17 TESI THESES tesidiperfezionamentoinMatematicasostenutail2novembre2012 COMMISSIONEGIUDICATRICE LuigiAmbrosio,Presidente GiuseppeButtazzo LuisCaffarelli ValentinoMagnani TommasoPacini AlessandroProfeti AngeloVistoli GuidoDePhilippis HausdorffCenterforMathematics VillaMaria EndenicherAllee62 D-53115Bonn,Germany RegularityofOptimalTransportMapsandApplications Guido De Philippis Regularity of Optimal Transport Maps and Applications EDIZIONI DELLA NORMALE (cid:2)c 2013ScuolaNormaleSuperiorePisa ISBN978-88-7642-456-4 ISBN978-88-7642-458-8(eBook) Contents Introduction vii 1. Regularityofoptimaltransportmapsandapplications . . . vii 2. Otherpapers . . . . . . . . . . . . . . . . . . . . . . . . . xii 1 Anoverviewonoptimaltransportation 1 1.1. ThecaseofthequadraticcostandBrenierPolar FactorizationTheorem . . . . . . . . . . . . . . . . . . 2 1.2. Breniervs. Aleksandrovsolutions . . . . . . . . . . . . 12 1.2.1. Breniersolutions . . . . . . . . . . . . . . . . . 12 1.2.2. Aleksandrovsolutions . . . . . . . . . . . . . . 14 1.3. Thecaseofageneralcostc(x,y). . . . . . . . . . . . . 22 1.3.1. Existenceofoptimalmaps . . . . . . . . . . . . 22 1.3.2. RegularityofoptimalmapsandtheMTWcondition 26 2 TheMonge-Ampe`reequation 29 2.1. Aleksandrovmaximumprinciple . . . . . . . . . . . . . 30 2.2. SectionsofsolutionsandCaffarellitheorems . . . . . . 33 2.3. ExistenceofsmoothsolutionstotheMonge-Ampe`re equation . . . . . . . . . . . . . . . . . . . . . . . . . . 51 3 SobolevregularityofsolutionstotheMongeAmpe`reequation 55 3.1. ProofofTheorem3.1 . . . . . . . . . . . . . . . . . . . 56 3.2. ProofofTheorem3.2 . . . . . . . . . . . . . . . . . . . 64 3.2.1. AdirectproofofTheorem3.8 . . . . . . . . . . 66 3.2.2. Aproofbyiterationofthe LlogL estimate . . . 69 3.3. AsimpleproofofCaffarelliW2,p estimates . . . . . . . 71 4 SecondorderstabilityfortheMonge-Ampe`reequation andapplications 73 4.1. ProofofTheorem4.1 . . . . . . . . . . . . . . . . . . . 75 vi GuidoDePhilippis 4.2. ProofofTheorem4.2 . . . . . . . . . . . . . . . . . . . 78 5 Thesemigeostrophicequations 81 5.1. Thesemigeostrophicequationsinphysicalanddual variables . . . . . . . . . . . . . . . . . . . . . . . . . . 81 5.2. The2-dimensionalperiodiccase . . . . . . . . . . . . . 86 5.2.1. Theregularityofthevelocityfield . . . . . . . . 90 5.2.2. ExistenceofanEuleriansolution . . . . . . . . 97 5.2.3. Existence of a Regular Lagrangian Flow for the semigeostrophicvelocityfield . . . . . . . . . . 99 5.3. The3-dimensionalcase . . . . . . . . . . . . . . . . . . 103 6 Partialregularityofoptimaltransportmaps 119 6.1. Thelocalizationargumentandproofoftheresults . . . . 120 6.2. C1,β regularityandstrictc-convexity . . . . . . . . . . . 125 6.3. ComparisonprincipleandC2,α regularity . . . . . . . . 138 A Propertiesofconvexfunctions 147 B AproofofJohnlemma 157 References 159 Introduction This thesis is devoted to the regularity of optimal transport maps. We provide new results on this problem and some applications. This is part of the work done by the author during his PhD studies. Other papers written during the PhD studies and not completely related to this topic aresummarizedinthesecondpartoftheintroduction. 1. Regularityofoptimaltransportmapsandapplications Monge optimal transportation problem goes back to 1781 and it can be statedasfollows: Giventwoprobabilitydensitiesρ andρ onRn (originallyrepresenting 1 2 theheightofapileofsoilandthedepthofanexcavation),letuslookfor amapT movingρ ontoρ ,i.e. suchthat1 1 2 (cid:2) (cid:2) ρ (x)dx = ρ (y)dy forallBorelsets A, (1) 1 2 T−1(A) A andminimizingthetotalcostofsuchprocess: (cid:2) (cid:3)(cid:2) (cid:4) c(x,T(x))ρ (x)dx=inf c(x,S(x))ρ (x)dx : S satisfies(1) . (2) 1 1 Here c(x,y) represent the “cost” of moving a unit of mass from x to y (theoriginalMonge’sformulationthecostc(x,y)wasgivenby|x−y|). Conditions for the existence of an optimal map T are by now well understood (and summarized without pretending to be aexhaustive in Chapter1,see[95,Chapter10]foramorerecentaccountofthetheory). Once the existence of an optimal map has been established a natural question is about its regularity. Informally the question can be stated as follows: Giventwosmoothdensities, ρ andρ supportedongoodsets, it istrue 1 2 theT issmooth? 