Variational Methods in Nonlinear Analysis: With Applications in Optimization and Partial Differential Equations

482 Pages·2019·6.033 MB·English

by  Dimitrios C Kravvaritis, Athanasios N Yannacopoulos

DimitriosC.Kravvaritis,AthanasiosN.Yannacopoulos
VariationalMethodsinNonlinearAnalysis
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DimitriosC. Kravvaritis,
AthanasiosN. Yannacopoulos
Variational
Methods in
Nonlinear Analysis
|
With Applications in Optimization and Partial
Differential Equations
MathematicsSubjectClassification2010
35-02,65-02,65C30,65C05,65N35,65N75,65N80
Authors
Prof.Dr.DimitriosC.Kravvaritis Prof.Dr.AthanasiosN.Yannacopoulos
NationalTechnicalUniversityofAthens AthensUniversityof
SchoolofAppliedMathematicaland EconomicsandBusiness
PhysicalSciences DepartmentofStatistics
HeroonPolytechniou9 Patission7610434Athens,Greece
ZografouCampus [email protected]
15780Athens,Greece
[email protected]
ISBN978-3-11-064736-5
e-ISBN(PDF)978-3-11-064738-9
e-ISBN(EPUB)978-3-11-064745-7
LibraryofCongressControlNumber:2020934276
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DCK:TomywifeKaterinaandmytwosonsChristosandNikos.
ANY:Tomyfixed-pointElectraandtoNikos,JennyandHelen

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Spentthemorningputtinginacommaandtheafternoonremovingit.
G.FlaumbertorO.Wilde
Ifyoucan’tproveyourtheorem,keepshiftingpartsoftheconclusiontotheassump-
tions,untilyoucan.
E.DeGiorgi(accordingtoWikipedia)

