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Springer Monographs in Mathematics Nikolaos S. Papageorgiou Vicenţiu D. Rădulescu Dušan D. Repovš Nonlinear Analysis– Theory and Methods Springer Monographs in Mathematics Editors-in-Chief Isabelle Gallagher, Paris, France Minhyong Kim, Oxford, UK Series Editors Sheldon Axler, San Francisco, USA Mark Braverman, Princeton, USA Maria Chudnovsky, Princeton, USA Tadahisa Funaki, Tokyo, Japan Sinan C. Güntürk, New York, USA Claude Le Bris, Marne la Vallée, France Pascal Massart, Orsay, France Alberto Pinto, Porto, Portugal Gabriella Pinzari, Padova, Italy Ken Ribet, Berkeley, USA René Schilling, Dresden, Germany Panagiotis Souganidis, Chicago, USA Endre Süli, Oxford, UK Shmuel Weinberger, Chicago, USA Boris Zilber, Oxford, UK This series publishes advanced monographs giving well-written presentations of the “state-of-the-art” in fields of mathematical research that have acquired the maturity needed for such a treatment. They are sufficiently self-contained to be accessible to more than just the intimate specialists of the subject, and sufficiently comprehensive to remain valuable references for many years. Besides the current stateofknowledgeinitsfield,anSMMvolumeshouldideallydescribeitsrelevance to and interaction with neighbouring fields of mathematics, and give pointers to future directions ofresearch. More information about this series at http://www.springer.com/series/3733 Nikolaos S. Papageorgiou (cid:129) ţ ă Vicen iu D. R dulescu (cid:129) š š Du an D. Repov – Nonlinear Analysis Theory and Methods 123 NikolaosS. Papageorgiou VicenţiuD.Rădulescu Department ofMathematics Institute of Mathematics, Physics National Technical University andMechanics Athens, Greece Ljubljana, Slovenia Institute of Mathematics, Physics Faculty of AppliedMathematics andMechanics AGH University of Science Ljubljana, Slovenia andTechnology Kraków,Poland DušanD.Repovš Faculty of Education, Faculty of Mathematics andPhysics University of Ljubljana Ljubljana, Slovenia Institute of Mathematics, Physics andMechanics Ljubljana, Slovenia ISSN 1439-7382 ISSN 2196-9922 (electronic) SpringerMonographs inMathematics ISBN978-3-030-03429-0 ISBN978-3-030-03430-6 (eBook) https://doi.org/10.1007/978-3-030-03430-6 LibraryofCongressControlNumber:2018968386 MathematicsSubjectClassification(2010): 35-02,49-02,58-02 ©SpringerNatureSwitzerlandAG2019 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 orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Contents 1 Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Definitions, Density, and Approximation Results . . . . . . . . . . . . 1 1.2 The One-Dimensional Case. . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3 Duals of Sobolev Spaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4 Absolute Continuity on Lines, the Chain Rule and Consequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.5 Trace Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.6 The Extension Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1.7 The Rellich–Kondrachov Theorem. . . . . . . . . . . . . . . . . . . . . . . 38 1.8 The Poincaré and Poincaré–Wirtinger Inequalities. . . . . . . . . . . . 43 1.9 The Sobolev Embedding Theorem. . . . . . . . . . . . . . . . . . . . . . . 45 1.10 Capacities. Miscellaneous Results . . . . . . . . . . . . . . . . . . . . . . . 58 1.11 Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2 Compact Operators and Operators of Monotone Type. . . . . . . . . . . 71 2.1 Compact and Completely Continuous Maps. . . . . . . . . . . . . . . . 71 2.2 Proper Maps and Gradient Maps . . . . . . . . . . . . . . . . . . . . . . . . 77 2.3 Linear Compact Operators. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 2.4 Spectral Theory of Compact Linear Operators . . . . . . . . . . . . . . 89 2.5 Multifunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 2.6 Monotone Maps: Definition and Basic Results . . . . . . . . . . . . . . 113 2.7 The Subdifferential and Duality Maps . . . . . . . . . . . . . . . . . . . . 118 2.8 Surjectivity and Characterizations of Maximal Monotonicity. . . . 130 2.9 Regularizations and Linear Monotone Operators. . . . . . . . . . . . . 139 2.10 Operators of Monotone Type. . . . . . . . . . . . . . . . . . . . . . . . . . . 149 2.11 Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160 3 Degree Theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 3.1 Brouwer Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 3.2 The Leray–Schauder Degree . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 v vi Contents 3.3 Degree for Multifunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 3.4 Degree for ðSÞþ-Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203 3.5 Degree for Maximal Monotone Perturbation of ðSÞþ-Maps. . . . . 214 3.6 Degree for Subdifferential Operators . . . . . . . . . . . . . . . . . . . . . 221 3.7 Some Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235 3.8 Index of a n-Point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 3.9 Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257 4 Partial Order, Fixed Point Theory, Variational Principles . . . . . . . . 263 4.1 Cones and Partial Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 264 4.2 Metric Fixed Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 4.3 Topological Fixed Points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 4.4 Order Fixed Points and the Fixed Point Index . . . . . . . . . . . . . . 