the Young’s function c∗ , and a constant Kc , c∗ (−∇F (ρ))ρdx + Kc .  H F+nPF (ρ) ≤ Ω  2  (1.2)  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 inequality shown in [10], that involves a Young’s function c and its dual c∗ , energy functionals HVF,W , relative entropy production Ic∗ (ρ0 | ρV ), Wasserstein distance dw2 (ρ0, ρ1 ) and barycentre b(ρ0 ), namely λ +ν 2 ν dw (ρ0, ρ1 ) − |b(ρ0 ) − b(ρ1 )|2 2 2 −nPF ,2x·∇W ≤ Hc+∇V ·x (ρ0 ) + Ic∗ (ρ0 | ρV ).  HVF,W +c (ρ0 | ρ1 ) +  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 measures. 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, as P(X) , and PAC (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 π+ : M + × M − → M + , where π+ (x, y) = x, and π− : M + × M − → M − , where π− (x, y) = y. So we define Γ(µ + , µ − ) = γ ∈ P(M + × M − ) | (π+ )# γ = µ + and (π− )# γ = µ − . Let c : M + ×M − → 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) =  2.0.1  inf  γ∈Γ(µ + ,µ − ) M + ×M −  c(x, y)dγ .  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  N  ∑ c(xi, yi ) ≤ ∑ c(xi , yi+1 ). (with N+1=1) i  i  Definition 2.3 A function φ : X → R∪ {+∞} is said to be c-convex if it is not identical to {+∞} , and there exists a function g : Y → R∪ {+∞} , such that φ (x) = sup(g(y) − c(x, y)) ∀x ∈ X. y∈Y  It´s c-transform is the function φ c (y) := inf (φ (x) + c(x, y)), x∈X  and it’s c-subdifferential is the c-cyclically monotone set ∂ c φ := {(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 = ∂ c φ 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  N  ∑ c(xi, yi ) − ∑ c(xi , yi+1 ) > 0. i  i  Even more, since c is continuous we can find a set of open neighbourhood Ui ×Vi , of (xi , yi ) such that N  N  ∑ c(ui, vi ) − ∑ c(ui , vi+1 ) > 0 i  ∀ (ui, vi ) ∈ Ui ×Vi .  i  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 γ =γ+  λ n  N  ∑( fi+1 × gi )# η − ( fi × gi )# η, i  where λ = inf γ(Ui ×Vi ) > 0. We check that (π+ )# γ = µ + + λn ∑Ni ( fi+1 )# η −( fi )# η = µ + , similarly (π− )# γ = µ − , and γ(X ×Y ) = 1 + λn ∑Ni  γ((Ui+1 ×Vi )) γ(Ui+1 ×Vi )  i ×Vi )) − γ((U γ(Ui ×Vi ) = 1.  Finally we compute C (γ) − C (γ) =  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 = ∂ c φ (x), by definition of c-convexity we have that  φ (x) = sup {−c(x, y) − φ c (y)} = sup y∈M −  y∈M −  = sup y∈M −  1 − |x − y|2 − φ c (y) 2  1 1 − |x|2 − |y|2 + xy − φ c (y) . 2 2  Now we can define a function ϕ, 1 1 ϕ(x) := φ (x) + |x|2 = sup − |y|2 + xy − φ c (y) . 2 2 − y∈M This function is convex and bounded and moreover, ∂ ϕ(x) = ∂ c φ . 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 D2 ϕ exists almost everywhere if ϕ is convex, so if we do a change of variables y = ∇ ϕ(x) we get that ψ(∇ϕ(x))g(∇ϕ(x)) det D2 ϕdx.  ψ(y)g(y)dy =  From this we get the Monge-Ampere’s equation g(∇ϕ) det D2 ϕ = 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 space Pn (X) is defined as follows, 1/n  |d(x, y)|n dλ (x, y)  dn (µ, η) = min λ  .  X×X  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 → 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) → 0 if and only if f dµn → 8  f dµ  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 → for each bounded continuous function f .  