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1、ProceedingsoftheSecondInternationalConferenceonNonlinearSystems(BulletinoftheMarathwadaMathematicalSocietyVol.8,No.2,December2007,Pages147{158.)ANINTRODUCTIONANDAPPLICATIONSTOHOMOGENIZATIONA.K.NandakumaranDepartmentofMathematics,IndianInstituteofScience,Bangalore{560012,IND
2、IA.Email:nands@math.iisc.ernet.inAbstractHomogenizationofpartialdi®erentialequationsisrelativelyanewareaandhastremendousapplicationsinvariousbranchesofengineeringscienceslike:materialsci-ence,porousmedia,studyofvibrationsofthinstructures,compositematerialstonameafew.Thought
3、hematerialscientistsandothershadreasonableideaaboutthehomoge-nizationprocess,itwaslackingagoodmathematicaltheorytillearlyseventies.The¯rstpropermathematicalprocedurewasdevelopedintheseventiesandlaterinthelast30yearsorsoithas°ourishedinvariouswaysbothapplicationwiseandmathem
4、atically.Iplantogiveanintroductorypresentationwiththeaimofcateringtoawideraudience.Wegothroughfewexamplestounderstandhomogenizationprocedureinageneralperspectivetogetherwithapplications.Wealsopresentcertainmathematicaltechniquesavailableandifpossiblesomedetails.Apossiblede¯
5、nitionofhomogenizationwouldbethatitisaprocessofunderstandingaheterogeneous(in{homogeneous)media,wheretheheterogenetiesareatthemicroscopiclevel,likeincompositematerials,byahomogeneousmedia.Inotherwords,onewouldliketoobtainahomogeneousdescriptionofahighlyoscillatingin{homogen
6、eousmedia.Wealsopresentothergeneralizationstononlinearproblems,porousmediaandsoon.Finally,wewillliketoseeacloselyrelatedissueofoptimalboundswhichitselfisanindependentareaofresearch.1INTRODUCTION.Themathematicaltheoryofhomogenizationgaineditssigni¯canceintheseventieswhenrigo
7、rousmathematicstounderstandtheprocedurewasintroduced.Themainmotivationwasfromthestudyofcompositematerials,moregenerallyanymediumordomainwhichinvolvesmicrostructures.Simplyspeaking,homogenizationisamathematicalproceduretounderstandheterogeneousmaterials(ormedia)withhighlyosc
8、illatingheterogeneties(atthemicroscopiclevel)viaahomogeneousmaterial.Mathematicall
