Differential Equations on Fractals

"DifferentialEquationsonFractals"opensthedoortounderstandingtherecentlydevelopedareaofanalysisonfractals,focusingontheconstructionofaLaplacianontheSierpinskigasketandrelatedfractals.Writteninalivelyandinformalstyle,withlotsofintriguingexercisesonalllevelsofdifficulty,thebookisaccessibletoadvancedundergraduates,graduatestudents,andmathematicianswhoseekanunderstandingofanalysisonfractals.RobertStrichartztakesthereadertothefrontiersofresearch,startingwithcarefullymotivatedexamplesandconstructions.OneofthegreataccomplishmentsofgeometricanalysisinthenineteenthandtwentiethcenturieswasthedevelopmentofthetheoryofLaplaciansonsmoothmanifolds.Butwhathappenswhentheunderlyingspaceisrough?Fractalsprovidemodelsofroughspacesthatneverthelesshaveastrongstructure,specificallyself-similarity.Exploitingthisstructure,researchersinprobabilitytheoryinthe1980swereabletoprovetheexistenceofBrownianmotion,andthereforeofaLaplacian,oncertainfractals.Anexplicitanalyticconstructionwasprovidedin1989byJunKigami."DifferentialEquationsonFractals"explainsKigami'sconstruction,showswhyitisnaturalandimportant,andunfoldsmanyoftheinterestingconsequencesthathaverecentlybeendiscovered.Thisbookcanbeusedasaself-studyguideforstudentsinterestedinfractalanalysis,orasatextbookforaspecialtopicscourse.