9  f dµ  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 πs : Rn × Rn → Rn , as πs (x, y) := (1 − s)x + sy, and we will call µs = (πs )# 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 πst = (πs × πt )# γ0 = ((1 − s)x + sy), (1 − t)x + ty)# γ0 ∈ Γ(µs , µt ). So that dw2 (µt , µs ) ≤  |x − y|2 dπst =  = (t − s)2  |(1 − s)x + sy − ((1 − t)x + ty)|2 dγ0  |x − y|2 dγ0 = (t − s)2 dw2 (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 H F (ρ) :=  F(ρ(x))dx . Ω  Proposition 3.6 Let H F be the internal energy functional. If we suppose F : [0, ∞) → Rn is differentiable with F(0) = 0, and x → xn F( xrn ) is convex and non increasing for all r > 0, then H F 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 H F (ρs ) =  F(ρs (x))dx = Ω  = Ω∩Supp(ρ0 )  Ω∩Supp(ρs )  F(ρs (x))dx  F(ρs (1 − s)x + s∇ψ(x)) det((1 − s)I + s∇ψ)dx. 1  Using the Monge-Ampere formula, and defining λ (s) = det((1 − s)I + s∇ψ) n we can conclude that ρ0 (x) ) det((1 − s)I + s∇ψ)dx det((1 − s)I + s∇ψ) Ω∩Supp(ρ0 ) ρ0 (x) = F( n )λ n dx λ Ω∩Supp(ρ0 )  H F (ρs ) =  F(  12  Using the fact that λ (s) is concave, the remark and the lemma, we see that s → n F( ρ0λ(x) n )λ is convex, this means  H F (ρs ) ≤ (1 − s)H F (ρ0 ) + sH F (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  dµt dt < ∞.  ·∂t µ + ∇ · (µV ) = 0 in a weak sense. Theorem 3.8 Let X be a complete smooth manifold, let µ0 be a probability measure 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 (x) = Vt (T (x)), dt 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 (ϕ ◦ Tt )dµ  ϕdµt = so for h > 0 we can write 1 ( h  ϕdµt+h −  ϕdµt ) =  13  ϕ ◦ Tt+x (x) − ϕ ◦ Tt dµ. h  Since Tt−1 is continuous, then ϕ ◦ Tt is Lipschitz and compactly supported uniformly 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 ∂ ∂ (ϕ ◦ Tt ) = (∇ϕ ◦ Tt ) · Tt = (∇ϕ ◦ Tt ) · (vt ◦ Tt ). ∂t ∂t By Lebesgue´s dominated convergence theorem we deduce that for almost all t we have  d dt  (∇ϕ ◦ Tt ) · (vt ◦ Tt ) =  ϕdµt =  ∇ϕ · vt dµt .  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 + vt · ∇ϕ = 0 ∂t ϕ |t=T = ϕT . Where ϕT ∈ D(X), the space of distribution, so we can compute for almost all t d dt  ϕt dµt = =−  ∂ ϕt dµt + ∂t vt · ∇ϕ +  ϕt d(  ∂ µt ) ∂t  ϕt d(∇ · vt µt ) = 0.  Since µ0 = 0, then ϕT dµT = 0 =⇒ µT = 0. Finally we can check that ϕt = ϕT ◦ TT ◦ Tt−1 Lipschitz with compact support, and is a solution of  d ∂ϕ ϕt (Tt x) = + v · ∇ϕ = 0. dt ∂t  We will need the following lemma to prove the Benamou-Brenier theorem.  14  Lemma 3.9 Let σ be a measure in Rn , f ∈ L2 (σ ), and T a map such that T# ( f σ ) = hT# (σ ) . Then h  ≤ f  L2 (T# σ )  L2 (σ )  Proof. Let g ∈ L2 (T# σ ), computing T# ( f σ ), g = g ◦ T, f σ ≤ 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# ( f σ ), 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 Wasserstein 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 dw2 (µ0 , µ1 ) =  1  inf  Vt ,µt admissible 0  |Vt |2 dµt dt.  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 interpolation 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 Tt (x) = ∇ψ(x) − x dt 15  We claim that Vt (Tt )# µ0 = Vt µt = (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 = dw2 (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 (x) = Vt (T (x)). dt T0 (x) = x We know the unique solution of the continuity equation is a displacement interpolation so µt = (Tt )# µ0 . We can compute 1 0  |Vt |2 dµt dt =  1 0  |Vt (Tt (x))|2 dµ0 dt ≥  16  |T1 (x) − x|2 dµ0 ≥ dw (µ0 , µ1 ).  