Hamilton-Jacobi Equations: Approximations, Numerical Analysis and Applications

316 Pages·2013·3.82 MB·English

Lecture Notes in Mathematics 2074
CIME Foundation Subseries
Yves Achdou · Guy Barles
Hitoshi Ishii · Grigory L. Litvinov
Hamilton-Jacobi
Equations:
Approximations,
Numerical Analysis
and Applications
Cetraro, Italy 2011
Editors: Paola Loreti,
Nicoletta Anna Tchou
Lecture Notes in Mathematics 2074
Editors:
J.-M.Morel,Cachan
B.Teissier,Paris
Forfurthervolumes:
http://www.springer.com/series/304
FondazioneC.I.M.E.,Firenze
C.I.M.E.stands forCentroInternazionale Matematico Estivo,thatis,International Mathe-
maticalSummerCentre.Conceivedintheearlyfifties,itwasbornin1954inFlorence,Italy,
andwelcomedbytheworldmathematicalcommunity:itcontinuessuccessfully,yearforyear,
tothisday.
Many mathematicians from all over the world have been involved in a way oranother in
C.I.M.E.’sactivitiesovertheyears.ThemainpurposeandmodeoffunctioningoftheCentre
maybesummarisedasfollows:everyyear,duringthesummer,sessionsondifferentthemes
from pure and applied mathematics are offered byapplication to mathematicians from all
countries. ASessionis generally basedonthree orfourmaincourses given byspecialists
ofinternationalrenown,plusacertainnumberofseminars,andisheldinanattractiverural
locationinItaly.
TheaimofaC.I.M.E.sessionistobringtotheattentionofyoungerresearcherstheorigins,
development, and perspectives of some very active branch of mathematical research. The
topicsofthecoursesaregenerallyofinternationalresonance.Thefullimmersionatmosphere
ofthecoursesandthedailyexchangeamongparticipantsarethusaninitiationtointernational
collaborationinmathematicalresearch.
C.I.M.E.Director C.I.M.E.Secretary
PietroZECCA ElviraMASCOLO
DipartimentodiEnergetica“S.Stecco” DipartimentodiMatematica“U.Dini”
Universita`diFirenze Universita`diFirenze
ViaS.Marta,3 vialeG.B.Morgagni67/A
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Italy Italy
e-mail:zecca@unifi.it e-mail:[email protected]fi.it
FormoreinformationseeCIME’shomepage:http://www.cime.unifi.it
CIMEactivityiscarriedoutwiththecollaborationandfinancialsupportof:
-INdAM(IstitutoNazionalediAltaMatematica)
-MIUR(Ministerodell’Istruzione,dell’Universita`edellaRicerca)
Co-fundedbytheEuropeanUnionunderthe7thFrameworkProgramme
FP7-PEOPLE-2010-ITN
Grantagreementnumber264735-SADCO
Yves Achdou Guy Barles Hitoshi Ishii
(cid:2) (cid:2)
Grigory L. Litvinov
Hamilton-Jacobi Equations:
Approximations, Numerical
Analysis and Applications
Cetraro, Italy 2011
Editors:
Paola Loreti
Nicoletta Anna Tchou
123
YvesAchdou GuyBarles
LaboratoireJacques-LouisLions-UPMC LaboratoiredeMathe´matiquesetPhysique
Paris,France The´orique
Tours,France
HitoshiIshii
DepartmentofMathematics GrigoryL.Litvinov
WasedaUniversity InstitutionforInformationTransmission
Shinjuku-ku,Tokyo,Japan Problems
RusssianAcademyofSciences
Moscow,Russia
ISBN978-3-642-36432-7 ISBN978-3-642-36433-4(eBook)
DOI10.1007/978-3-642-36433-4
SpringerHeidelbergNewYorkDordrechtLondon
LectureNotesinMathematicsISSNprintedition:0075-8434
ISSNelectronicedition:1617-9692
LibraryofCongressControlNumber:2013937326
MathematicsSubjectClassification(2010):49-02,35F21,49L25,35B40,49N70,91A13,14T05,15A80
(cid:2)c Springer-VerlagBerlinHeidelberg2013
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Foreword
These lecture notescontainthe materialrelative to the coursesgivenat the CIME
SummerSchoolheldinCetraro,Italy,fromAugust29toSeptember3, 2011.The
topic was Hamilton–Jacobi Equations: Approximations, Numerical Analysis and
Applications
The courses dealt mostly with the following subjects: first-order and second-
order Hamilton–Jacobi–Bellman equations, properties of viscosity solutions,
homogenizationandasymptoticbehaviours,meanfield games,approximationand
numericalmethods,andidempotentanalysis.Thecontentofthecourseswentfrom
an introductionto viscosity solutionsto quite advancedtopics, at the cutting edge
of the research in the field. We believe that they opened perspectiveson new and
delicateissues.
Thisvolumecontainsfourcourses
• FiniteDifferenceMethodsforMeanFieldGames
YvesAchdou
• AnIntroductiontotheTheoryofViscositySolutionsforFirst-OrderHamilton–
JacobiEquationsandApplications
GuyBarles
• A Short Introduction to Viscosity Solutions and the Large Time Behavior of