1FromthemathematicalpointofviewwearerequiringthatT(cid:5)(ρ1Ln)=ρ2Ln,seeChapter1. viii GuidoDePhilippis Or, somehow more precisely, one can investigate how much is the “gain” in regularity from the densities to T. As we will see in a mo- ment,anaturalguessisthatT shouldhave“onederivative”morethanρ 1 andρ . 2 Tostartinvestigatingregularity,noticethat(1)canbere-writtenas ρ (x) |det∇T(x)| = 1 , (3) ρ (T(x))) 2 which turns out to be a very degenerate first order PDE. As we already said,theaboveequationcouldleadtotheguessthatT hasonederivative more than the densities. Notice however that the above equation is sat- isfied by every map which satisfies (1). Thus, by simple examples, we cannot expect solutions of (3) to be well-behaved. Indeed, consider for instance the case in which ρ = 1 and ρ = 1 with A and B smooth 1 A 2 B open sets. If we right (respectively left) compose T with a map S sat- isfying det∇S = 1 and S(A) = A (resp. S(B) = B) we still obtain a solutionof(3)whichisnomoreregularthan S. Itisatthispointthatcondition(2)comesintoplay. Toseehow,letus focus on the quadratic case, c(x,y) = |x − y|2/2. In this case Brenier Theorem 1.8, ensures that the optimal T is given by the gradient of a convex function, T = ∇u. Plugging this information into (3) we obtain thatu solvesthefollowingMonge-Ampe`reequation ρ (x) det∇2u(x) = 1 . (4) ρ (∇u(x))) 2 In this way we have obtained a (degenerate) elliptic second order PDE, and there is hope to obtain regularity of T = ∇u from the regularity of thedensities.2 Inspiteoftheabovediscussion,alsoequation(4)itisnot enoughtoensureregularityofu. Asimpleexampleisgivenbythecase inwhichthesupportofthefirstdensityisconnectedwhilethesupportof thesecondisnot. Indeed,sinceby(1)itfollowseasilythat T(sptρ ) = sptρ , 1 2 2Oneshouldcomparethiswiththefollowingfact:thereisnohopetogetregularityofavectorfield vsatisfying ∇·v=0, whileifweaddtheadditionalconditionv=∇uweobtaintheLaplaceequation (cid:6)v=0. ix RegularityofOptimalTransportMapsandApplications we immediately see that, even if the densities are smooth on their sup- ports, T has to be discontinuous (cp. Example 1.16). It was noticed by Caffarelli, [21], that the right assumption to be made on the support of ρ is convexity. In this case any solution of (4) arising from the optimal 2 transportationproblemturnsouttobeastrictlyconvexAleksandrovsolu- tiontotheMonge-Ampe`reequation3 ρ (x) detD2u = 1 onInt(sptρ ). (5) ρ (∇u(x))) 1 2 Asaconsequence,underthepreviousassumptions,wecantranslateany regularityresultsforAleksandrovsolutionstotheMonge-Ampe`reequa- tion to solution to the optimal transport problem. In particular, by the theorydevelopedin[18,19,20,89](seealso[66, Chapter17])wehave thefollowing(seeChapter2foramoreprecisediscussion): - Ifρ andρ areboundedawayfromzeroandinfinityontheirsupport 1 2 andsptρ isconvex,thenu ∈C1,α (andhenceT ∈Cα ). 2 loc loc - If, in addition, ρ and ρ are continuous, then T ∈ W2,p for every 1 2 loc p ∈ [1,∞). - Ifρ andρ areCk,β and,again,sptρ isconvex,thenT ∈Ck+2,β. 1 2 2 loc AnaturalquestionwhichwasleftopenbytheabovetheoryistheSobolev regularity of T under the only assumptions that ρ and ρ are bounded 1 2 away from zero and infinity on their support and sptρ is convex. In 2 [93], Wang shows with a family of counterexamples that the best one can expect is T ∈ W1,1+ε with ε = ε(n,λ), where λ is the “pinching” (cid:6)log(ρ1/ρ2(∇u))(cid:6)∞,seeExample2.21. ApartfrombeingaverynaturalquestionfromthePDEpointofview, Sobolev regularity of optimal transport maps (or equivalently of Alek- sandrovsolutionstotheMonge-Ampe`reequation)hasarelevantapplic- ation to the study of the semigeostrophic system, as was pointed out by Ambrosio in [4]. This is a system of equations arising in study of large oceanic and atmospheric flows. Referring to Chapter 5 for a more ac- curate discussion we recall here that the system can be written, after a 3ThiskindofsolutionshavebeenintroducedbyAleksandrovinthestudyoftheMinkowskiProb- lem:givenafunctionκ:Sn−1→[0,∞)findaconvexbodyKsuchthattheGausscurvatureofits boundaryisgivenbyκ◦ν∂K.AlltheresultsofChapters2,3,4,applytothisproblemaswell.