Notes on the Infinity Laplace Equation PDF
Preview Notes on the Infinity Laplace Equation
SPRINGER BRIEFS IN MATHEMATICS Peter Lindqvist Notes on the Infinity Laplace Equation 123 SpringerBriefs in Mathematics Series editors Nicola Bellomo, Torino, Italy Michele Benzi, Atlanta, USA Palle E.T. Jorgensen, Iowa City, USA Tatsien Li, Shanghai, China Roderick Melnik, Waterloo, Canada Lothar Reichel, Kent, USA Otmar Scherzer, Vienna, Austria Benjamin Steinberg, New York, USA Yuri Tschinkel, New York, USA G. George Yin, Detroit, USA Ping Zhang, Kalamazoo, USA SpringerBriefsinMathematicsshowcasesexpositionsinallareasofmathematics andappliedmathematics.Manuscriptspresentingnewresultsorasinglenewresult inaclassicalfield,newfield,oranemergingtopic,applications,orbridgesbetween newresultsandalreadypublishedworks,areencouraged.Theseriesisintendedfor mathematicians and applied mathematicians. BCAM SpringerBriefs Editorial Board Enrique Zuazua Departamento de Mathemáticas Universidad Autónoma de Madrid Madrid, Spain Irene Fonseca Center for Nonlinear Analysis Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, USA Juan J. Manfredi Department of Mathematics University of Pittsburgh Pittsburgh, USA Emmanuel Trélat Laboratoire Jacques-Louis Lions Institut Universitaire de France Université Pierre et Marie Curie CNRS, UMR, Paris Xu Zhang School of Mathematics Sichuan University Chengdu, China BCAM SpringerBriefs aims to publish contributions in the following disciplines: Applied Mathematics, Finance, Statistics and Computer Science. BCAM has appointed an Editorial Board, who evaluate and review proposals. Typical topics include: a timely report of state-of-the-art analytical techniques, bridge between new research results published in journal articles and a contextual literature review, a snapshot of a hot or emerging topic, a presentation of core conceptsthatstudentsmustunderstandinordertomakeindependentcontributions. PleasesubmityourproposaltotheEditorialBoardortoFrancescaBonadei,Executive EditorMathematics,Statistics,andEngineering:[email protected] More information about this series at http://www.springer.com/series/10030 Peter Lindqvist fi Notes on the In nity Laplace Equation 123 PeterLindqvist Department ofMathematical Sciences NorwegianUniversity of Science andTechnology Trondheim Norway ISSN 2191-8198 ISSN 2191-8201 (electronic) SpringerBriefs inMathematics ISBN978-3-319-31531-7 ISBN978-3-319-31532-4 (eBook) DOI 10.1007/978-3-319-31532-4 LibraryofCongressControlNumber:2016933231 ©TheAuthor(s)2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodologynowknownorhereafterdeveloped. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt fromtherelevantprotectivelawsandregulationsandthereforefreeforgeneraluse. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained hereinorforanyerrorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland Preface Habent sua fata libelli pro captu lectoris. ThesenotesarewrittenupaftermylecturesattheUniversityofPittsburghinMarch 2014 and at Tsinghua University in May 2014. My objective is the 1-Laplace Equation, a marvellous kin to the ordinary Laplace Equation. The 1-Laplace Equation has delightful counterparts to the Dirichlet integral, the Mean Value Theorem, the Brownian Motion, Harnack’s Inequality, and so on. It has applica- tions to image processing and mass transfer problems and provides optimal Lipschitz extensions of boundary values. My treaty of this “fully non-linear” degenerate equation is far from complete and generalizations are deliberately avoided—“Less