303 4.5 Fixed Points for Multifunctions . . . . . . . . . . . . . . . . . . . . . . . . . 314 4.6 Abstract Variational Principles. . . . . . . . . . . . . . . . . . . . . . . . . . 319 4.7 Young Measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 340 4.8 Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356 5 Critical Point Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 5.1 Pseudogradients and Compactness Conditions . . . . . . . . . . . . . . 363 5.2 Critical Points via Minimization—The Direct Method. . . . . . . . . 373 5.3 Deformation Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379 5.4 Minimax Theorems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 397 5.5 Critical Points Under Constraints. . . . . . . . . . . . . . . . . . . . . . . . 420 5.6 Critical Points Under Symmetries . . . . . . . . . . . . . . . . . . . . . . . 427 5.7 The Structure of the Critical Set . . . . . . . . . . . . . . . . . . . . . . . . 444 5.8 Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 451 6 Morse Theory and Critical Groups. . . . . . . . . . . . . . . . . . . . . . . . . . 457 6.1 Elements of Algebraic Topology . . . . . . . . . . . . . . . . . . . . . . . . 458 6.2 Critical Groups, Morse Relations. . . . . . . . . . . . . . . . . . . . . . . . 477 6.3 Continuity and Homotopy Invariance of Critical Groups. . . . . . . 501 6.4 Extended Gromoll–Meyer Theory . . . . . . . . . . . . . . . . . . . . . . . 506 6.5 Local Extrema and Critical Points of Mountain Pass Type . . . . . 526 6.6 Computation of Critical Groups. . . . . . . . . . . . . . . . . . . . . . . . . 530 6.7 Existence and Multiplicity of Critical Points. . . . . . . . . . . . . . . . 545 6.8 Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553 References.... .... .... .... ..... .... .... .... .... .... ..... .... 557 Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 573 Introduction Insight must precede application. Max Planck (1858–1947), Nobel Prize in Physics 1918 Answeringtheneedsofspecificappliedproblems,NonlinearAnalysisemergedasa separate field of research within Mathematical Analysis immediately after World War II, when Linear Functional Analysis (primarily Banach space theory) had reached a rather mature stage. Years of accumulated experience has convinced people that theory can no longer afford the luxury of dealing with linear, smooth and well-posed models. In many applications of interest, such requirements either exclude many important aspects of the problem or, even more dramatically, fail completely to provide a satisfactory model for the phenomena under investigation. SuchconsiderationsledtothedevelopmentofNonlinearAnalysis,whichtodayhas developedsignificantlyandisoneofthemost activeareas ofresearch.Theadvent of Nonlinear Analysis led to unifying theories describing different classical problems and permitted the investigation of a whole new range of applications. The theories,methodsandtechniquesofNonlinearAnalysisprovedtobeindispensable tools in the analysis of various problems in many other fields. For this reason, Nonlinear Analysis eventually acquired an interdisciplinary character and it is a prerequisiteformanynonmathematicianswhowanttoconductanindepthanalysis ofproblemstheyface.Thisleadstoanincreasingdemandforbooksthatsummarize the recent developments in various parts of Nonlinear Analysis. Make no mistake, Nonlinear Analysis is a very broad subject and every such book focuses only on a part of it. Here the emphasis is on those aspects that are usefulinthestudyofboundaryvalueproblems.Infact,VolumeIIwillbedevoted to the study of such problems. Given the orientation of this book project, it is natural to start in Chap. 1 with Sobolev Spaces, which are the main tools in the analysis of both stationary and nonstationary problems. Sobolev spaces play a centralroleinthemoderntheoryofpartialdifferentialequationsandtheyleadtoa significantbroadeningofthenotionofsolutionofaboundaryvalueproblem.They provide a natural analytical framework for the study of linear and nonlinear boundary value problems. We provide a concise but complete introduction to the subject, emphasizing those parts of the theory which are relevant to the study of vii viii Introduction boundaryvalueproblems.Wedealwithbothfunctionsofoneandseveralvariables. Inthelast section wealso discusscapacities,whichariseinthestudy ofsmall sets inRN andofthefinepropertiesofSobolevfunctions.Wealsopresentsomerelated results which will be of interest to people dealing with boundary value problems. InChap.2 wedeal with CompactOperatorsandOperatorsofMonotoneType. Compact operators are in fact the starting point of Nonlinear Analysis, going back to the celebrated work of Leray and Schauder in the 1930s. Compactness was introduced as a first attempt to deal with infinite-dimensional nonlinear operator equations, since by its nature compactness (in all its forms) approximates infinite objects by finite objects. In parallel we develop the corresponding linear theory, leading to the spectral theorem for compact self-adjoint operators on a Hilbert space. This theorem is the basis of the spectral analysis of linear elliptic operators under different boundary conditions. Of course, compact operators have serious limitations, which researchers tried to overcome by introducing new classes of nonlinear maps. A broader framework for the analysis of infinite-dimensional