{"Affiliation":[{"label":"Affiliation","value":"Science, Faculty of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."},{"label":"Affiliation","value":"Mathematics, Department of","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","classmap":"vivo:EducationalProcess","property":"vivo:departmentOrSchool"},"iri":"http:\/\/vivoweb.org\/ontology\/core#departmentOrSchool","explain":"VIVO-ISF Ontology V1.6 Property; The department or school name within institution; Not intended to be an institution name."}],"AggregatedSourceRepository":[{"label":"AggregatedSourceRepository","value":"DSpace","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","classmap":"ore:Aggregation","property":"edm:dataProvider"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/dataProvider","explain":"A Europeana Data Model Property; The name or identifier of the organization who contributes data indirectly to an aggregation service (e.g. Europeana)"}],"Campus":[{"label":"Campus","value":"UBCV","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","classmap":"oc:ThesisDescription","property":"oc:degreeCampus"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeCampus","explain":"UBC Open Collections Metadata Components; Local Field; Identifies the name of the campus from which the graduate completed their degree."}],"Creator":[{"label":"Creator","value":"Garcia Ramos Aguilar, Felipe","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/creator","classmap":"dpla:SourceResource","property":"dcterms:creator"},"iri":"http:\/\/purl.org\/dc\/terms\/creator","explain":"A Dublin Core Terms Property; An entity primarily responsible for making the resource.; Examples of a Contributor include a person, an organization, or a service."}],"DateAvailable":[{"label":"DateAvailable","value":"2010-10-28T18:34:55Z","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"edm:WebResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"DateIssued":[{"label":"DateIssued","value":"2010","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/issued","classmap":"oc:SourceResource","property":"dcterms:issued"},"iri":"http:\/\/purl.org\/dc\/terms\/issued","explain":"A Dublin Core Terms Property; Date of formal issuance (e.g., publication) of the resource."}],"Degree":[{"label":"Degree","value":"Master of Science - MSc","attrs":{"lang":"en","ns":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","classmap":"vivo:ThesisDegree","property":"vivo:relatedDegree"},"iri":"http:\/\/vivoweb.org\/ontology\/core#relatedDegree","explain":"VIVO-ISF Ontology V1.6 Property; The thesis degree; Extended Property specified by UBC, as per https:\/\/wiki.duraspace.org\/display\/VIVO\/Ontology+Editor%27s+Guide"}],"DegreeGrantor":[{"label":"DegreeGrantor","value":"University of British Columbia","attrs":{"lang":"en","ns":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","classmap":"oc:ThesisDescription","property":"oc:degreeGrantor"},"iri":"https:\/\/open.library.ubc.ca\/terms#degreeGrantor","explain":"UBC Open Collections Metadata Components; Local Field; Indicates the institution where thesis was granted."}],"Description":[{"label":"Description","value":"In this thesis we will review some recent results of Optimal Mass Transportation emphasizing on the role of displacement interpolation and displacement\nconvexity. We will show some of its recent applications, specially the ones by Bernard, and Agueh-Ghoussoub-Kang.","attrs":{"lang":"en","ns":"http:\/\/purl.org\/dc\/terms\/description","classmap":"dpla:SourceResource","property":"dcterms:description"},"iri":"http:\/\/purl.org\/dc\/terms\/description","explain":"A Dublin Core Terms Property; An account of the resource.; Description may include but is not limited to: an abstract, a table of contents, a graphical representation, or a free-text account of the resource."}],"DigitalResourceOriginalRecord":[{"label":"DigitalResourceOriginalRecord","value":"https:\/\/circle.library.ubc.ca\/rest\/handle\/2429\/29637?expand=metadata","attrs":{"lang":"en","ns":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","classmap":"ore:Aggregation","property":"edm:aggregatedCHO"},"iri":"http:\/\/www.europeana.eu\/schemas\/edm\/aggregatedCHO","explain":"A Europeana Data Model Property; The identifier of the source object, e.g. the Mona Lisa itself. This could be a full linked open date URI or an internal identifier"}],"FullText":[{"label":"FullText","value":"Mass transport and geometric inequalities by Felipe Garc\u0131\u0301a Ramos Aguilar B. Sc. Mathematics, Universidad Nacional Auto\u0301noma de Me\u0301xico, 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\u00a9 Felipe Garc\u0131\u0301a 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\u2019s 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; Ramo\u0301n Za\u0301rate 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 llego\u0301, dejo\u0301 su legado y se fue\u0301. ...y al Pollo tambie\u0301n. vi Chapter 1 Introduction Mathematics is the art of giving the same name to different things. \u2014 J.H. Poincare\u0301 The most basic problem of modern mass transportation is the Monge-Kantorovich problem for quadratic cost function, that is, given the measures \u00b50 and \u00b51 of Rn find a measure \u03b30 of Rn\u00d7Rn, that satisfies inf \u03b3\u2208\u0393(\u00b50,\u00b51) \u222b Rn\u00d7Rn |x\u2212 y|2 d\u03b3(x,y) = \u222b Rn\u00d7Rn |x\u2212 y|2 d\u03b30(x,y), where \u0393(\u00b50,\u00b51) is the set of measures in Rn\u00d7Rn with marginals \u00b50 and \u00b51. This actually defines a distance dw(\u00b50,\u00b51) 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 (\u00b5t ,Vt) , that minimize inf \u222b 1 0 \u222b |Vt |2 d\u00b5tdt, where \u00b5t 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 \u00b5t +div(V\u00b5) = 0. 1 In Chapter 2 we will show the proof of Brenier\u2019s 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\u2019s problem by considering instead of classical flows Vt , generalized flows \u03b7t,x(v), which are probability mea- sures that indicate the probability of having velocity v \u2208 T M at the point x in time t. To generalize the continuity equation we will define a Transport measure, as a Young measure \u03b7 that, for a set of test functions g, satisfies\u222b I\u00d7T M [\u2202tg+\u2202xg \u00b7 v]d\u03b7t,x(v)d\u00b5(t,x) = \u222b I\u00d7T M [\u2202tg+\u2202xg \u00b7 v]d\u03b7(v, t,x) = 0. (1.1) For certain functionals L we will show existence of minimizers of \u222b Ld\u03b7 , first in the case where \u03b7 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 \u03c1 that represents density, force F , pressure PF , the internal energy functional H, the Young\u2019s function c\u2217, and a constant Kc, HF+nPF (\u03c1)\u2264 \u222b \u2126 c\u2217(\u2212\u2207F \u2032(\u03c1))\u03c1dx+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\u2019s function c and its dual c\u2217, energy functionals HF,WV , relative entropy production Ic\u2217(\u03c10 | \u03c1V ), Wasserstein distance d2w(\u03c10,\u03c11) and barycentre b(\u03c10), namely HF,WV+c(\u03c10 | \u03c11)+ \u03bb +\u03bd 2 d2w(\u03c10,\u03c11)\u2212 \u03bd 2 |b(\u03c10)\u2212b(\u03c11)|2 \u2264 H\u2212nPF ,2x\u00b7\u2207Wc+\u2207V \u00b7x (\u03c10)+Ic\u2217(\u03c10 | \u03c1V ). 