Preface
Nonlinearanalysisisavastfield,whichoriginatedfromtheneedofextendingclas-
sicallinearstructuresinanalysisinordertotreatnonlinearproblems.Itcanbecon-
sideredasgeneraltermfordescribingawidespectrumofabstractmathematicaltech-
niques,rangingfromgeometry,topologyandanalysis,whichneverthelessfindim-
portantapplicationsina varietyoffields,suchasmathematical modeling(e.g.,in
economics,engineering,biology,decisionscience,etc.),optimizationorinapplied
mathematics(e.g.,integral,differentialequationsandpartialdifferentialequations).
Itistheaimofthisbooktoselectoutofthiswidespectrum,certainaspectsthatthe
authorsconsiderasmostusefulinavarietyofapplications,andinparticularaspects
ofthetheoryrelatedwithvariationalmethods,whicharemostcloselyrelatedtoap-
plicationsinoptimizationandPDEs.
Thebookisintendedtograduatestudents,lecturersorindependentresearchers
whowishaconciseintroductiontononlinearanalysis,inordertolearnthematerial
independently,useitasaguidelinefortheorganizationofacourse,oruseitasaref-
erencewhentryingtomastercertaintechniquestouseintheirownresearch.Wehope
thatwehaveachievedagoodbalancebetweentheoryandapplications.Allthetheo-
reticalresultsareprovedinfulldetail,emphasizingcertaindelicatepoints,andmany
examplesareprovidedguidingthereadereithertowardsextensionsofthefundamen-
talresultsortowardsimportantapplicationsinavarietyoffields.Largesectionsare
devotedtotheapplicationsofthetheoreticalresultsinoptimization,PDEsandvaria-
tionalinequalities,includinghintstowardamorealgorithmicapproach.Keepingthe
balanceright,thevolumemanageablebutatthesametimestrivingtoprovideaswide
aspectrumoftechniquesandsubjectsaspossiblehasbeenaverytrickybusiness.
We hope that we have managed to present a good variety of these parts of nonlin-
earanalysiswhichfindthemajorityofapplicationsinthefieldswhicharethewithin
thecoreinterestofthisbook,i.e.,optimizationandPDEs.Naturally,wehadtodraw
linesinourindulgencetothebeautifulsubjectswehavedecidedtopresent(other-
wisewewouldhaveresultsinaninevolumetreatiseratherthananinechapterbook)
butatthesametimewehavetocompletelyomitimportantfieldssuchas,e.g.,degree
theory,andweapologizeforthis,hopingtomakeupforthisomissioninthefuture!
Atanyrate,wehopethatoureffort(whichbythewaywasextremelyenjoyable)will
makeitselfusefultothosewishingaconciseintroductiontononlinearanalysiswith
anemphasistowardapplications,andwillhelpnewcomersofanymathematicalage
tothefield,colleaguesinacademiatoorganizetheircourses,andpractitionersorre-
searcherstowardobtainingthemethodologicalframeworkthatwillhelpthemtoward
reachingtheirgoal.
Thisbookconsistsofninechapters,whichwethinkcoverawidespectrumofcon-
ceptsandtechniquesofnonlinearanalysisandtheirapplications.
https://doi.org/10.1515/9783110647389-201
X | Preface
Chapter1collects,mostlywithoutproofs,somepreliminarymaterialfromtopol-
ogy,linearfunctionalanalysis,Sobolevspacesandmultivaluedmaps,whicharecon-
sideredasnecessaryinordertoproceedfurthertodevelopingthematerialcoveredin
thisbook.Eventhoughproofsarenotprovidedforthisintroductorymaterial,wetry
toillustrateandclarifytheconceptsusingnumerousexamples.
Chapter2developscalculusinBanachspacesandfocusesonconvexityasafun-
damentalpropertyofsubsetsofBanachspacesaswellasonthepropertiesofcon-
vexfunctionsonBanachspaces.Thismaterialisdevelopedindetail,startingfrom
topologicalpropertiesofconvexsetsinBanachspace,inparticular,theirremarkable
propertieswithrespecttotheweaktopology,andthenfocusesonpropertiesofcon-
vexfunctionswithrespecttocontinuityandsemicontinuityorwithrespecttotheir
GâteauxorFréchetderivatives.Extendedcommentsonthedeepconnectionsofcon-
vexitywithoptimizationaremade.
Chapter 3 is devoted to an important tool of nonlinear analysis, which is used
throughoutthisbook,fixed-pointtheory.Onecoulddevoteawholebooktothesub-
ject, so here we take a minimal approach, of presenting a selection of fixed-point
theoremswhichwefindindispensable.ThislistconsistsoftheBanachcontraction
theorem, Brouwer’s fixed-point theorem which, although finite dimensional in na-
ture, forms the basis for developing a large number of useful fixed-point theorems
ininfinitedimensionalspaces,Schauder’sfixed-pointtheoremandBrowder’sfixed-
pointtheorem.Specialattentionispaidtotheconstructionofiterativeschemesfor
theapproximationoffixedpoints,andinparticulartheKrasnoselskii–Manniterative
schemewhichfindsinterestingapplicationsinnumericalanalysis.Wealsointroduce
theCaristifixed-pointtheoremanditsdeepconnectionwithaveryusefultool,the
Ekelandvariationalprinciple.Thechaptercloseswithafixed-pointtheoremformul-
tivaluedmaps.
Chapter4providesanintroductiontononsmoothconvexfunctionsandthecon-
cept of the subdifferential. The theory of the subdifferential as well as subdifferen-
tialcalculusisdevelopedgradually,leadingtoitsapplicationstooptimization.The
Moreau proximity operator in Hilbert spaces is introduced and its use in a class of
popularoptimizationalgorithms,whichgounderthegeneralnameofproximalalgo-
rithmsispresented.
Chapter5isdevotedtothetheoryofdualityinconvexoptimization.Thechap-
ter begins with an indispensable tool, a version of the minimax theorem which al-
lowsustocharacterizesaddlepointsforfunctionalsofspecificstructureinBanach
spaces.WethendefineandprovidethepropertiesoftheFenchel–Legendreconjugate
aswellasthebiconjugate,andillustratethemwithnumerousexamples.Tofurtheril-
lustratethepropertiesoftheseimportanttransforms,weintroducetheconceptofthe
inf-convolution,aconceptofinterestinitsownrightinoptimization.Havingdevel-
opedthenecessarymachinery,wepresenttheimportantcontributionoftheminimax
approach in optimization starting with a detailed presentation of Fenchel’s duality