Nonlinear Analysis - Theory and Methods

586 Pages·2019·6.576 MB·English

by  Papageorgiou, Rǎdulescu, Repovš

Springer Monographs in Mathematics
Nikolaos S. Papageorgiou
Vicenţiu D. Rădulescu
Dušan D. Repovš
Nonlinear Analysis–
Theory and Methods
Springer Monographs in Mathematics
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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
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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 prob-
lems 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 oper-
ators (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 general-
izations 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 topo-
logical 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 mul-
tifunctions. 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