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\u00b4s, 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 ,\u00b5) to arbitrary space Y , we can equip Y with the pushforward measure T#\u00b5 , where T#\u00b5(B) = \u00b5(T\u22121(B)), for every set B\u2282 Y. We will denote the space of probability measures of X , asP(X) , andPAC(X)\u2282 P(X) the space of absolutely continuous probability measures. Let M\u00b1 \u2282 Rn be two compact sets , and \u00b5\u00b1 \u2208P(M\u00b1). We denote the projection functions as pi+ : M+ \u00d7M\u2212 7\u2192 M+, where pi+(x,y) = x, and pi\u2212 : M+ \u00d7M\u2212 7\u2192 M\u2212, where pi\u2212(x,y) = y. So we define \u0393(\u00b5+,\u00b5\u2212) = { \u03b3 \u2208P(M+\u00d7M\u2212) | (pi+)#\u03b3 = \u00b5+ and (pi\u2212)#\u03b3 = \u00b5\u2212 } . Let c : M+\u00d7M\u2212 7\u2192R, be a continuous cost function. The Kantorovich problem is to minimize the total cost defined as C (\u03b3) = \u222b M+\u00d7M\u2212 c(x,y)d\u03b3, where \u03b3 \u2208 \u0393(\u00b5+,\u00b5\u2212) . As we shall see the existence of minimizers is not hard, as we have that C is linear 4 in \u03b3 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 \u0393(\u00b5+,\u00b5\u2212) is closed, it is weak-* compact. Since C (\u03b3) is continuous in the weak-* topology, it attains a minimum in the compact set. We will say \u03b3 is optimal if \u03b3 \u2208 \u0393op(\u00b5+,\u00b5\u2212) := { \u03b3 \u2208 \u0393(\u00b5+,\u00b5\u2212) | C (\u03b3) = inf \u03b3\u2208\u0393(\u00b5+,\u00b5\u2212) \u222b M+\u00d7M\u2212 c(x,y)d\u03b3 } . 2.0.1 Brenier\u2019s theorem First we state two definitions and their relationship. Definition 2.2 Given a set \u0393 \u2282 X \u00d7Y, and a cost function c(x,y), we say that \u0393 is c-cyclically monotone if for any finite set of pairs {(xi,yi) | 1\u2264 i\u2264 N} \u2282 \u0393. we have that N \u2211 i c(xi,yi)\u2264 N \u2211 i c(xi,yi+1). (with N+1=1) Definition 2.3 A function \u03c6 : X 7\u2192 R\u222a{+\u221e} is said to be c-convex if it is not identical to {+\u221e} , and there exists a function g : Y 7\u2192 R\u222a{+\u221e} , such that \u03c6(x) = sup y\u2208Y (g(y)\u2212 c(x,y)) \u2200x \u2208 X . It\u00b4s c-transform is the function \u03c6 c(y) := inf x\u2208X (\u03c6(x)+ c(x,y)), and it\u2019s c-subdifferential is the c-cyclically monotone set \u2202 c\u03c6 := {(x,y) \u2208 X\u00d7Y | \u03c6 c(y)\u2212\u03c6(x) = c(x,y)} . 5 Remark 2.4 The fact that a set \u0393 is c-cyclically monotone if and only if \u0393 = \u2202 c\u03c6 for a c-convex function \u03c6 is called Rockafellers\u2019s theorem, a proof can be found in [13]. Remark 2.5 If we take X =Y =Rn and c(x,y) =\u2212x \u00b7y, 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 \u03b3 is c-cyclically monotone. Proof. If supp \u03b3 is not c-cyclically monotone then we have that, this means there exists a set of pairs {(xi,yi) | 1\u2264 i\u2264 N} \u2282 \u0393 such that N \u2211 i c(xi,yi)\u2212 N \u2211 i c(xi,yi+1)> 0. Even more, since c is continuous we can find a set of open neighbourhood Ui\u00d7Vi, of (xi,yi) such that N \u2211 i c(ui,vi)\u2212 N \u2211 i c(ui,vi+1)> 0 \u2200 (ui,vi) \u2208Ui\u00d7Vi. We define the measures \u03b3i(E) = \u03b3(E\u2229(Ui\u00d7Vi))\u03b3(Ui\u00d7Vi) ; let \u03b7 = \u03a0\u03b3i a measure of (X \u00d7Y )N and Hi = ( fi,gi ) the projections such that \u03b3i = Hi#\u03b7 . We define \u03b3\u0303 = \u03b3+ \u03bb n N \u2211 i ( fi+1\u00d7gi)#\u03b7\u2212 ( fi\u00d7gi)#\u03b7 , where \u03bb = inf\u03b3(Ui\u00d7Vi)> 0.We check that (pi+)#\u03b3\u0303 = \u00b5++ \u03bbn \u2211Ni ( fi+1)#\u03b7\u2212( fi)#\u03b7 = \u00b5+, similarly (pi\u2212)#\u03b3\u0303 = \u00b5\u2212, and \u03b3\u0303(X\u00d7Y ) = 1+ \u03bbn \u2211Ni \u03b3((Ui+1\u00d7Vi))\u03b3(Ui+1\u00d7Vi) \u2212 \u03b3((Ui\u00d7Vi)) \u03b3(Ui\u00d7Vi) = 1. Finally we compute C (\u03b3)\u2212C (\u03b3\u0303) = \u222b M+\u00d7M\u2212\u2211c( fi+1,gi)\u2212 c( fi,gi)d\u03b7 > 0 Now we proceed to prove a theorem by Brenier, that applies to quadratic cost functions, i.e. c(x,y) = 12 |x\u2212 y|2 . 6 Theorem 2.7 Let M\u00b1 \u2282 Rn be open and bounded sets, and \u00b5\u00b1 \u2208 PAC(M\u00b1). Hence there exists an optimal mapping T#\u00b5+ = \u00b5\u2212, and \u03d5 convex function such that T = \u2207\u03d5 almost everywhere. Proof. By the previous results, we know there exists an optimal measure \u03b3op \u2208 \u0393op(\u00b5+,\u00b5\u2212), let S be a maximal c-cyclical monotone set, hence supp\u03b3op \u2282 S . By Rockafellar theorem, there exists a c-convex function \u03c6 , such that S = \u2202 c\u03c6(x), by definition of c-convexity we have that \u03c6(x) = sup y\u2208M\u2212 {\u2212c(x,y)\u2212\u03c6 c(y)}= sup y\u2208M\u2212 { \u22121 2 |x\u2212 y|2\u2212\u03c6 c(y) } = sup y\u2208M\u2212 { \u22121 2 |x|2\u2212 1 2 |y|2+ xy\u2212\u03c6 c(y) } . Now we can define a function \u03d5, \u03d5(x) := \u03c6(x)+ 1 2 |x|2 = sup y\u2208M\u2212 { \u22121 2 |y|2+ xy\u2212\u03c6 c(y) } . This function is convex and bounded and moreover, \u2202\u03d5(x) = \u2202 c\u03c6 . This means \u03d5 is Lipschitz and hence differentiable almost everywhere by Rademacher\u2019s theorem So almost everywhere \u2202\u03d5(x) = \u2207\u03d5(x). Monge-Ampere equation Let M\u00b1 \u2282 Rn be open and bounded sets, and \u00b5\u00b1 \u2208PAC(M\u00b1), that is \u00b5+ = f dx and \u00b5\u2212 = gdy, from the last theorem there exists an optimal mapping T#\u00b5+ = \u00b5\u2212, and \u03d5 convex function such that T = \u2207\u03d5, in other words we have that\u222b \u03c8(y)g(y)dy = \u222b \u03c8(\u2207\u03d5(x)) f (x)dx for all test functions \u03c8. 7 Alexandrov\u2019s theorem says that D2\u03d5 exists almost everywhere if \u03d5 is convex, so if we do a change of variables y = \u2207 \u03d5(x) we get that\u222b \u03c8(y)g(y)dy = \u222b \u03c8(\u2207\u03d5(x))g(\u2207\u03d5(x))detD2\u03d5dx. From this we get the Monge-Ampere\u2019s equation g(\u2207\u03d5)detD2\u03d5 = 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 \u00b5 \u2208 X with finite nth moment, which means\u222b |d(x0,x)|n d\u00b5(x)< \u221e for some x0 \u2208 X , and hence for any x0 \u2208 X . Definition 2.8 We can define a distance in the spacePn(X) is defined as follows, dn(\u00b5,\u03b7) = \u2223\u2223\u2223\u2223min\u03bb \u222b X\u00d7X |d(x,y)|n d\u03bb (x,y) \u2223\u2223\u2223\u22231\/n . Where \u03bb runs along de probability measures on X\u00d7X, whose marginals are \u00b5 and \u03b7 . 