problems is provided by monotone operators, which extend to an infinite- dimensional context the classical notion of an increasing real function. Monotone operators are rooted in variational problems. Of special interest are the so-called maximal monotone operators, which exhibit remarkable surjectivity properties. However, the development of a coherent theory of maximal monotone maps leads necessarily to multivalued maps (multifunctions). For this reason, in Sect. 2.5 we have a detour to Set-Valued Analysis. We point out that multivalued analysis provides basic tools in many applied areas such as optimization, optimal control, mathematicaleconomics,gametheory,etc.The“differential”theoryofnonsmooth convex functions leads to a special class of multivalued maximal monotone operators (convex subdifferential). At the end of the chapter we also discuss useful generalizations of the notion of monotonicity. In Chap. 3, we conduct a detailed study of the main degree theories. We start with Brouwer’s theory (finite-dimensional spaces), following the analytical approach that goes back to the work of Nagumo in the 1950s. Brouwer’s original approach to the definition of the degree was based on combinatorial and algebraic topology.Sincemostproblemsofinterestareinfinite-dimensional,itwasnecessary to extend Brouwer’s theory to infinite-dimensional maps. This was done by Leray and Schauder in the 1930s, who used compact operators (namely operators of the form I(cid:2)K with I being the identity map and K a compact operator). The Leray–Schauder degree theory and its consequences are examined in Sect. 3.2. After that we examine degree theories for set-valued maps and for operators of monotone type. All these are recent theories and were introduced to deal with infinite-dimensional nonlinearproblemsforwhichtheLeray–Schauder theoryfails to address. At the end of the chapter, we also discuss some alternative generalizations of the Leray–Schauder theory using measures of noncompactness (con- densing maps) and we examine the index of a n-point. Introduction ix Chapter 4 deals with Fixed Point Theory and with some important Variational Principles. There is an informal classification of fixed-point theorems to “metric fixed points”, “topological fixed points” and “order fixed points”. Since the latter class involves some order structure in the underlying space (usually a Banach space), in Sect. 4.1 we start with a general discussion of cones and of the partial order they induce on the ambient space. Then we start discussing the three afore- mentioned classes offixed points. Metric fixed-point theorems are always formu- lated in a metric space setting and the methods involved in their study exploit the metric structure and geometry of the spaces involved together with the metric properties of the maps. In contrast to the topological fixed point theory, the topological properties of the spaces and/or of the maps are involved. In particular, the notionofcompactnessisimportantinourconsiderationsthere.In“orderfixedpoint theory”, the order on the space induced by a cone is the main ingredient and the basichypothesesandconditionsarebasedaroundthisnotion,aswellasthenotion of the “Leray–Schauder fixed point index”. We also discuss fixed points of multifunctions. In Sect. 4.6 we discuss some important abstract variational principles. Special emphasis isgiven on thecelebrated “Ekeland variational principle” and its many interesting consequences. We conclude the chapter with a discussion of Youngmeasuresthatariseinalargeclassofvariationalproblems.Whenthedirect methodofthecalculusofvariationsfails,theminimizingsequences(orappropriate subsequences of them) have a limit behavior (usually more and more oscillatory), which is captured by embedding the original functions in the space of Young measures (or parametric probabilities). This is the process of “relaxation”, familiar to people studying optimal control problems. InChap.5westudyCriticalPointTheory.Whenusingvariationalmethods,we are trying to find solutions of a given nonlinear equation, by looking for critical stationarypointsofafunctional(energyorEulerfunctional)definedonthefunction space in which we want the solutions to be. If this functional is bounded from below, as above, we can look for local extrema and the direct method enters into play. If the functional is indefinite, we cannot expect to have local extrema and so other methods for locating critical points need to be found. These methods are basedonminimaxprinciples,whichleadtocriticalpoints.Theseminimaxmethods are derived either using the deformation approach or the Ekeland variational principle. Here we follow the deformation approach which uses the change of the topological structure of the sublevel sets of the Euler functional along the flow producedbyakindofnegativegradientvectorfield.Weconductadetailedstudyof critical point theory including an analysis of the structure of the critical set at the end of the chapter. In Chap. 6, continuing the theme of locating and counting critical points of a givenfunctional,wediscussMorseTheoryandCriticalGroups,whichprovidethe tools to prove multiplicity theorems. Since these topics make use of tools from AlgebraicTopology,inSect.6.1wereviewtheneededbackgroundfromthatfield. Then we proceed with a self-contained presentation of the Morse theory related to thestudyoftheexistenceandmultiplicityofsolutionsforvariationalproblems.Our