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 definition 2.8). Note that Y1 (I, X) is closed in P1 (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 f (t, x)dη = I×X  f (t, x)dηt dλ . I X  17  (4.1)  Now we would like to study some properties of the map η→  f (t, x)dη.  (4.2)  I×X  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 → R, which is continuous in the second variable. A normal integrand is a Borel function f (t, x) : I × X → (−∞, ∞] , 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 → 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 → [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, Savar´e [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 ) → 1 as n → ∞. 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 →  I×X f n (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 integrand 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→  (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 → R, is an integrable function, and l : X → [0, ∞) is a proper function. Then for each C ∈ R, the set η ∈ Y1 (I, X) | is compact. 19  f dη ≤ C  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) |  4.2  f dη ≤ C} is closed.  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 → 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 Tx M such that η = µ ⊗ ηt,x . We define the vector field V (t, x) : I × M → T M by the expression V (t, x) =  vdηt,x (v). Tx M  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, ∂t µ + div(V µ) = 0, in the sense of distributions. We have the following characterization result.  20  (4.3)  Lemma 4.10 The measure µ satisfies equation (4.3) if and only if [∂t g + ∂x g · v] dη(t, x, v) = 0  (4.4)  I×T M  for all smooth compactly supported test functions g ∈ Cc∞ ((a, b) × M). Proof. If we disintegrate η, we have that  [∂t g + ∂x g · v] dη(t, x, v) = I×T M  [∂t g + ∂x g · vd] ηt,x (v)dµ(t, x), I×T M  for each test function. Considering the definition of V , we have the equality ∂x g · vdηt,x (v) = ∂x g ·V (t, x). TM  This means η satisfies equation (4.4) if and only if ∂t g + ∂x g ·V (t, x)dµ(t, x) = 0. I×M  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 measure, and we will denote the space of transport measures as T (I, M). For the boundary conditions we do the following, given two probability measures µi and µ f on M, we say η is a transport measure between µi and µ f , if in addition we have that [∂t g + ∂x g · v] dη(t, x, v) = I×T M  M  gb (x)dµ f −  ga (x)dµi , M  for all g : [a, b] × M → R, smooth compactly supported function. We denote by Tµi f (I, M), the set of transport measures between µi and µ f . µ  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 → 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 1  f (t, x, v)dΓ(t, x, v) =  f (t, γ(t), v)dΓt (v)dt  0  I×T M  ∀ f ∈ L1 (Γ).  Tγ(t) M  Now, for each f ∈ Cc∞ (a, b) and ϕ ∈ Cc∞ (M), let´s apply the equation (4.4) to the function g(t, x) = f (t)ϕ(x), to get f (t)ϕ(x) + f (t)dϕx · v dη(t, x, v)  0= I×T M 1  =  1  f´´(t)ϕ(γ(t))dt + 0  f (t) 0  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 ∈ Cc∞ (M).  Hence γ is absolutely continuous and .  vdΓt (v) = γ(t).  (4.5)  Tγ(t) M  Theorem 4.13 The set G (I, M) is closed in Y1 (I, T M), and the map Γ→γ 22  (4.6)  is continuous. Proof. Let Γn be a sequence of generalized curves converging to η in P1 (I, T M). The set {Γn } ∪ η is compact hence, it has uniformly integrable first moment, so if the Γn´s are generalized curves over γn , then the sequence γn is absolutely equicontinuous. Hence there exists a subsequence γnm and a curve γ0 absolutely continuous, 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 x  will consider the space ACxif of absolutely continuous curves γ : I → M, such that γ(a) = xi and γ(b) = x f , and the set x  Gxi f = Tµi f (I, M) ∩ G (I, M), µ  x  of generalized curves above elements of ACxif . We will notice convexity of L is not x  x  needed for the result in Gxi f , but it is for ACxif , which is one of the advantages of working with generalized curves. For the following results, we suppose L : [a, b] × T M → R ∪ {+∞} is a normal integrand. Definition 4.14 We say L is fiber-wise convex if, the function v → L(t, x, v), is convex on Tx M, 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+ → R, such that limr→∞ l(r)/r = ∞ and such that L(t, x, v) ≥ l( v x ) for every (t, x, v) ∈ [a, b] × Tk M. Lemma 4.16 Let L be a fiber-wise convex normal integrand. If Γ is a generalized curve above γ, then 1  L(t, γ(t), γ(t))dt ≤ 0  23  LdΓ.  Proof. Using equation (4.5) and Jensen´s inequality we have vdΓt (v)) ≤  L(t, γ(t), γ(t)) = L(t, γ(t), Tγ(t) M  L(t, γ(t), v)dΓt (v). Tγ(t) M  Hence 1  1  L(t, γ(t), γ(t))dt ≤ 0  0  L(t, γ(t), v)dΓt (v)dt =  LdΓ.  Tγ(t) M  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 x  ACg := Γ ∈ Gxi f |  LdΓ ≤ C  x  is compact in Gxi f . If L is fiber-wise convex, the set b  x  AC := γ ∈ ACxif |  L(t, γ(t), γ(t))dt ≤ C a  x  is compact in ACxif for the uniform topology. Proof. The compactness of ACg follows from theorem 4.9. convex, using lemma 4.16 we know the image of  ACg  If L is fiber-wise  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 Then we have the same conclusion as in the last theorem. 24  x −1  .  Proof. If Γ is a generalized curve over γ such that  LdΓ ≤ C, using v  x  ≤  (L(t, x, v) + 1)/c b  dt ≤  γ(t) a  γ(t)  C+b−a . c  , xi ) , which is compact since M has This means the curve γ lies in the ball B( C+b−a c 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−a , xi ) c if x ∈ / B( C+b−a , xi ) c  ∞ we have that Γ satisfies  LdΓ ≤ C if and only if  uniformly super-linear on  B( C+b−a , xi ), c  ,  LB dΓ ≤ C. Using the fact L is  we see that the quotient  LB (t, x, v) 1+ v x is bounded below. So we can use the previous theorem.  25  (4.8)  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 inequalities one can prove using mass transportation techniques is the isoperimetric inequality. Using only the arithmetic-geometric inequality in the following sense 1  n(det D2 ϕ) n ≤ tr(D2 ϕ) = ∆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 det D2 ϕ = 1. Since |∇ϕ| ≤ 1, using Gauss theorem we can compute |∂ Ω| =  − ∇ϕ · → n ds =  1ds ≥ ∂Ω  ∂Ω  1  n(det D2 ϕ) n = n |Ω| = n.  ∆ϕdx ≥ Ω  26  Ω  It has been known that there is a relationship between the isoperimetric inequality 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 BrunnMonkowski, HWI, Log-sobolev, and Gagliardo-Nirenberg. See [3],[2],[11]. Recently 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 result emphasizing on displacement convexity by proving first a general Sobolev inequality that can be used to obtain Log-Sobolev, Sobolev and Gagliardo-Nirenberg inequalities. Afterwards we will prove the Agueh-Ghoussoub-Kang’s general inequality , 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 H F (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 → xn F( xrn ) is convex and non increasing for all r > 0, then we have that H F (ρ1 ) − H F (ρ0 ) ≥  ρo (T − 1) · ∇F (ρ0 )dx, Ω  for all ρ0 , ρ1 ∈ P2,AC . Proof. Since HF (ρt ) is convex then we obtain H F (ρ1 ) − H F (ρ0 ) d F ≥ H (ρt ) 1 dt t=0 d = F(((1 − t)I + tT )# ρ0 )dx dt Ω =−  t=0  F (ρ0 )div(ρ0 (T − I))dx = Ω  ρo (T − 1) · ∇F (ρ0 )dx. Ω  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 c∗ (−∇F (ρ))ρdx + Kc .  