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\u00b4s consider C1(X), the space of functions f : X 7\u2192 R, such that sup x\u2208X | f (x)| 1+d(x0,x) < \u221e for one (and hence for all) x0 \u2208 X . Then we have that d1(\u00b5n,\u00b5) 7\u2192 0 if and only if\u222b f d\u00b5n 7\u2192 \u222b f d\u00b5 8 for every f \u2208C1(X). We also define a weaker topology , called the narrow topology Definition 2.9 We say \u00b5n converges narrowly to \u00b5 if\u222b f d\u00b5n 7\u2192 \u222b f d\u00b5 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 \u00b5+,\u00b5\u2212 \u2208P2(Rn) and \u03b30 \u2208 \u0393(\u00b5+,\u00b5\u2212) be the solution to the Kantorovich problem with quadratic cost . For every s \u2208 [0,1] we define pis : Rn\u00d7 Rn\u2192 Rn, as pis(x,y) := (1\u2212 s)x+ sy, and we will call \u00b5s = (pis)#\u03b30 the displacement interpolation between \u00b5+ and \u00b5\u2212. 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 \u00b5s be the displacement interpolation between \u00b50 and \u00b51 then 10 \u2200s, t \u2208 [0,1] , we have that dw(\u00b5t ,\u00b5s) = |t\u2212 s|dw(\u00b50,\u00b51) . Proof. First we take pist = (pis\u00d7pit)# \u03b30 = ((1\u2212 s)x+ sy),(1\u2212 t)x+ ty)#\u03b30 \u2208 \u0393(\u00b5s,\u00b5t). So that d2w(\u00b5t ,\u00b5s)\u2264 \u222b |x\u2212 y|2 dpist = \u222b |(1\u2212 s)x+ sy\u2212 ((1\u2212 t)x+ ty)|2 d\u03b30 = (t\u2212 s)2 \u222b |x\u2212 y|2 d\u03b30 = (t\u2212 s)2d2w(\u00b50,\u00b51). To get the equality use the triangle inequality dw(\u00b50,\u00b51)\u2264 dw(\u00b50,\u00b5s)+dw(\u00b5s,\u00b5t)+dw(\u00b5t ,\u00b51) \u2264 sdw(\u00b50,\u00b51)+dw(\u00b5s,\u00b5t)+(1\u2212 t)dw(\u00b50,\u00b51). So we conclude that dw(\u00b5t ,\u00b5s) = |t\u2212 s|dw(\u00b50,\u00b51) . 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)\u2282P2\u2192 R is displacement convex if H(\u03c1s)\u2264 (1\u2212 s)H(\u03c10)+ sH(\u03c11), for all \u03c1s displacement interpolation of \u03c10 and \u03c11 \u2208 dom(H). 11 Lemma 3.4 Suppose h:(0,\u221e)\u2192R\u222a{\u221e} is convex and non increasing, and g:[0,1]\u2192(0,\u221e) is concave. Then h\u25e6g will be convex. Proof. Let s, t0, t1 \u2208 [0,1] , then h\u25e6g((1\u2212 s)t0+ st1)\u2264 h((1\u2212 s)g(t0)+ sg(t1)) \u2264 (1\u2212 s)h\u25e6g(t0)+ sh\u25e6g(t1). Definition 3.5 Let F : [0,\u221e)\u2192Rn differentiable, then we can define the associated Internal Energy Functional as HF(\u03c1) := \u222b \u2126 F(\u03c1(x))dx . Proposition 3.6 Let HF be the internal energy functional. If we suppose F : [0,\u221e)\u2192 Rn is differentiable with F(0) = 0, and x 7\u2192 xnF( rxn ) is convex and non increasing for all r > 0, then HF is displacement convex. Proof. Let \u03c10 and \u03c11 \u2208P2,ac, , \u2207\u03c8 be the optimal mapping, and the displacement interpolation of \u03c10 and \u03c11. Since Supp(\u03c1s) = Supp(\u03c10)((1\u2212 s)I+ s\u2207\u03c8) we have that HF(\u03c1s) = \u222b \u2126 F(\u03c1s(x))dx = \u222b \u2126\u2229Supp(\u03c1s) F(\u03c1s(x))dx = \u222b \u2126\u2229Supp(\u03c10) F(\u03c1s(1\u2212 s)x+ s\u2207\u03c8(x))det((1\u2212 s)I+ s\u2207\u03c8)dx. Using the Monge-Ampere formula, and defining \u03bb (s) = det((1\u2212 s)I+ s\u2207\u03c8) 1n we can conclude that HF(\u03c1s) = \u222b \u2126\u2229Supp(\u03c10) F( \u03c10(x) det((1\u2212 s)I+ s\u2207\u03c8))det((1\u2212 s)I+ s\u2207\u03c8)dx = \u222b \u2126\u2229Supp(\u03c10) F( \u03c10(x) \u03bb n )\u03bb ndx 12 Using the fact that \u03bb (s) is concave, the remark and the lemma, we see that s 7\u2192 F(\u03c10(x)\u03bb n )\u03bb n is convex, this means HF(\u03c1s)\u2264 (1\u2212 s)HF(\u03c10)+ sHF(\u03c11). 3.3 Benamou-Brenier formula In physics if we have a density \u00b5t 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 (\u00b5t ,Vt), an admissible pair if \u00b7t\u2192 \u00b5t is weak* continuous \u00b7t\u2192 \u222b |x|d\u00b5t is continuous \u00b7\u222b \u2016V (t,x)\u20162 d\u00b5tdt < \u221e. \u00b7\u2202t\u00b5+\u2207 \u00b7 (\u00b5V ) = 0 in a weak sense. Theorem 3.8 Let X be a complete smooth manifold, let \u00b50 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\u2264t\u2264T , such that dTt dt (x) =Vt(T (x)), then (\u00b5t ,Vt) is an admissible pairing, where \u00b5t = Tt#\u00b5 is the unique solution to the continuity equation . Proof. Let \u03d5 be a test function and t \u2208 (0, t), by definition of push-forward we have \u222b \u03d5d\u00b5t = \u222b (\u03d5 \u25e6Tt)d\u00b5 so for h > 0 we can write 1 h ( \u222b \u03d5d\u00b5t+h\u2212 \u222b \u03d5d\u00b5t) = \u222b \u03d5 \u25e6Tt+x(x)\u2212\u03d5 \u25e6Tt h d\u00b5. 13 Since T\u22121t is continuous, then \u03d5 \u25e6 Tt is Lipschitz and compactly supported uni- formly for t \u2208 [0, t] , so the right hand of the equation is uniformly bounded for t \u2208 [0, t\u2212h] and for almost all t,x converges point-wise to \u2202 \u2202 t (\u03d5 \u25e6Tt) = (\u2207\u03d5 \u25e6Tt) \u00b7 \u2202\u2202 t Tt = (\u2207\u03d5 \u25e6Tt) \u00b7 (vt \u25e6Tt). By Lebesgue\u00b4s dominated convergence theorem we deduce that for almost all t we have d dt \u222b \u03d5d\u00b5t = \u222b (\u2207\u03d5 \u25e6Tt) \u00b7 (vt \u25e6Tt) = \u222b \u2207\u03d5 \u00b7 vtd\u00b5t . To prove uniqueness we will prove that if \u00b5t satisfies the continuity equation then for any T \u2208 [0, t] , if \u00b50 = 0 then \u00b5T = 0. We first assume we can find a Lipschitz compactly supported function \u03d5(t,x) that satisfies \u2202\u03d5 \u2202 t + vt \u00b7\u2207\u03d5 = 0 \u03d5 |t=T= \u03d5T . Where \u03d5T \u2208D(X), the space of distribution, so we can compute for almost all t d dt \u222b \u03d5td\u00b5t = \u222b \u2202\u03d5t \u2202 t d\u00b5t + \u222b \u03d5td( \u2202\u00b5t \u2202 t ) =\u2212 \u222b vt \u00b7\u2207\u03d5+ \u222b \u03d5td(\u2207 \u00b7 vt\u00b5t) = 0. Since \u00b50 = 0, then \u222b \u03d5T d\u00b5T = 0 =\u21d2 \u00b5T = 0. Finally we can check that \u03d5t = \u03d5T \u25e6TT \u25e6T\u22121t Lipschitz with compact support, and is a solution of d dt \u03d5t(Ttx) = \u2202\u03d5 \u2202 t + v \u00b7\u2207\u03d5 = 0. We will need the following lemma to prove the Benamou-Brenier theorem. 14 Lemma 3.9 Let \u03c3 be a measure inRn, f \u2208L2(\u03c3), and T a map such that T#( f\u03c3)= hT#(\u03c3) . Then \u2016h\u2016L2(T#\u03c3) \u2264 \u2016 f\u2016L2(\u03c3) Proof. Let g \u2208 L2(T#\u03c3), computing \u3008T#( f\u03c3),g\u3009= \u3008g\u25e6T, f\u03c3\u3009 \u2264 \u2016 f\u2016L2(\u03c3) \u2016g\u25e6T\u2016L2(\u03c3) = \u2016 f\u2016L2(\u03c3) \u2016g\u2016L2(T#\u03c3) . Using Riesz representation theorem we know the continuous linear functional F such that F(g) = \u3008T#( f\u03c3),g\u3009= \u3008hT#(\u03c3),g\u3009 , has norm \u2016h\u2016L2(T#\u03c3) . This means \u2016h\u2016L2(T#\u03c3) \u2264 \u2016 f\u2016L2(\u03c3) . 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 \u00b50, \u00b51 \u2208P2,AC, then we have the equality d2w(\u00b50,\u00b51) = infVt ,\u00b5t admissible \u222b 1 0 \u222b |Vt |2 d\u00b5tdt. Proof. Since we are assuming absolute continuity for \u00b50 and \u00b51, we know there is a convex funtion \u03c8 such that \u2207\u03c8#\u00b50 = \u00b51 a.e. Let \u00b5t be the displacement inter- polation function between \u00b50 and \u00b51. For 0\u2264 t \u2264 1, let \u00b5t = (Tt)#\u00b50 where Tt = (1\u2212 t)Id+ t\u2207\u03c8. So we define Vt(x) := d dt Tt(x) = \u2207\u03c8(x)\u2212 x 15 We claim that Vt(Tt)#\u00b50 =Vt\u00b5t = (Tt)#((\u2207\u03c8\u2212 Id)\u00b50). So using lemma 3.9 we have that \u2016Vt\u2016L2(Tt#\u00b5) \u2264 \u2016\u2207\u03c8\u2212 Id\u2016L2(\u00b5) , this means \u222b |Vt |2 d\u00b5t \u2264 \u222b |x\u2212\u03c8(x)|2 d\u00b50 = d2w(\u00b50,\u00b51). 