is more.” Trondheim Peter Lindqvist February v Acknowledgments IthankErikLindgren,whodetectedaseriousflawinthefirstversionofChap.10, Mikko Parviainen, and Nikolai Ubostad for their comments and help with proof- reading.IamverygratefultotheUniversityofPittsburghandTsinghuaUniversity, for giving me the opportunity to lecture about the topic. Finally, I thank Juan Manfredi for valuable suggestions. vii Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3 Variational Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4 Viscosity Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 5 An Asymptotic Mean Value Formula . . . . . . . . . . . . . . . . . . . . . . 33 6 Comparison with Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 7 From the Theory of Viscosity Solutions . . . . . . . . . . . . . . . . . . . . 45 8 Uniqueness of Viscosity Solutions . . . . . . . . . . . . . . . . . . . . . . . . . 49 9 Tug-of-War . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 10 The Equation Δ1v¼F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 ix Chapter 1 Introduction Thesenotesareaboutthecelebrated∞-LaplaceEquation,whichsofarasIknow first was derived by G. Aronsson in 1967, see [A1]. This fascinating equation has later been rediscovered in connexion with Image Processing and in Game Theory. Intwovariablesitreads u2u +2u u u +u2u =0, x xx x y xy y yy whereu = u(x,y).1 Forafunctionu = u(x ,x ,...,x )inn variablestheequa- 1 2 n tionis (cid:2)n ∂u ∂u ∂2u (cid:2)∞u ≡ = 0. ∂x ∂x ∂x ∂x i,j=1 i j i j Itisthelimitequationofthe p-LaplaceEquations (cid:3) (cid:4) (cid:2)pu ≡|∇u|p−4 |∇u|2(cid:2)u+(p−2)(cid:2)∞u = 0 (1.1) asp →∞.Toseethis,divideoutthefactor|∇u|p−4,dividebyp−2andletp →∞ intheresultingexpression |∇u|2(cid:2)u p−2 +(cid:2)∞u = 0. Our derivation leaves much to be desired, but the limit equation is the correct one.—Thesolutionsareoftencalled∞-harmonicfunctions. 1Formallyitwouldbeclassifiedasparabolic,sinceitsdiscriminantisu2xu2y−(uxuy)2=0,butthe mainfeaturesare“elliptic”.Thecharacteristiccurvesareuxdy−uydx =0.Inseveralvariables, itisdegenerateelliptic. ©TheAuthor(s)2016 1 P.Lindqvist,NotesontheInfinityLaplaceEquation, SpringerBriefsinMathematics,DOI10.1007/978-3-319-31532-4_1 2 1 Introduction ClassicalSolutionsAfunctionu ∈C2((cid:3)), (cid:3)denotingadomaininn-dimensional EuclideanspaceRn,whichsatisfiestheequationpointwiseiscalledaclassicalsolu- tion.Afewexplicitexamplesare (cid:5) b|x −a|+c, a x2+x2+···+x2+b, (cid:7)a,x(cid:8)+b, 1 2 k arctan x2, arctan (cid:5) x3 , x43 −x43, and x1 x2+x2 1 2 1 2 a x43 +a x43 +···+a x43 where a3+a3+···+a3 =0, 1 1 2 2 n n 1 2 n indomainswheretheyareofclassC2.Since 1 (cid:2)∞u = (cid:7)∇u,∇|∇u|2(cid:8) 2 all solutions u ∈ C1 of the eikonal equation |∇u|2 = C can be included. Finally, solutionsindisjointvariablescanbesuperposed: (cid:5) (cid:5) x2+x2−7 x2+x2+x +x43 −x43, 1 2 3 4 5 6 7 forexample.Alsosolutionslike |xi21 +···xi2ν|23 −|x2j1 +···x2jμ|23 arepossible.Interestingexplicitexamplesintwovariablesareconstructedin[A3]. —Asweshallsee,theclassicalsolutionsaretoofewtosolvetheDirichletproblem. Gradient Flow Let x = x(t) = (x (t),x (t),...,x (t)) be a sufficiently smooth 1 2 n curve.Differentiating|∇u(x(t))|2 alongthecurve,weobtain d (cid:2)n ∂u ∂2u dx |∇u(x(t))|2 = 2 i. dt ∂x ∂x ∂x dt i,j=1 j i j Selectthecurvesothatitisasolutiontothesystem dx =∇u(x(t)). dt Alongthisso-calledgradientflow,whichyieldsthestreamlines,2wehave d |∇u(x(t))|2 = 2(cid:2)∞u(x(t)). dt 2Thestreamlinesareorthogonaltothelevelsurfacesu=constant.
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