H F+nPF (ρ) ≤  (5.1)  Ω  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 H F (ρc ) − H F (ρ) ≥  ρ(T x − x) · ∇F (ρ). Ω  We note that since ρ∇(F (ρ)) = ∇(PF (ρ)) we have that −nPF (ρ) = H −nPF (ρ).  ρ∇(F (ρ)) · x = Ω  Ω  We obtain H F (ρ) − H F (ρc ) ≤ ≤ H −nPF (ρ) −  ρ(x − T x) · ∇(F (ρ))  ρ∇(F (ρ)) · T xdx. Ω  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 H F (ρ) − H F (ρc ) ≤ H −nPF (ρ) +  c∗ (−∇F (ρ))ρdx  c(T x)ρdx + Ω  = H −nPF (ρ) +  Ω  c∗ (−∇F (ρ))ρdx.  c(x)ρc dx + Ω  Ω  Finally we get c∗ (−∇F (ρ))ρdx +  H F+nPF (ρ) ≤ Ω  (c(x) + F (ρc ))ρc dx − H PF (ρc ) Ω  We name the constant c(x) + F (ρc ) = Kc , and we note that H PF (ρc ) ≥ 0 to conclude the proof. In the following pages we will see that using different F’s this inequality generalizes 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  where σc =  ρ log ρdx ≤  n p log( p/n p ne p−1 σc  Rn  ρc∗ (−  ∇ρ )dx), ρ  −c(x) dx. Rn e  Proof. Let F(x) = x log(x), and F(0) = 0. We check that x → xn F(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 , H PF = ρ = 1. We take inequality (5.1), c∗ (−∇F (ρ))ρdx +  H F+nPF (ρ) ≤ Ω  29  (F (ρc ) + c)ρc dx,  (5.2)  where ρc is a solution of the equation ∇(log ρc + c) = 0, which we take ρc (x) = e−c(x) /σc , so we get  ρ log ρ + n ≤  c∗ (−  ∇ρ )ρdx + ρ  log e−c(x) − log(  Rn  e−c(x) dx) + c ρc dx (5.3)  =  c∗ (−  ∇ρ )ρdx − log( ρ  e−c(x) dx).  (5.4)  Rn  Let cλ (x) := c(λ x), hence c∗λ (y) = c∗ ( λy ). If we apply the inequality to this Young function we get ρ log ρ + n ≤ = Considering that c∗ ( λy ) =  ∇ρ )ρdx − log( λρ ∇ρ c∗ (− )ρdx − log( λρ c∗ (−  1 ∗ λ p c (y),  λop =  e−c(λ x) dx) Rn  Rn  e−c(x) dx) + n log λ .  the infimum over λ is attained when  p n  c∗ (−  ∇ρ )ρdx. ρ  So we get the inequality for all probability densities ρ, with supp(ρ) ⊂ Ω, and ρ ∈ W 1,∞ (Rn ) ρ log ρ ≤  n p  c∗ (− ∇ρ λ ρ )ρdx  c∗ (−  ∇ρ n p )ρdx − log(σc ) + log ρ p n  n p ∇ρ n log c∗ (− )ρdx − log(σc ) − n + p n λρ p n p ∇ρ = log( ρc∗ (− )dx). p/n Rn p−1 p ρ ne σc ≤  30  c∗ (−  ∇ρ )ρdx − n λρ  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]. np Corollary 5.6 (Gagliardo-Nirenberg) Let 1 < p < n and r ∈ (0, n−p ) such that  r = p. We define γ := 1r + 1q , where  1 p  + 1q = 1. For any f ∈ W 1,p (Rn ) we have that  there exists θ such that f  r  ≤ C(p, r) ∇ f  θ p  f  Proof. We will use inequality (5.1) with F(x) = γ = 1 and since r ∈ (0,  np n−p  1−θ rγ  .  xγ γ−1 .  Since r = p we have that  ), we have that 1 > γ > 1 − 1n . To use the inequality xn−nγ γ−1 rγ q |x| , q  we check that F(0) = 0, and x → xn F(x−n ) =  is convex and non increasing  since n − nγ < 1 and γ − 1 < 0. Let c(x) =  so c∗ (x) =  1 p(rγ) p−1  |x| p . Using  inequality (5.1) we get  F(ρ) + nρF (ρ) − nF(ρ)dx ≤  1 (−∇F (ρ)) p ρdx + Kc . p(rγ) p−1  ρ Ω  Making the substitution of F(x) = 1 γ −1  xγ γ−1  ρ γ − nρ γ + nγρ γ dx ≤  we get  1 (−γρ γ−2 ∇ρ) p ρdx + Kc , p(rγ) p−1  ρ Ω  and rearranging the equation we get (  1 + n) γ −1  If we suppose that f (  r  ρ γ dx ≤  ρ Ω  rγ (∇ρ) p ρdx + Kc . p(r) p  = 1, we take ρ = | f |r to get  1 + n) γ −1  | f |rγ dx ≤  and for general f we get this inequality  31  rγ |∇ f | p ρdx + Kc , Ω p  (5.5)  p 1 rγ ∇ f p +n p − p f r γ −1  f  rγ rγ  f  r  ≥ −Kc .  