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 \u00b5t = (Tt)#\u00b50. We can compute \u222b 1 0 \u222b |Vt |2 d\u00b5tdt = \u222b 1 0 \u222b |Vt(Tt(x))|2 d\u00b50dt \u2265 \u222b |T1(x)\u2212 x|2 d\u00b50 \u2265 dw(\u00b50,\u00b51). 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\u2019s 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\u2019s 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\u00d7X), where I = [a,b] with \u03bb the normalized Lebesgue measure, and (X ,d) is a complete and separable metric space. Definition 4.1 A Young Measure in (I\u00d7X), is a positive measure \u03b7 on (I\u00d7X), such that for any measurable set A \u2282 I,\u03b7(A\u00d7X) = \u03bb (A). We denote the set of Young measures as Y1(I,X) \u2282P1(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 {\u03b7t}t\u2208I in X , such that\u222b I\u00d7X f (t,x)d\u03b7 = \u222b I \u222b X f (t,x)d\u03b7td\u03bb . (4.1) 17 Now we would like to study some properties of the map \u03b7 7\u2192 \u222b I\u00d7X f (t,x)d\u03b7 . (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\u00d7X 7\u2192 R, which is continuous in the second variable. A normal integrand is a Borel function f (t,x) : I\u00d7X 7\u2192 (\u2212\u221e,\u221e] , which is lower semi-continuous in the second variable. Definition 4.3 We say Y \u2282P(X) has uniformly integrable first moment if for every \u03b5 > 0 there exists a ball B \u2282 X such that\u222b X\u2212B d(x0,x)d\u00b5 \u2264 \u03b5 \u2200\u00b5 \u2208 Y, for one and hence for all x0 \u2208 X .We will use the following result in the proposition. Definition 4.4 A set Y \u2282P(X) is called tight if for every \u03b5 > 0 \u2203 K\u03b5 compact such that \u00b5(X\u2212K\u03b5)\u2264 \u03b5 \u2200\u00b5 \u2208 Y. Theorem 4.5 The function g(t,x) : I\u00d7X 7\u2192 R is a normal integrand if and only if g = supn\u2208N gn(t,x), where gn is a sequence of Caratheodory integrands. Proof. See Berliocchi, Lasry [7]. Theorem 4.6 (Prokhorov) Let K \u2282P(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 \u00b7 The family Y is tight with uniformly integrable first moment. 18 \u00b7There exists a function f ;X 7\u2192 [0,\u221e] whose sub-levels are compact, a constant C and a point x0 such that\u222b X (1+d(x0,x) f (x)d\u00b5 \u2264C \u2200\u00b5 \u2208 Y. Proof. See Ambrosio, Gigli, Savare\u0301 [9] Proposition 4.8 The map (4.2) is continuous on Y1(I\u00d7X) if f is a Caratheodory integrand such that | f (t,x)|\/(1+d(x0,x)) is bounded for some x0 \u2208 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 \u2208 X . Proof. Using the Scorza-Dragoni Theorem [12], we know there exists a sequence of compact sets Jn \u2282 I, such that f is continuous on Jn\u00d7X , and \u03bb (Jn) 7\u2192 1 as n 7\u2192 \u221e. 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 \u03b7 7\u2192 \u222bI\u00d7X fn(t,x)d\u03b7 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\u2208N 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 \u03b7 7\u2192 \u222b (1+d(x0,x))gn(t,x)d\u03b7 . 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)\u2265 l(x)(1+d(x0,x))+ g(t), where g : I 7\u2192R, is an integrable function, and l : X 7\u2192 [0,\u221e) is a proper func- tion. Then for each C \u2208 R, the set{ \u03b7 \u2208 Y1(I,X) | \u222b f d\u03b7 \u2264C } is compact. 19 Proof. The set{ \u03b7 \u2208 Y1(I,X) | \u222b l(x)(1+d(x0,x))d\u03b7 \u2264C } \u2283 { \u03b7 \u2208 Y1(I,X) | \u222b f d\u03b7 \u2264C } is closed and 1-tight by the equivalence results, hence it is compact. Since the map (4.2) is lower semi continuous the set {\u03b7 \u2208 Y1(I,X) | \u222b f d\u03b7 \u2264C} is closed. 4.2 Transport measures In this section we will consider Young measures acting on I\u00d7T 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+\u2016v\u2016x , and its inverse are bounded for any point x0 \u2208 X . If \u03b7 \u2208 Y1(I,T M) is a Young measure, the image of \u03b7 of the projection I\u00d7 T M 7\u2192 I\u00d7M will be denoted as \u00b5. We can think of \u00b5 as a density in M. Using the disintegration theorem [9] with respect to this projection, we obtain the measurable family \u03b7t,x of probability measures on TxM such that \u03b7 = \u00b5\u2297\u03b7t,x. We define the vector field V (t,x) : I\u00d7M 7\u2192 T M by the expression V (t,x) = \u222b TxM vd\u03b7t,x(v). We note that V (t,x) is a Borel vector field, that satisfies the integrability condition \u222b \u2016V (t,x)\u2016x d\u00b5(t,x)< \u221e. We would like to know wheter \u00b5 satisfies the continuity equation, \u2202t\u00b5+div(V\u00b5) = 0, (4.3) in the sense of distributions. We have the following characterization result. 20 Lemma 4.10 The measure \u00b5 satisfies equation (4.3) if and only if\u222b I\u00d7T M [\u2202tg+\u2202xg \u00b7 v]d\u03b7(t,x,v) = 0 (4.4) for all smooth compactly supported test functions g \u2208C\u221ec ((a,b)\u00d7M). Proof. If we disintegrate \u03b7 , we have that \u222b I\u00d7T M [\u2202tg+\u2202xg \u00b7 v]d\u03b7(t,x,v) = \u222b I\u00d7T M [\u2202tg+\u2202xg \u00b7 vd]\u03b7t,x(v)d\u00b5(t,x), for each test function. Considering the definition of V , we have the equality\u222b T M \u2202xg \u00b7 vd\u03b7t,x(v) = \u2202xg \u00b7V (t,x). This means \u03b7 satisfies equation (4.4) if and only if\u222b I\u00d7M \u2202tg+\u2202xg \u00b7V (t,x)d\u00b5(t,x) = 0. Which is equivalent to say \u00b5 satisfies equation (4.3). Any \u03b7 \u2208 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 \u00b5i and \u00b5 f on M, we say \u03b7 is a transport measure between \u00b5i and \u00b5 f , if in addition we have that\u222b I\u00d7T M [\u2202tg+\u2202xg \u00b7 v]d\u03b7(t,x,v) = \u222b M gb(x)d\u00b5 f \u2212 \u222b M ga(x)d\u00b5i, for all g : [a,b]\u00d7M 7\u2192 R, smooth compactly supported function. We denote by T \u00b5 f \u00b5i (I,M), the set of transport measures between \u00b5i and \u00b5 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 \u03b7 is called a generalized curve if for each t \u2208 I we have that \u00b5t = \u03b4\u03b3(t), for a continuous curve \u03b3(t) : I 7\u2192 M. We say \u03b7 is a generalized curve over \u03b3, 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 \u0393 \u2208 T (I,M) be a generalized curve over \u03b3, then \u03b3 is absolutely continuous. Proof. By the disintegration theorem, the measure \u0393 can be written in the from d\u0393 = dt\u2297 \u03b4\u03b3(t)\u2297 d\u0393t , with some measurable family {\u0393t} of probability measure in T\u03b3(t)M. In other words \u222b I\u00d7T M f (t,x,v)d\u0393(t,x,v) = \u222b 1 0 \u222b T\u03b3(t)M f (t,\u03b3(t),v)d\u0393t(v)dt \u2200 f \u2208 