Table Of Content17
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.