SolutionsofHamilton–JacobiEquations
HitoshiIshii
• Idempotent/TropicalAnalysis,theHamilton–JacobiandBellmanEquations
GrigoryL.Litvinov
A fifth courseheld atthe workshopbyPanagiotisE. Souganidisofthe University
of Chicago (Homogenizationand Approximationfor Hamilton–JacobiEquations)
isnotincludedinthisvolume.
The participants came from several countries (ordered decreasingly with the
numberof participants):Italy, France, the USA, Argentina,Austria, Chile, China,
Germany,Japan,Greece,Iran,Rumania,Russia,SwedenandVietnam.
v
vi Foreword
OnSeptember1st,PaolaLoreti,ElviraMascoloandNicolettaTchouorganized
asessionopentotheyoungerresearchers.This“CIME-young”sessionallowedthe
doctoralstudentsandposdoctoralresearcherstopresenttheirnewresults.
YoungSpeakers
• MorenoConcezzi
Universita`DegliStudiRomaTre,Italy
Numerical methodsandapplications-dynamicprogrammingfor HCS andfrac-
tionarylaplacianapproximation
• Jean-PaulDaniel
LaboratoireJacques-LouisLions—Universite´Paris6,France
AgameinterpretationforfullynonlinearequationswithNeumanncondition
• TizianoDeAngelis
SapienzaUniversita`diRoma,Italy
OptimalstoppingofaHilbertspacevalueddiffusionprocess
• JoschaDiehl
UniversityofBerlin,Germany
PathwiseapproachtoroughBurger’sPDEs
• BenjaminFehrman
UniversityofChicago,USA
HomogenizationofsystemsofviscousHamilton–Jacobiequations
• GiulioGalise
Universita`degliStudidiSalerno,Italy
Viscositysolutionsofuniformlyellipticequationswithoutboundaryandgrowth
conditionsatinfinity
• AnnaChiaraLai
DipartimentodiScienzediBaseeApplicateperl’Ingegneria,SapienzaUniver-
sita` diRoma,Italy
Amulti-phalanxself-similarrobotfinger
• RobertoMecca
Dipartimento di Matematica “G. Castelnuovo”, Sapienza Universita` di Roma,
Italy
Shapefromshadingviaphotometricstereotechniqueanewdifferentialapproach
• CristinaPocci
DipartimentodiScienzediBaseeApplicateperl’Ingegneria,SapienzaUniver-
sita` diRoma,Italy
Propagationoffrontsinnonlineardiffusionequations
• F.J.Silva,
Dipartimento di Matematica “G. Castelnuovo”, Sapienza Universita` di Roma,
Italy
A semi-Lagrangian scheme for a 1st order-infinite horizon mean field game
model
Foreword vii
Authors’ShortVitae
• Yves Achdou has been a professor at Universite´ Paris-Diderot since 2000.
His research is mainly devoted to the numerical analysis of partial differential
equationswithseveralapplicationsrangingfromfluiddynamicstomathematical
finance.
• Guy Barles has been a professor at Universite´ Franc¸ois Rabelais Tours since
1990. He has worked on viscosity solutions of fully nonlinear elliptic and
parabolic equations, including in particular Hamilton–Jacobi equations and
variousapplicationslikedislocations.
• Hitoshi Ishii was a professor at Chuo University (Tokyo) (April 1989–March
1996)and atTokyoMetropolitanUniversity(April1996–March2001).He has
beenaprofessoratWasedaUniversity,Tokyo,sinceApril2001andalsoadjunct
professoratKingAbdulazizUniversity,Jeddah,sinceAugust2011.Hisresearch
interesthasbeeninnonlinearpartialdifferentialequationsandtheirapplications.
• GrigoryL. Litvinovhas beena research fellow and lecturerat the Independent
UniversityofMoscowsince2002.Since2006heisaresearcherattheRussian–
French Laboratory J.-V. Poncelet. Since 2011 he is a leading scientist at the
Institute for Information Transmission Problems of the Russian Academy of
Sciences and part-time professor at the National Research University Higher
School of Economics, Moscow, Russia. His research interest has been in
functionalanalysis, Lie groupsand hypergroups,idempotentmathematics, and
computertechnology:algorithmsandspecialhardwareandsoftware.
PaolaLoreti
DipartimentodiScienzediBasee
Applicateperl’Ingegneria
SapienzaUniversita`diRoma
ViaA.Scarpan.16
00161Roma
Italy
NicolettaTchou
IRMAR
Universite´deRennes1
CampusdeBeaulieu
35042RennesCedex
France

Acknowledgements
The editors wish to thank all the people who contributed to the success of the
school.TheyareparticularlygratefultotheCIMEandespeciallytoElviraMascolo
for her kindness and efficiency. The European Project: Initial Training Network
SensitivityAnalysisforDeterministicControllerDesign—SADCO(2011–2014)—
has financed the participation of 15 young researchers. Sapienza Universita` di
Roma(Italy)andUniversite´deRennes1(France)havepartiallyfinancedthetravel
expenses.
ix