If we have the function fλ (x) = f (λ x), with a change of variables we get the following equalities p r fλ rγ rγ  fλ  fλ  r p ∇ fλ p  = λ −np/r f = λ −n f  p r ,  rγ rγ ,  = λ −n/r f  r,  = λ ∇ f (xλ )  p p  p p.  = λ p−n ∇ f  So the inequality becomes λ p−n+np/r We take λ = ∇ f  p rγ ∇ f p − λ −n+n/r p f rp a p  f  b r  f  c rγ ,  1 +n γ −1  f  rγ rγ  f  r  ≥ −Kc .  and we pick a, b, and c, so that the powers of  the norms are the same in both terms, that is pr , pr + np − n (p − 1)r b= , pr + np − n r c= . pr + np − n a=  So we obtain 1 Kc  −  rγ 1 + −n p γ −1  ∇f  32  a p  f  c rγ  ≥ f  b r  ,  where −npr + np pr + np − n (p − 1)r(−n + n/r) −1 b = pr + np − n rp − nr + np c = pr + np − n a =  Finally we note that if we take the limit as r → p∗ = and a , b →  np(n−np−p) (n−p)(pr+np−n)  we have that c → 0,  so we get the Sobolev inequality f  5.2  np n−p ,  p∗  ≤ C(p, n) ∇ f  p.  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 in P2,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 HVF,W (ρ) := H F (ρ) + HV (ρ) + H W (ρ). Where we have ·Internal energy H F (ρ) :=  F(ρ(x))dx, . Ω  ·Potential energy HV (ρ) =:  ρ(x)V (x)dx. Ω  ·Interaction energy H W (ρ) =:  1 2 33  ρ(W ∗ ρ), Ω  where ∗ denotes the convolution product. Furthermore we define the relative energy of ρ0 with respect to ρ1 as HVF,W (ρ0 | ρ1 ) := HVF,W (ρ1 ) − HVF,W (ρ0 ), and the relative entropy production of ρ with respect to ρV as 2  I2 (ρ | ρV ) :=  ∇(F (ρ) +V +W ∗ ρ) ρdx. Ω  So if ρV is a probability density that satisfies ∇(F (ρV ) +V +W ∗ ρ) = 0, then 2  I2 (ρ | ρV ) :=  ∇(F (ρ) − F (ρV ) +W ∗ (ρ − ρV )) ρdx. Ω  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 measured against c∗ as c∗ −∇(F (ρ0 ) +V +W ∗ ρ0 ) ρ0 dx,  Ic∗ (ρ0 | ρV ) := Ω  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 ) ≥  ρo (T − 1) · ∇V dx + Ω  λ 2 d (ρ0 , ρ1 ), 2 w  for all ρ0 , ρ1 ∈ P2,AC . Proof. Expanding and using D2V ≥ λ I, we obtain V (b) −V (a) ≥ ∇V (a) · (b − a) +  34  λ |a − b|2 . 2  This means that V (T x) −V (x) ≥ ∇V (x) · (T x − x) +  λ |x − T x|2 . 2  Hence integrating we obtain HV (ρ1 ) − HV (ρ0 ) ≥  V (T x)ρ0 −V (x)ρ0 dx  λ |x − T x|2 ρ0 dx 2 Ω λ = ρo (T − 1) · ∇V dx + dw2 (ρ0 , ρ1 ). 2 Ω ≥  ∇V (x) · (T x − x) +  Lemma 5.9 Assume W:Rn → R is even and satisfies that D2W ≥ νI, for some ν ∈ R, then we have that H W (ρ1 )−H W (ρ0 ) ≥  ν ρo (T −1)·∇(W ∗ρ0 )dx+ (dw2 (ρ0 , ρ1 )−|b(ρ0 ) − b(ρ1 )|2 ), 2 Ω  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 1 2 1 = 2 1 = 2  H W (ρ1 ) =  W (x − y)ρ1 (x)ρ1 (y)dxdy Ω×Ω  W (T x − Ty)ρ0 (x)ρ0 (y)dxdy Ω×Ω  W (x − y + (T − I)(x) − (T − I)(y))ρ0 (x)ρ0 (y)dxdy Ω×Ω  35  Since D2W ≥ νI we obtain 1 2 ν + 4  H W (ρ1 ) ≥  [W (x − y) + ∇W (x − y) · ((T − I)(x) − (T − I)(y))] ρ0 (x)ρ0 (y)dxdy Ω×Ω  |(T − I)(x) − (T − I)(y)|2 ρ0 (x)ρ0 (y)dxdy Ω×Ω  = H W (ρ0 ) + +  ν 4  1 2  ∇W (x − y) · ((T − I)(x) − (T − I)(y))ρ0 (x)ρ0 (y)dxdy Ω×Ω  |(T − I)(x) − (T − I)(y)|2 ρ0 (x)ρ0 (y)dxdy. Ω×Ω  Now we note the following equalities, for the last term |(T − I)(x) − (T − I)(y)|2 ρ0 (x)ρ0 (y)dxdy Ω×Ω 2  |(T − I)(x)|2 ρ0 (x)dx − 2  =2 Ω×Ω  (T − I)(x)ρ0 (x)dx Ω×Ω  |(T − I)(x)|2 ρ0 (x)dx − |b(ρ1 ) − b(ρ0 )|2 .  =2 Ω×Ω  For the second term we consider that ∇W is odd ∇W (x − y) · ((T − I)(x) − (T − I)(y))ρ0 (x)ρ0 (y)dxdy Ω×Ω  ∇W (x − y) · ((T − I)(x))ρ0 (x)ρ0 (y)dydx  =2 Ω×Ω  (∇W ∗ ρ0 ) · (T − I)(x))ρ0 (x)dx.  =2 Ω×Ω  Using these two equalities we get H W (ρ1 )−H W (ρ0 ) ≥  ν ρo (T −1)·∇(W ∗ρ0 )dx+ (dw2 (ρ0 , ρ1 )−|b(ρ0 ) − b(ρ1 )|2 ). 2 Ω  Theorem 5.10 (Basic inequality) Under the hypotheses of the three previous lemmas, 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 λ +ν 2 ν dw (ρ0, ρ1 ) − |b(ρ0 − b(ρ1 )|2 2 2 −nPF ,2x·∇W ≤ Hc+∇V ·x (ρ0 ) + Ic∗ (ρ0 | ρV ).  HVF,W +c (ρ0 | ρ1 ) +  Proof. First we note that since ρ0 ∇(F (ρ0 )) = ∇(PF (ρ0 )) we have that ρ0 ∇(F (ρ0 ) +V +W ∗ ρ0 ) · x Ω  −nPF (ρ0 ) + ρ0 [∇(V + ∇W ∗ ρ0 )] · x  = Ω  =  1 −nPF (ρ0 ) + ρ0 ∇V · x + ρ0 (2x · ∇W ∗ ρ0 )dx 2 Ω  −nPF ,2x·∇W = H∇V (ρ0 ). ·x  If we add the inequalities from the previous lemmas we get HVF,W (ρ1 ) − HVF,W (ρ0 ) ≥  ρ0 (T x − x) · ∇(F (ρ0 ) +V +W ∗ ρ0 )dx + Ω  λ 2 d (ρ0 , ρ1 ) 2 w  ν + (dw2 (ρ0 , ρ1 ) − |b(ρ0 ) − b(ρ1 )|2 ). 2 Rearranging and using the first inequality we have HVF,W (ρ0 ) − HVF,W (ρ1 ) + ≤  λ +ν 2 dw (ρ0 , ρ1 ) − |b(ρ0 ) − b(ρ1 )|2 ) 2  ρ0 (x − T x) · ∇(F (ρ0 ) +V +W ∗ ρ0 ) Ω  −nPF ,2x·∇W ≤ H∇V (ρ0 ) − ·x  ρ0 ∇(F (ρ0 ) +V +W ∗ ρ0 ) · T xdx. Ω  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 HVF,W (ρ0 ) − HVF,W (ρ1 ) + −nPF ,2x·∇W ≤ H∇V (ρ0 ) + ·x  =  −nPF ,2x·∇W H∇V (ρ0 ) + ·x  λ +ν 2 dw (ρ0 , ρ1 ) − |b(ρ0 ) − b(ρ1 )|2 ) 2 c∗ (−∇(F (ρ0 ) +V +W ∗ ρ0 )ρ0 dx  c(T x)ρ0 dx + Ω  Ω  c∗ (−∇(F (ρ0 ) +V +W ∗ ρ0 )ρ0 dx.  c(x)ρ1 dx + Ω  Ω  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 F ,W −2x·∇W HVF+nP (ρ) ≤ −H PF ,W (ρV +c ) + Ic∗ (ρ | ρV ) + KV +c . −∇V ·x  (5.6)  Furthermore if we set W = V = 0, since H PF (ρc ) ≥ 0, we obtain H F+nPF (ρ) ≤ −H PF (ρc ) + Ic∗ (ρ | ρV ) + KV +c c∗ (−∇F (ρ))ρdx + KV +c .  ≤  (5.7) (5.8)  Ω  Hence recovering inequality (5.1). Proof. Let´s consider the inequality we just proved ν λ +ν 2 dw (ρ0, ρ1 ) − |b(ρ0 − b(ρ1 )|2 2 2 −nPF ,2x·∇W ≤ Hc+∇V ·x (ρ0 ) + Ic∗ (ρ0 | ρV ).  HVF,W +c (ρ0 | ρ1 ) +  (5.9) (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 F ,W −2x·∇W HVF+nP (ρ) + −∇V ·x  λ +ν 2 ν dw (ρ, ρV +c ) − |b(ρ − b(ρ1 )|2 2 2  ≤ −H PF ,W (ρV +c ) + Ic∗ (ρ | ρV ) +  (5.11)  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 ν λ +ν 2 dw (ρ, ρV +c ) − |b(ρ) − b(ρV +c )|2 2 2 2 λ +ν ν 2 |T x − x| ρ0 (x)dx − (T x − x)ρ0 (x)dx ≥ 0. = 2 2 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 function.  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 1 1 σ HUF (ρ0 | ρ1 ) + (µ − )W22 (ρ0 , ρ1 ) ≤ 2 σ 2  2  ρ ∇(F ◦ ρ0 +U dx. Ω  Proof. If we take the basic inequality with c(x) = V = U − c. Hence we have that  c∗ (p)  =  1 2σ  2  |σ p| =  2 1 2σ |x| , W 2 σ 2 |p| , so  = 0, and we set, using the general  inequality we get HUF (ρ0 | ρ1 )+  (µ − σ −1 ) 2 σ −nPF dw (ρ0, ρ1 ) ≤ Hc+∇(U−c)·x (ρ0 )+ 2 2 39  ρ0 ∇(F ◦ρ0 +U −c)dx. Ω  We can compute σ 2  2  ρ0 ∇(F ◦ ρ0 +U − c) dx Ω  σ = 2  2  ρ ∇(F ◦ ρ0 +U dx + Ω  1 2σ  ρ0 |x|2 dx − Ω  xρ0 · ∇(F ◦ ρ0 +U)dx, Ω  and nPF Hc+∇(U−c)·x (ρ0 ) = H nPF (ρ0 ) −  ρx · ∇Udx + Ω  1 2σ  |x|2 ρ0 dx. Ω  By combining the two and using integration by parts we get that −nPF Hc+∇(U−c)·x (ρ0 ) +  = = = = =  σ 2 σ 2 σ 2 σ 2 σ 2  σ 2  2  ρ0 ∇(F ◦ ρ0 +U − c) dx Ω 2  xρ0 · ∇(F ◦ ρ0 )dx − H nPF (ρ0 )  ρ ∇(F ◦ ρ0 +U dx − Ω  Ω 2  div(xρ0 ) · (F ◦ ρ0 )dx − H nPF (ρ0 )  ρ ∇(F ◦ ρ0 +U dx + Ω  Ω 2  ρ ∇(F ◦ ρ0 +U dx + Ω 2  ρ ∇(F ◦ ρ0 +U dx + Ω  x · ∇F(ρ0 )dx − H nPF (ρ0 )  nρ0 · (F ◦ ρ0 )dx + Ω  Ω  x · ∇F(ρ0 )dx + Ω  n(F ◦ ρ0 )dx Ω  2  ρ ∇(F ◦ ρ0 +U dx. Ω  Returning to the first inequality we get that HUF (ρ0 | ρ1 ) +  (µ − σ −1 ) 2 σ dw (ρ0, ρ1 ) ≤ 2 2  2  ρ ∇(F ◦ ρ0 +U) dx. Ω  Corollary 5.13 Furthermore if we take µ > 0, that is, take U is uniformly convex, take σ = µ1 , we can get the Generalized Log-Sobolev inequality: HUF (ρ0 | ρ1 ) ≤  1 2µ  2  ρ ∇(F ◦ ρ0 +U) dx = Ω  1 I2 (ρ0 | ρU ). 2µ  Corollary 5.14 (HWI ) Finally we can obtain the generalized HW I-inequality,  40  which is originally due to Otto and Villani (see [5]). HUF (ρ0 | ρ1 ) +  µ 2 d (ρ0, ρ1 ) ≤ 2 w  I2 (ρ0 | ρU )dw (ρ0, ρ1 ).  Proof. If we write the inequality of the last corollary as HUF (ρ0 | ρ1 ) +  µ 2 σ 1 2 dw (ρ0, ρ1 ) ≤ I2 (ρ0 | ρU ) + d (ρ0, ρ1 ) 2 2 2σ w  d (ρ ρ ) and minimize over σ , we obtain the minimum when σ = √ w 0, 1 , we can write I2 (ρ0 |ρU )  the inequality as HUF (ρ0 | ρ1 ) +  5.2.2  µ 2 d (ρ0, ρ1 ) ≤ 2 w  I2 (ρ0 | ρU )dw (ρ0, ρ1 ).  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 f ρU = 1, we have that 1 σ 1 f log( f )ρU + (µ − )W22 (ρ0 , ρ1 ) ≤ 2 σ 2  ρU Ω  |∇ f |2 dx. f  Where ρU = e−U / e−U dx. Proof. The proof follows from corollary 5.12, taking ρ0 = ρU , ρ1 = f ρU , and F(x) = x log x. So we compute  HUF (ρU ) = =  ρU log ρU +UρU dx = 1 e−U dx  e−U (− log  e−U / e−U ) = − log  41  e−U dx log e−U / e−U , and  e−U dx +U e−U /  e−U dx  HUF ( f ρU ) =  f ρU log f ρU +U f ρU =  =  f log( f )ρU − log  =  f log( f )ρU − log  Hence HUF (ρ0 | ρ1 ) =  1 e−U dx  e−U  e−U f (log f − log  e−U )  f ρU  e−U  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 ≤  1 µ  42  ρU Ω  |∇ f |2 dx. f  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. Proceedings of the workshop: Mass Transportation Methods in Kinetic Theory and Hydrodynamics, 2003. → pages 2, 27 [4] L. Evans. Partial Differential Equations. AMS, 2005. → pages 31 [5] C. V. F. Otto. Generalization of an inequality by talagrand and links with the logarithmic sobolev inequality. J. Funct. Anal., 173:361–400, 2000. → pages 41 [6] L. Gross. Logarithmic sobolev inequa. Amer. J. Math, 97:1061–1083, 1975. → pages 29 [7] J.-M. L. H. Berliocchi. Intgrands normales and mesures paramtres en calcul des variations. Bull. Soc. Math. France, 101:129–184, 1973. → pages 18 [8] Y. B. J.-D Benamou. A computational fluid mechics solution to the monge-kantorovich mass transfer problem. Numerische Mathematik, 84(3): 375–393, 2000. → pages 1, 10 [9] G. S. L. Ambrosio, N. Gigli. Gradient Flows in Metric Spaces and in the Space of Probability Measures. Lectures in Math. ETH Zurich, 2005. → pages 17, 18, 19, 20  43  [10] X. K. M. Agueh, N. Ghossoub. Geometric inequalities via a general comparison principle for interacting gases. Geometric and Functional Analysis, 14(1):215–244, 2004. → pages 2, 3, 27, 33 [11] R. McCann. A canvexity principle for interacting gases. Adv. Math, 128(1): 153–179, 1997. → pages 2, 27 [12] N.-S. P. S. Hu. Handbook of Multivalued Analysis. Kluwer Academic Publishers, 1975. → pages 19 [13] C. Villani. Topics in Optimal Transportation. AMS, 2003. → pages 1, 4, 6, 26 [14] L. Young. Lectures on the Calculus of Variations and Optimal Control Theory. Chelsea, 2005. → pages 17  44  

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