L1(\u0393). Now, for each f \u2208 C\u221ec (a,b) and \u03d5 \u2208 C\u221ec (M), let\u00b4s apply the equation (4.4) to the function g(t,x) = f (t)\u03d5(x), to get 0 = \u222b I\u00d7T M [ f \u2032(t)\u03d5(x)+ f (t)d\u03d5x \u00b7 v ] d\u03b7(t,x,v) = \u222b 1 0 f (\u0301t)\u03d5(\u03b3(t))dt+ \u222b 1 0 f (t) \u222b T\u03b3(t)M d\u03d5\u03b3(t) \u00b7 vd\u0393t(v)dt. This means, in the sense of distributions, that (\u03d5 \u25e6 \u03b3 )\u0301(t) = \u222b T\u03b3(t)M d\u03d5\u03b3(t) \u00b7 vd\u0393t(v) \u2200\u03d5 \u2208C\u221ec (M). Hence \u03b3 is absolutely continuous and\u222b T\u03b3(t)M vd\u0393t(v) = . \u03b3(t). (4.5) Theorem 4.13 The set G (I,M) is closed in Y1(I,T M), and the map \u0393 7\u2192 \u03b3 (4.6) 22 is continuous. Proof. Let \u0393n be a sequence of generalized curves converging to \u03b7 inP1(I,T M). The set {\u0393n}\u222a\u03b7 is compact hence, it has uniformly integrable first moment, so if the \u0393n\u0301s are generalized curves over \u03b3n , then the sequence \u03b3n is absolutely equi- continuous. Hence there exists a subsequence \u03b3nm and a curve \u03b30 absolutely con- tinuous, such that \u03b3nm \u2192 \u03b30. 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 {\u0393 | \u222b Ld\u0393\u2264C} are compact. We will consider the space ACx fxi of absolutely continuous curves \u03b3 : I 7\u2192M, such that \u03b3(a) = xi and \u03b3(b) = x f , and the set G x f xi =T \u00b5 f \u00b5i (I,M)\u2229G (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]\u00d7T M 7\u2192 R\u222a{+\u221e} is a normal integrand. Definition 4.14 We say L is fiber-wise convex if, the function v 7\u2192 L(t,x,v), is con- vex on TxM, for every t \u2208 [a,b], and x \u2208M. Definition 4.15 We say L is uniformly super-linear over a compact K, if there exists a function l :R+ 7\u2192R, such that limr 7\u2192\u221e l(r)\/r =\u221e and such that L(t,x,v)\u2265 l(\u2016v\u2016x) for every (t,x,v) \u2208 [a,b]\u00d7TkM. Lemma 4.16 Let L be a fiber-wise convex normal integrand. If \u0393 is a generalized curve above \u03b3, then \u222b 1 0 L(t,\u03b3(t), \u0005 \u03b3(t))dt \u2264 \u222b Ld\u0393. 23 Proof. Using equation (4.5) and Jensen\u00b4s inequality we have L(t,\u03b3(t), \u0005 \u03b3(t)) = L(t,\u03b3(t), \u222b T\u03b3(t)M vd\u0393t(v))\u2264 \u222b T\u03b3(t)M L(t,\u03b3(t),v)d\u0393t(v). Hence \u222b 1 0 L(t,\u03b3(t), \u0005 \u03b3(t))dt \u2264 \u222b 1 0 \u222b T\u03b3(t)M L(t,\u03b3(t),v)d\u0393t(v)dt = \u222b Ld\u0393. Theorem 4.17 Let L be a normal integrand such that the quotient L(t,x,v) 1+\u2016v\u2016x (4.7) is bounded from below. Conclusion: for each C \u2208 R, the set A gC := { \u0393 \u2208 G x fxi | \u222b Ld\u0393\u2264C } is compact in G x fxi . If L is fiber-wise convex, the set AC := { \u03b3 \u2208 ACx fxi | \u222b b a L(t,\u03b3(t), \u0005 \u03b3(t))dt \u2264C } 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 \u00b7L is uniformly super-linear over each compact subset of M. \u00b7There exists a positive constant such that L(t,x,v)\u2265 c\u2016v\u2016x\u22121 . Then we have the same conclusion as in the last theorem. 24 Proof. If \u0393 is a generalized curve over \u03b3 such that \u222b Ld\u0393 \u2264 C, using \u2016v\u2016x \u2264 (L(t,x,v)+1)\/c \u222b b a \u2225\u2225\u2225 \u0005\u03b3(t)\u2225\u2225\u2225 \u03b3(t) dt \u2264 C+b\u2212a c . This means the curve \u03b3 lies in the ball B(C+b\u2212ac ,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 \u2208 B(C+b\u2212ac ,xi) \u221e if x \/\u2208 B(C+b\u2212ac ,xi) , we have that \u0393 satisfies \u222b Ld\u0393 \u2264C if and only if \u222b LBd\u0393 \u2264C. Using the fact L is uniformly super-linear on B(C+b\u2212ac ,xi), we see that the quotient LB(t,x,v) 1+\u2016v\u2016x (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(detD2\u03d5) 1 n \u2264 tr(D2\u03d5) = \u2206\u03d5, we give a sketch, ignoring subtle analytic issues, of the original proof due to M. Gromov (see [13]) Theorem 5.1 Let \u2126 be an open set, such that |\u2126| = 1, then we have that |\u2202\u2126| \u2265 |\u2202B|= n, where B is the ball with area one. Sketch. We take the unitary functions in \u2126 and B , 1\u2126 and 1B. Both are probability functions so we can take the optimal transport \u2207\u03d5, from \u2126 onto B. Hence this function satisfies the Monge-Ampere equation detD2\u03d5 = 1. Since |\u2207\u03d5| \u2264 1, using Gauss theorem we can compute |\u2202\u2126|= \u222b \u2202\u2126 1ds\u2265 \u222b \u2202\u2126 \u2207\u03d5 \u00b7\u2212\u2192n ds = \u222b \u2126 \u2206\u03d5dx\u2265 \u222b \u2126 n(detD2\u03d5) 1 n = n |\u2126|= 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\u2019s 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 \u03c10 to \u03c11, and \u03c1t := ((1\u2212 t)I+T )#\u03c10. Lemma 5.2 Suppose F : [0,\u221e)\u2192 Rn is differentiable with F(0) = 0, and x 7\u2192 xnF( rxn ) is convex and non increasing for all r > 0, then we have that HF(\u03c11)\u2212HF(\u03c10)\u2265 \u222b \u2126 \u03c1o(T \u22121) \u00b7\u2207F \u2032(\u03c10)dx, for all \u03c10,\u03c11 \u2208P2,AC. Proof. Since HF(\u03c1t) is convex then we obtain HF(\u03c11)\u2212HF(\u03c10) 1 \u2265 [ d dt HF(\u03c1t) ] t=0 = [ d dt \u222b \u2126 F(((1\u2212 t)I+ tT )#\u03c10)dx ] t=0 =\u2212 \u222b \u2126 F \u2032(\u03c10)div(\u03c10(T \u2212 I))dx = \u222b \u2126 \u03c1o(T \u22121) \u00b7\u2207F \u2032(\u03c10)dx. 27 Definition 5.3 We will call a Young function, any strictly convex super-linear C1- function c :Rn\u2192R, such that c(0) = 0, and we will denote by c\u2217 its Legendre dual, as defined in remark 2.5. Theorem 5.4 (General Sobolev inequality) Under the hypotheses of the previous lemma, let \u2126 be any open bounded convex set, then for any \u03c1 \u2208P2,AC, satisfying supp\u03c1 \u2282\u2126 and PF(x) := xF \u2032(x)\u2212F(x) \u2208W 1,\u221e(\u2126) , we have that HF+nPF (\u03c1)\u2264 \u222b \u2126 c\u2217(\u2212\u2207F \u2032(\u03c1))\u03c1dx+Kc. (5.1) Proof. Using the previous lemma for \u03c10 = \u03c1, and \u03c11 = \u03c1c, where \u03c1c \u2208P2,AC is a solution of \u2207(F \u2032(\u03c1c)+ c) = 0, we get that HF(\u03c1c)\u2212HF(\u03c1)\u2265 \u222b \u2126 \u03c1(T x\u2212 x) \u00b7\u2207F \u2032(\u03c1). We note that since \u03c1\u2207(F \u2032(\u03c1)) = \u2207(PF(\u03c1)) we have that\u222b \u2126 \u03c1\u2207(F \u2032(\u03c1)) \u00b7 x = \u222b \u2126 \u2212nPF(\u03c1) = H\u2212nPF (\u03c1). We obtain HF(\u03c1)\u2212HF(\u03c1c)\u2264 \u222b \u03c1(x\u2212T x) \u00b7\u2207(F \u2032(\u03c1)) \u2264 H\u2212nPF (\u03c1)\u2212 \u222b \u2126 \u03c1\u2207(F \u2032(\u03c1)) \u00b7T xdx. For the last term we can use the generalized Young\u2019s inequality to obtain that \u2212\u2207(F \u2032(\u03c1)) \u00b7T x\u2264 c(T x)+ c\u2217(\u2212\u03c1\u2207(F \u2032(\u03c1)). 28 Integrating this to the inequality we have HF(\u03c1)\u2212HF(\u03c1c) \u2264 H\u2212nPF (\u03c1)+ \u222b \u2126 c(T x)\u03c1dx+ \u222b \u2126 c\u2217(\u2212\u2207F \u2032(\u03c1))\u03c1dx = H\u2212nPF (\u03c1)+ \u222b \u2126 c(x)\u03c1cdx+ \u222b \u2126 c\u2217(\u2212\u2207F \u2032(\u03c1))\u03c1dx. Finally we get HF+nPF (\u03c1)\u2264 \u222b \u2126 c\u2217(\u2212\u2207F \u2032(\u03c1))\u03c1dx+ \u222b \u2126 (c(x)+F \u2032(\u03c1c))\u03c1cdx\u2212HPF (\u03c1c) We name the constant c(x)+F \u2032(\u03c1c) = Kc, and we note that HPF (\u03c1c) \u2265 0 to conclude the proof. In the following pages we will see that using different F\u2019s 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 \u2126 \u2282 Rn be an open bounded and convex set, and let c be a Young functional , such that c\u2217 is p-homogeneous, for p > 1, we have that for all probability densities \u03c1, with supp(\u03c1)\u2282\u2126, and \u03c1 \u2208W 1,\u221e(Rn) \u222b Rn \u03c1 log\u03c1dx\u2264 n p log( p nep\u22121\u03c3 p\/nc \u222b Rn \u03c1c\u2217(\u2212\u2207\u03c1 \u03c1 )dx), where \u03c3c = \u222b Rn e \u2212c(x)dx. Proof. Let F(x)= x log(x), and F(0)= 0.We check that x 7\u2192 xnF(x\u2212n)=\u2212n log(x) is convex and non increasing. Considering that in this case PF(x) = x, we get that for any probability measure \u03c1 , HPF = \u222b \u03c1 = 1. We take inequality (5.1), HF+nPF (\u03c1)\u2264 \u222b \u2126 c\u2217(\u2212\u2207F \u2032(\u03c1))\u03c1dx+ \u222b (F \u2032(\u03c1c)+ c)\u03c1cdx, (5.2) 29 where \u03c1c is a solution of the equation \u2207(log\u03c1c+ c) = 0, which we take \u03c1c(x) = e\u2212c(x)\/\u03c3c, so we get \u222b \u03c1 log\u03c1+n\u2264 \u222b c\u2217(\u2212\u2207\u03c1 \u03c1 )\u03c1dx+ \u222b ( loge\u2212c(x)\u2212 log( \u222b Rn e\u2212c(x)dx)+ c ) \u03c1cdx (5.3) = \u222b c\u2217(\u2212\u2207\u03c1 \u03c1 )\u03c1dx\u2212 log( \u222b Rn e\u2212c(x)dx). (5.4) Let c\u03bb (x) := c(\u03bbx), hence c\u2217\u03bb (y) = c \u2217( y\u03bb ). If we apply the inequality to this Young function we get\u222b \u03c1 log\u03c1+n\u2264 \u222b c\u2217(\u2212\u2207\u03c1 \u03bb\u03c1 )\u03c1dx\u2212 log( \u222b Rn e\u2212c(\u03bbx)dx) = \u222b c\u2217(\u2212\u2207\u03c1 \u03bb\u03c1 )\u03c1dx\u2212 log( \u222b Rn e\u2212c(x)dx)+n log\u03bb . Considering that c\u2217( y\u03bb ) = 1 \u03bb p c \u2217(y), the infimum over \u03bb is attained when \u03bb po = p n \u222b c\u2217(\u2212\u2207\u03c1 \u03c1 )\u03c1dx. So we get the inequality for all probability densities \u03c1, with supp(\u03c1) \u2282 \u2126, and \u03c1 \u2208W 1,\u221e(Rn) \u222b \u03c1 log\u03c1 \u2264 n p \u222b c\u2217(\u2212\u2207\u03c1\u03bb\u03c1 )\u03c1dx \u222b c\u2217(\u2212\u2207\u03c1 \u03c1 )\u03c1dx\u2212 log(\u03c3c)+ np log ( p n \u222b c\u2217(\u2212\u2207\u03c1 \u03bb\u03c1 )\u03c1dx ) \u2212n \u2264 n p log ( p n \u222b c\u2217(\u2212\u2207\u03c1 \u03bb\u03c1 )\u03c1dx ) \u2212 log(\u03c3c)\u2212n+ np = n p log( p nep\u22121\u03c3 p\/nc \u222b Rn \u03c1c\u2217(\u2212\u2207\u03c1 \u03c1 )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 \u2208 (0, npn\u2212p) such that r 6= p. We define \u03b3 := 1r + 1q , where 1p + 1q = 1. For any f \u2208W 1,p(Rn) we have that there exists \u03b8 such that \u2016 f\u2016r \u2264C(p,r)\u2016\u2207 f\u2016\u03b8p \u2016 f\u20161\u2212\u03b8r\u03b3 . Proof. We will use inequality (5.1) with F(x) = x \u03b3 \u03b3\u22121 . Since r 6= p we have that \u03b3 6= 1 and since r \u2208 (0, npn\u2212p ), we have that 1 > \u03b3 > 1\u2212 1n . To use the inequality we check that F(0) = 0, and x 7\u2192 xnF(x\u2212n) = xn\u2212n\u03b3\u03b3\u22121 is convex and non increasing since n\u2212 n\u03b3 < 1 and \u03b3 \u2212 1 < 0. Let c(x) = r\u03b3q |x|q , so c\u2217(x) = 1p(r\u03b3)p\u22121 |x|p . Using inequality (5.1) we get \u222b F(\u03c1)+n\u03c1F \u2032(\u03c1)\u2212nF(\u03c1)dx\u2264 \u222b \u2126 \u03c1 1 p(r\u03b3)p\u22121 (\u2212\u2207F \u2032(\u03c1))p\u03c1dx+Kc. (5.5) Making the substitution of F(x) = x \u03b3 \u03b3\u22121 we get 1 \u03b3\u22121 \u222b \u03c1\u03b3 \u2212n\u03c1\u03b3 +n\u03b3\u03c1\u03b3dx\u2264 \u222b \u2126 \u03c1 1 p(r\u03b3)p\u22121 (\u2212\u03b3\u03c1\u03b3\u22122\u2207\u03c1)p\u03c1dx+Kc, and rearranging the equation we get ( 1 \u03b3\u22121 +n) \u222b \u03c1\u03b3dx\u2264 \u222b \u2126 \u03c1 r\u03b3 p(r)p (\u2207\u03c1)p\u03c1dx+Kc. If we suppose that \u2016 f\u2016r = 1, we take \u03c1 = | f |r to get ( 1 \u03b3\u22121 +n) \u222b | f |r\u03b3 dx\u2264 \u222b \u2126 r\u03b3 p |\u2207 f |p\u03c1dx+Kc, and for general f we get this inequality 31 r\u03b3 p \u2016\u2207 f\u2016pp \u2016 f\u2016pr \u2212 ( 1 \u03b3\u22121 +n ) \u2016 f\u2016r\u03b3r\u03b3 \u2016 f\u2016r \u2265\u2212Kc. If we have the function f\u03bb (x) = f (\u03bbx), with a change of variables we get the following equalities \u2016 f\u03bb\u2016pr = \u03bb\u2212np\/r \u2016 f\u2016pr , \u2016 f\u03bb\u2016r\u03b3r\u03b3 = \u03bb\u2212n \u2016 f\u2016r\u03b3r\u03b3 , \u2016 f\u03bb\u2016r = \u03bb\u2212n\/r \u2016 f\u2016r , \u2016\u2207 f\u03bb\u2016pp = \u2016\u03bb\u2207 f (x\u03bb )\u2016pp = \u03bb p\u2212n \u2016\u2207 f\u2016pp . So the inequality becomes \u03bb p\u2212n+np\/r r\u03b3 p \u2016\u2207 f\u2016pp \u2016 f\u2016pr \u2212\u03bb\u2212n+n\/r ( 1 \u03b3\u22121 +n ) \u2016 f\u2016r\u03b3r\u03b3 \u2016 f\u2016r \u2265\u2212Kc. We take \u03bb = \u2016\u2207 f\u2016ap \u2016 f\u2016br \u2016 f\u2016cr\u03b3 , 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\u2212n , b = (p\u22121)r pr+np\u2212n , c = r pr+np\u2212n . So we obtain 1 Kc ( \u2212r\u03b3 p + 1 \u03b3\u22121 \u2212n ) \u2016\u2207 f\u2016a\u2032p \u2016 f\u2016c \u2032 r\u03b3 \u2265 \u2016 f\u2016b \u2032 r , 32 where a\u2032 = \u2212npr+np pr+np\u2212n b\u2032 = (p\u22121)r(\u2212n+n\/r) pr+np\u2212n \u22121 c\u2032 = rp\u2212nr+np pr+np\u2212n Finally we note that if we take the limit as r\u2192 p\u2217 = npn\u2212p , we have that c\u2032\u2192 0, and a\u2032,b\u2032\u2192 np(n\u2212np\u2212p)(n\u2212p)(pr+np\u2212n) so we get the Sobolev inequality \u2016 f\u2016p\u2217 \u2264C(p,n)\u2016\u2207 f\u2016p . 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,\u221e)\u2192 Rn differentiable, and V,W : R\u2192 [0,\u221e) , twice differentiable, then we can define the associated Free Energy Functional as HF,WV (\u03c1) := H F(\u03c1)+HV (\u03c1)+HW (\u03c1). Where we have \u00b7Internal energy HF(\u03c1) := \u222b \u2126 F(\u03c1(x))dx, . \u00b7Potential energy HV (\u03c1) =: \u222b \u2126 \u03c1(x)V (x)dx. \u00b7Interaction energy HW (\u03c1) =: 1 2 \u222b \u2126 \u03c1(W \u2217\u03c1), 33 where \u2217 denotes the convolution product. Furthermore we define the relative energy of \u03c10 with respect to \u03c11 as HF,WV (\u03c10 | \u03c11) := HF,WV (\u03c11)\u2212HF,WV (\u03c10), and the relative entropy production of \u03c1 with respect to \u03c1V as I2(\u03c1 | \u03c1V ) := \u222b \u2126 \u2223\u2223\u2207(F \u2032(\u03c1)+V +W \u2217\u03c1)\u2223\u22232\u03c1dx. So if \u03c1V is a probability density that satisfies \u2207(F \u2032(\u03c1V )+V +W \u2217\u03c1) = 0, then I2(\u03c1 | \u03c1V ) := \u222b \u2126 \u2223\u2223\u2207(F \u2032(\u03c1)\u2212F \u2032(\u03c1V )+W \u2217 (\u03c1\u2212\u03c1V ))\u2223\u22232\u03c1dx. We will also work with non-quadratic versions of entropy, so we define the generalized relative entropy production-type function of \u03c1 with respect to \u03c1V mea- sured against c\u2217 as Ic\u2217(\u03c10 | \u03c1V ) := \u222b \u2126 c\u2217 (\u2212\u2207(F \u2032(\u03c10)+V +W \u2217\u03c10))\u03c10dx, where c\u2217 is the Legendre conjugate of c. Lemma 5.8 Assume V:Rn\u2192 R satisfies that D2V \u2265 \u03bb I, for some \u03bb \u2208 R, then we have that HV (\u03c11)\u2212HV (\u03c10)\u2265 \u222b \u2126 \u03c1o(T \u22121) \u00b7\u2207V dx+ \u03bb2 d 2 w(\u03c10,\u03c11), for all \u03c10,\u03c11 \u2208P2,AC. Proof. Expanding and using D2V \u2265 \u03bb I, we obtain V (b)\u2212V (a)\u2265 \u2207V (a) \u00b7 (b\u2212a)+ \u03bb 2 |a\u2212b|2 . 34 This means that V (T x)\u2212V (x)\u2265 \u2207V (x) \u00b7 (T x\u2212 x)+ \u03bb 2 |x\u2212T x|2 . Hence integrating we obtain HV (\u03c11)\u2212HV (\u03c10)\u2265 \u222b V (T x)\u03c10\u2212V (x)\u03c10dx \u2265 \u222b \u2126 \u2207V (x) \u00b7 (T x\u2212 x)+ \u03bb 2 |x\u2212T x|2\u03c10dx = \u222b \u2126 \u03c1o(T \u22121) \u00b7\u2207V dx+ \u03bb2 d 2 w(\u03c10,\u03c11). Lemma 5.9 Assume W:Rn \u2192 R is even and satisfies that D2W \u2265 \u03bdI, for some \u03bd \u2208 R, then we have that HW (\u03c11)\u2212HW (\u03c10)\u2265 \u222b \u2126 \u03c1o(T\u22121)\u00b7\u2207(W \u2217\u03c10)dx+ \u03bd2 (d 2 w(\u03c10,\u03c11)\u2212|b(\u03c10)\u2212b(\u03c11)|2), for all \u03c10,\u03c11 \u2208P2,AC, and where b represents the centre of mass denoted by b(\u03c1)=\u222b x\u03c1(x)dx. Proof. First we note that we can write the interaction energy as follows HW (\u03c11) = 1 2 \u222b \u2126\u00d7\u2126 W (x\u2212 y)\u03c11(x)\u03c11(y)dxdy = 1 2 \u222b \u2126\u00d7\u2126 W (T x\u2212Ty)\u03c10(x)\u03c10(y)dxdy = 1 2 \u222b \u2126\u00d7\u2126 W (x\u2212 y+(T \u2212 I)(x)\u2212 (T \u2212 I)(y))\u03c10(x)\u03c10(y)dxdy 35 Since D2W \u2265 \u03bdI we obtain HW (\u03c11)\u2265 12 \u222b \u2126\u00d7\u2126 [W (x\u2212 y)+\u2207W (x\u2212 y) \u00b7 ((T \u2212 I)(x)\u2212 (T \u2212 I)(y))]\u03c10(x)\u03c10(y)dxdy + \u03bd 4 \u222b \u2126\u00d7\u2126 |(T \u2212 I)(x)\u2212 (T \u2212 I)(y)|2\u03c10(x)\u03c10(y)dxdy = HW (\u03c10)+ 1 2 \u222b \u2126\u00d7\u2126 \u2207W (x\u2212 y) \u00b7 ((T \u2212 I)(x)\u2212 (T \u2212 I)(y))\u03c10(x)\u03c10(y)dxdy + \u03bd 4 \u222b \u2126\u00d7\u2126 |(T \u2212 I)(x)\u2212 (T \u2212 I)(y)|2\u03c10(x)\u03c10(y)dxdy. Now we note the following equalities, for the last term\u222b \u2126\u00d7\u2126 |(T \u2212 I)(x)\u2212 (T \u2212 I)(y)|2\u03c10(x)\u03c10(y)dxdy = 2 \u222b \u2126\u00d7\u2126 |(T \u2212 I)(x)|2\u03c10(x)dx\u22122 \u2223\u2223\u2223\u2223\u222b\u2126\u00d7\u2126(T \u2212 I)(x)\u03c10(x)dx \u2223\u2223\u2223\u22232 = 2 [\u222b \u2126\u00d7\u2126 |(T \u2212 I)(x)|2\u03c10(x)dx\u2212|b(\u03c11)\u2212b(\u03c10)|2 ] . For the second term we consider that \u2207W is odd\u222b \u2126\u00d7\u2126 \u2207W (x\u2212 y) \u00b7 ((T \u2212 I)(x)\u2212 (T \u2212 I)(y))\u03c10(x)\u03c10(y)dxdy = 2 \u222b \u2126\u00d7\u2126 \u2207W (x\u2212 y) \u00b7 ((T \u2212 I)(x))\u03c10(x)\u03c10(y)dydx = 2 \u222b \u2126\u00d7\u2126 (\u2207W \u2217\u03c10) \u00b7 (T \u2212 I)(x))\u03c10(x)dx. Using these two equalities we get HW (\u03c11)\u2212HW (\u03c10)\u2265 \u222b \u2126 \u03c1o(T\u22121)\u00b7\u2207(W \u2217\u03c10)dx+ \u03bd2 (d 2 w(\u03c10,\u03c11)\u2212|b(\u03c10)\u2212b(\u03c11)|2). Theorem 5.10 (Basic inequality) Under the hypotheses of the three previous lem- mas, let \u2126 be any open bounded convex set, then for any \u03c10,\u03c11 \u2208P2,AC, satisfying 36 supp\u03c10 \u2282\u2126 and PF(x) := xF \u2032(x)\u2212F(x) \u2208W 1,\u221e(\u2126) , we have that HF,WV+c(\u03c10 | \u03c11)+ \u03bb +\u03bd 2 d2w(\u03c10,\u03c11)\u2212 \u03bd 2 |b(\u03c10\u2212b(\u03c11)|2 \u2264 H\u2212nPF ,2x\u00b7\u2207Wc+\u2207V \u00b7x (\u03c10)+Ic\u2217(\u03c10 | \u03c1V ). Proof. First we note that since \u03c10\u2207(F \u2032(\u03c10)) = \u2207(PF(\u03c10)) we have that\u222b \u2126 \u03c10\u2207(F \u2032(\u03c10)+V +W \u2217\u03c10) \u00b7 x = \u222b \u2126 \u2212nPF(\u03c10)+\u03c10 [\u2207(V +\u2207W \u2217\u03c10)] \u00b7 x = \u222b \u2126 \u2212nPF(\u03c10)+\u03c10\u2207V \u00b7 x+ 12\u03c10(2x \u00b7\u2207W \u2217\u03c10)dx = H\u2212nPF ,2x\u00b7\u2207W\u2207V \u00b7x (\u03c10). If we add the inequalities from the previous lemmas we get HF,WV (\u03c11)\u2212HF,WV (\u03c10) \u2265 \u222b \u2126 \u03c10(T x\u2212 x) \u00b7\u2207(F \u2032(\u03c10)+V +W \u2217\u03c10)dx+ \u03bb2 d 2 w(\u03c10,\u03c11) + \u03bd 2 (d2w(\u03c10,\u03c11)\u2212|b(\u03c10)\u2212b(\u03c11)|2). Rearranging and using the first inequality we have HF,WV (\u03c10)\u2212HF,WV (\u03c11)+ \u03bb +\u03bd 2 d2w(\u03c10,\u03c11)\u2212|b(\u03c10)\u2212b(\u03c11)|2) \u2264 \u222b \u2126 \u03c10(x\u2212T x) \u00b7\u2207(F \u2032(\u03c10)+V +W \u2217\u03c10) \u2264 H\u2212nPF ,2x\u00b7\u2207W\u2207V \u00b7x (\u03c10)\u2212 \u222b \u2126 \u03c10\u2207(F \u2032(\u03c10)+V +W \u2217\u03c10) \u00b7T xdx. For the last term we can use the generalized Young\u2019s inequality to obtain that \u2212\u2207(F \u2032(\u03c10)+V +W \u2217\u03c10) \u00b7T x \u2264 c(T x)+ c\u2217(\u2212\u2207(F \u2032(\u03c10)+V +W \u2217\u03c10). 37 Integrating this to the inequality we have HF,WV (\u03c10)\u2212HF,WV (\u03c11)+ \u03bb +\u03bd 2 d2w(\u03c10,\u03c11)\u2212|b(\u03c10)\u2212b(\u03c11)|2) \u2264 H\u2212nPF ,2x\u00b7\u2207W\u2207V \u00b7x (\u03c10)+ \u222b \u2126 c(T x)\u03c10dx+ \u222b \u2126 c\u2217(\u2212\u2207(F \u2032(\u03c10)+V +W \u2217\u03c10)\u03c10dx = H\u2212nPF ,2x\u00b7\u2207W\u2207V \u00b7x (\u03c10)+ \u222b \u2126 c(x)\u03c11dx+ \u222b \u2126 c\u2217(\u2212\u2207(F \u2032(\u03c10)+V +W \u2217\u03c10)\u03c10dx. This proves the inequality. A simpler inequality is the one obtained when V and W are strictly convex hence \u03bd ,\u03bb \u2265 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\u2192 R, we have HF+nPF ,W\u22122x\u00b7\u2207WV\u2212\u2207V \u00b7x (\u03c1)\u2264\u2212HPF ,W (\u03c1V+c)+Ic\u2217(\u03c1 | \u03c1V )+KV+c. (5.6) Furthermore if we set W =V = 0, since HPF (\u03c1c)\u2265 0, we obtain HF+nPF (\u03c1)\u2264\u2212HPF (\u03c1c)+Ic\u2217(\u03c1 | \u03c1V )+KV+c (5.7) \u2264 \u222b \u2126 c\u2217(\u2212\u2207F \u2032(\u03c1))\u03c1dx+KV+c. (5.8) Hence recovering inequality (5.1). Proof. Let\u00b4s consider the inequality we just proved HF,WV+c(\u03c10 | \u03c11)+ \u03bb +\u03bd 2 d2w(\u03c10,\u03c11)\u2212 \u03bd 2 |b(\u03c10\u2212b(\u03c11)|2 (5.9) \u2264 H\u2212nPF ,2x\u00b7\u2207Wc+\u2207V \u00b7x (\u03c10)+Ic\u2217(\u03c10 | \u03c1V ). (5.10) In particular if we take \u03c10 = \u03c1 and \u03c11 = \u03c1V+c , where \u03c1V+c is a solution of \u2207(F \u2032(\u03c1V+c)+V + c+W \u2217\u03c1V+c) = 0. 38 Hence we have that for any \u03c1 \u2208Pc(\u2126),with supp \u03c1 \u2282\u2126, and PF(\u03c1)\u2208W 1,\u221e(\u2126) we have that HF+nPF ,W\u22122x\u00b7\u2207WV\u2212\u2207V \u00b7x (\u03c1)+ \u03bb +\u03bd 2 d2w(\u03c1,\u03c1V+c)\u2212 \u03bd 2 |b(\u03c1\u2212b(\u03c11)|2 (5.11) \u2264\u2212HPF ,W (\u03c1V+c)+Ic\u2217(\u03c1 | \u03c1V )+ \u222b ( F \u2032(\u03c1V+c)+V + c+W \u2217\u03c1V+c ) \u03c1V+c. (5.12) Where we can define the constant F \u2032(\u03c1V + c)+V + c+W \u2217\u03c1 := KV+c . Since \u03bd ,\u03bb \u2265 0 we get that \u03bb +\u03bd 2 d2w(\u03c1,\u03c1V+c)\u2212 \u03bd 2 |b(\u03c1)\u2212b(\u03c1V+c)|2 = \u03bb +\u03bd 2 \u222b |T x\u2212 x|2\u03c10(x)dx\u2212 \u03bd2 \u2223\u2223\u2223\u2223\u222b (T x\u2212 x)\u03c10(x)dx\u2223\u2223\u2223\u22232 \u2265 0. So we can remove the terms involving \u03bd and \u03bb 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 \u00b5 \u2208 R, and U :Rn\u2192 R be a C2 function such that D2U \u2265 \u00b5I, then for any \u03c3 > 0 we have that HFU (\u03c10 | \u03c11)+ 1 2 (\u00b5\u2212 1 \u03c3 )W 22 (\u03c10,\u03c11)\u2264 \u03c3 2 \u222b \u2126 \u03c1 \u2223\u2223\u2207(F \u2032 \u25e6\u03c10+U\u2223\u22232 dx. Proof. If we take the basic inequality with c(x) = 12\u03c3 |x|2 , W = 0, and we set, V = U \u2212 c. Hence we have that c\u2217(p) = 12\u03c3 |\u03c3 p|2 = \u03c32 |p|2 , so using the general inequality we get HFU (\u03c10 | \u03c11)+ (\u00b5\u2212\u03c3\u22121) 2 d2w(\u03c10,\u03c11)\u2264H\u2212nPFc+\u2207(U\u2212c)\u00b7x(\u03c10)+ \u03c3 2 \u222b \u2126 \u03c10\u2207(F \u2032\u25e6\u03c10+U\u2212c)dx. 39 We can compute \u03c3 2 \u222b \u2126 \u03c10 \u2223\u2223\u2207(F \u2032 \u25e6\u03c10+U\u2212 c)\u2223\u22232 dx = \u03c3 2 \u222b \u2126 \u03c1 \u2223\u2223\u2207(F \u2032 \u25e6\u03c10+U\u2223\u22232 dx+ 12\u03c3 \u222b \u2126 \u03c10 |x|2 dx\u2212 \u222b \u2126 x\u03c10 \u00b7\u2207(F \u2032 \u25e6\u03c10+U)dx, and HnPFc+\u2207(U\u2212c)\u00b7x(\u03c10) = H nPF (\u03c10)\u2212 \u222b \u2126 \u03c1x \u00b7\u2207Udx+ 1 2\u03c3 \u222b \u2126 |x|2\u03c10dx. By combining the two and using integration by parts we get that H\u2212nPFc+\u2207(U\u2212c)\u00b7x(\u03c10)+ \u03c3 2 \u222b \u2126 \u03c10 \u2223\u2223\u2207(F \u2032 \u25e6\u03c10+U\u2212 c)\u2223\u22232 dx = \u03c3 2 \u222b \u2126 \u03c1 \u2223\u2223\u2207(F \u2032 \u25e6\u03c10+U\u2223\u22232 dx\u2212\u222b \u2126 x\u03c10 \u00b7\u2207(F \u2032 \u25e6\u03c10)dx\u2212HnPF (\u03c10) = \u03c3 2 \u222b \u2126 \u03c1 \u2223\u2223\u2207(F \u2032 \u25e6\u03c10+U\u2223\u22232 dx+\u222b \u2126 div(x\u03c10) \u00b7 (F \u2032 \u25e6\u03c10)dx\u2212HnPF (\u03c10) = \u03c3 2 \u222b \u2126 \u03c1 \u2223\u2223\u2207(F \u2032 \u25e6\u03c10+U\u2223\u22232 dx+\u222b \u2126 n\u03c10 \u00b7 (F \u2032 \u25e6\u03c10)dx+ \u222b \u2126 x \u00b7\u2207F(\u03c10)dx\u2212HnPF (\u03c10) = \u03c3 2 \u222b \u2126 \u03c1 \u2223\u2223\u2207(F \u2032 \u25e6\u03c10+U\u2223\u22232 dx+\u222b \u2126 x \u00b7\u2207F(\u03c10)dx+ \u222b \u2126 n(F \u25e6\u03c10)dx = \u03c3 2 \u222b \u2126 \u03c1 \u2223\u2223\u2207(F \u2032 \u25e6\u03c10+U\u2223\u22232 dx. Returning to the first inequality we get that HFU (\u03c10 | \u03c11)+ (\u00b5\u2212\u03c3\u22121) 2 d2w(\u03c10,\u03c11)\u2264 \u03c3 2 \u222b \u2126 \u03c1 \u2223\u2223\u2207(F \u2032 \u25e6\u03c10+U)\u2223\u22232 dx. Corollary 5.13 Furthermore if we take \u00b5 > 0, that is, take U is uniformly convex, take \u03c3 = 1\u00b5 , we can get the Generalized Log-Sobolev inequality: HFU (\u03c10 | \u03c11)\u2264 1 2\u00b5 \u222b \u2126 \u03c1 \u2223\u2223\u2207(F \u2032 \u25e6\u03c10+U)\u2223\u22232 dx = 12\u00b5I2(\u03c10 | \u03c1U). Corollary 5.14 (HWI ) Finally we can obtain the generalized HWI-inequality, 40 which is originally due to Otto and Villani (see [5]). HFU (\u03c10 | \u03c11)+ \u00b5 2 d2w(\u03c10,\u03c11)\u2264 \u221a I2(\u03c10 | \u03c1U)dw(\u03c10,\u03c11). Proof. If we write the inequality of the last corollary as HFU (\u03c10 | \u03c11)+ \u00b5 2 d2w(\u03c10,\u03c11)\u2264 \u03c3 2 I2(\u03c10 | \u03c1U)+ 12\u03c3 d 2 w(\u03c10,\u03c11) and minimize over \u03c3 , we obtain the minimum when \u03c3 = dw(\u03c10,\u03c11)\u221a I2(\u03c10|\u03c1U ) , we can write the inequality as HFU (\u03c10 | \u03c11)+ \u00b5 2 d2w(\u03c10,\u03c11)\u2264 \u221a I2(\u03c10 | \u03c1U)dw(\u03c10,\u03c11). 5.2.2 Gaussian inequalities By taking a particular F we can prove Otto-Villani\u2019s HWI inequality. Corollary 5.15 Let \u00b5 \u2208R, and U :Rn\u2192R be a C2 function such that D2U \u2265 \u00b5I, then for any \u03c3 > 0 , and any non-negative function f such that f\u03c1U \u2208W 1,\u221e(Rn) and\u222b f\u03c1U = 1, we have that \u222b f log( f )\u03c1U + 1 2 (\u00b5\u2212 1 \u03c3 )W 22 (\u03c10,\u03c11)\u2264 \u03c3 2 \u222b \u2126 \u03c1U |\u2207 f |2 f dx. Where \u03c1U = e\u2212U\/ \u222b e\u2212U dx. Proof. The proof follows from corollary 5.12, taking \u03c10 = \u03c1U , \u03c11 = f\u03c1U , and F(x) = x logx. So we compute HFU (\u03c1U) = \u222b \u03c1U log\u03c1U +U\u03c1U dx = \u222b [( e\u2212U\/ \u222b e\u2212U dx ) log ( e\u2212U\/ \u222b e\u2212U dx ) +U ( e\u2212U\/ \u222b e\u2212U dx )] = 1\u222b e\u2212U dx \u222b e\u2212U(\u2212 log \u222b e\u2212U) =\u2212 log \u222b e\u2212U , and 41 HFU ( f\u03c1U) = \u222b f\u03c1U log f\u03c1U +U f\u03c1U = 1\u222b e\u2212U dx \u222b e\u2212U f (log f \u2212 log \u222b e\u2212U) = \u222b f log( f )\u03c1U \u2212 ( log \u222b e\u2212U )\u222b f\u03c1U = \u222b f log( f )\u03c1U \u2212 log \u222b e\u2212U Hence HFU (\u03c10 | \u03c11) = \u222b f log( f )\u03c1U . Furthermore if U is uniformly convex , we can consider \u00b5 > 0, so we can simplify the inequality to get the original Log-Sobolev inequality of Gross \u222b f log( f )\u03c1U \u2264 1\u00b5 \u222b \u2126 \u03c1U |\u2207 f |2 f dx. 42 Bibliography [1] P. Bernard. Young measures, supersposition and transport. Indiana University Mathematics Journal, 57(1):247\u2013246, 2008. \u2192 pages 2, 17 [2] C. V. D. Cordero-Erausquin, B. Nazaret. A mass-transportation approach to sharp sobolev and gagliardo-nirenberg inequalities. preprint, 2002. \u2192 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. \u2192 pages 2, 27 [4] L. Evans. Partial Differential Equations. AMS, 2005. \u2192 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\u2013400, 2000. \u2192 pages 41 [6] L. Gross. Logarithmic sobolev inequa. Amer. J. Math, 97:1061\u20131083, 1975. \u2192 pages 29 [7] J.-M. L. H. Berliocchi. Intgrands normales and mesures paramtres en calcul des variations. Bull. Soc. Math. France, 101:129\u2013184, 1973. \u2192 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\u2013393, 2000. \u2192 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. \u2192 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\u2013244, 2004. \u2192 pages 2, 3, 27, 33 [11] R. McCann. A canvexity principle for interacting gases. Adv. Math, 128(1): 153\u2013179, 1997. \u2192 pages 2, 27 [12] N.-S. P. S. Hu. Handbook of Multivalued Analysis. Kluwer Academic Publishers, 1975. \u2192 pages 19 [13] C. Villani. Topics in Optimal Transportation. AMS, 2003. \u2192 pages 1, 4, 6, 26 [14] L. Young. Lectures on the Calculus of Variations and Optimal Control Theory. 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