Table Of ContentTexts and Monographs in Physics
Series Editors:
W BeiglbOck E. H. Lieb W Thirring
Philippe Blanchard Erwin Briining
Variational Methods
in Mathetnatical Physics
A Unified Approach
With 7 Figures
Springer- Verlag
Berlin Heidelberg New York
London Paris Tokyo
Hong Kong Barcelona
Budapest
Professor Dr. Philippe Blanchard
Theoretische Physik, Fakultlit fiir Physik, Universitlit Bielefeld
Postfach 8640, WA800 Bielefeld, FRO
Dr. Erwin Bruning
Department of Mathematics, University of Cape Town
Private Bag, Rondebosch 7700, South Africa
Translator
Dr. Gillian M. Hayes
Department of Artificial Intelligence, University of Edinburgh
5 Forrest Hill, Edinburgh EHI 2QL, Scotland
Editors
Wolf Beiglbock Elliott H. Lieb
Institut fiir Angewandte Mathematik Jadwin Hall
Universitlit Heidelberg P. O. Box 708
1m Neuenheimer Feld 294 Princeton University
W-6900 Heidelberg I, FRO Princeton, NJ 08544-0708, USA
Walter Thirring
Institut fiir Theoretische Physik
der Universitlit Wien
Boltzmanngasse 5
A-1090 Wien, Austria
Title of the original German edition:
Philippe Blanchard, Erwin Bruning: Direkte Methodel' der VariatiollsrechlUlIlg - Ein Lehrbuch
© Springer-Verlag Wien 1982
ISBN-13:978-3-642-82700-6 e-ISBN -13 :978-3-642-82698-6
DOl: 10.1007/978-3-642-82698-6
Library of Congress Cataloging-in-Publication Data. Blanchard, Philippe [Direkte Methoden der
Variationsrechnung. English] Variational methods in mathematical physics: a unified approachl
Philippe Blanchard, Erwin Brtining. p. cm. - (Texts and monographs in physics). Translation of:
Direkte Methoden der Variationsrechnung: Includes bibliographical references and index.
ISBN-13:978-3-642-82700-6 I.Variational principles.2. Mathematical physics. I. Bruning, Erwin. II.
Title. III. Series. QCI74.17.V35B58 1992 530.1'5564-dc20 92-3355 CIP
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Preface
The first edition (in German) had the prevailing character of a textbook owing
to the choice of material and the manner of its presentation. This second
(translated, revised, and extended) edition, however, includes in its new parts
considerably more recent and advanced results and thus goes partially beyond
the textbook level.
We should emphasize here that the primary intentions of this book are
to provide (so far as possible given the restrictions of space) a selfcontained
presentation of some modern developments in the direct methods of the cal
culus of variations in applied mathematics and mathematical physics from a
unified point of view and to link it to the traditional approach. These modern
developments are, according to our background and interests:
(i) Thomas-Fermi theory and related theories, and
(ii) global systems of semilinear elliptic partial-differential equations and the
existence of weak solutions and their regularity.
Although the direct method in the calculus of variations can naturally be
considered part of nonlinear functional analysis, we have not tried to present
our material in this way. Some recent books on nonlinear functional analysis in
this spirit are those by K. Deimling (Nonlinear Functional Analysis, Springer,
Berlin Heidelberg 1985) and E. Zeidler (Nonlinear Functional Analysis and Its
Applications, Vols. 1-4; Springer, New York 1986-1990).
The reader with some background in the calculus of variations will cer
tainly miss many important aspects and results, for instance (locally) convex
analysis and duality theory, minimal surfaces, various recent results in critical
point theory and various classical methods and results.
To help to get a fairly complete picture of most of the fascinating aspects
of the calculus of variations we would like to give some references covering
many important topics not treated in this book.
Two recent books on (locally) convex analysis and duality in the calculus
of variations are
1. Ekeland, R. Temam: Convex Analysis and Variational Problems, North Hol
land, Amsterdam 1976, and
VI Preface
V. M. Alekseev, V. M. Tikhimirov, S. V. Fomin: Optimal Control, Plenum, New
York 1987.
Problems of minimal surfaces have played a prominent role in the devel
opment of the calculus of variations and of mathematics itself. A beautiful
nontechnical presentation of this subject has been given by S. Hildebrandt
and A. Tromba in the book Mathematics and Optimal Form, Freeman, New
York 1985. Further sources of information are, for instance, E. Gusti: Mini
mal Surfaces and Functions of Bounded Variation, Birkhauser, Basel 1984,
and S. Hildebrandt: Calculus of Variation Today, presented at the Oberwol
fach Meetings, in Perspectives in Mathematics, ed. by W. Jager, J. Moser,
R. Remmert, Birkhauser, Basel, 1983, pp.321-336.
Finally, concerning critical point theory, one should consult books on non
linear functional analysis and the recent book by J. Mawhin and M. Willem:
Critical Point Theory and Hamiltonian Systems, Applied Mathematical Sci
ences 74, Springer, Berlin Heidelberg, 1989.
The first German and this second edition differ by the following changes
and additions. The remark at the end of Section V.6 about the "principle
of symmetric criticality" as formulated and proved by R. S. Palais has been
replaced by a new Section V. 7 where this principle is discussed in some detail.
Section VIII.3 of the first edition about global nonlinear elliptic scalar
m.
field equations on d, d ~ 3, has been omitted. Now a new and long Chapter
IX contains various recent results on global solutions of systems of semilinear
elliptic partial-differential equations obtained by variational mathods. In par
m.
ticular we treat such equations on 2• Furthermore, one section presents the
basic results of the "elliptic regularity theory" for this type of equation. In
addition some recent results for local problems of this type involving "critical
Sobolev exponents" are discussed.
The former Chapter IX on Thomas-Fermi theory has now become Chap
ter X and has been supplemented by various remarks and comments mainly
concerning extensions of the results originally presented.
Finally, to Appendix 4 we have added some further basic facts about weak
derivatives, and there is a new technical Appendix 5, which has enabled us to
simplify some proofs in Chapter IX.
As in the first edition our lists of references only represent our sources of
information or suggest publications for further reading; we have not tried to
be complete.
However, we have tried to eliminate all the errors, omissions, and misprints
of the first edition, in particular those we learned about from colleagues and
students, especially T. Herb and N. Jakobowsky.
Moreover, various useful suggestions and criticism from readers of the
first edition have been taken into account. In this connection we would like to
thank first of all E. H. Lieb, who not only gave us many valuable comments on
our first version of Thomas-Fermi theory (now Chapter X) but also provided
Preface VII
helpful explanations of, and comments on, various important results presented
in Chapter IX.
Unfortunately, in the German edition, in the long formulae occurring in
Section V.6 on Noether's theorem in classical field theory some important
terms were omitted. This was brought to our attention by J. Messer. We would
like to thank him for this and several other comments on Thomas-Fermi the
ory.
And last but not least we would like to thank Dipl.-Math. U. Alfes who
found an error in the original German formulation of the hypotheses of The
orem 111.4.
The new part of the second edition have been written directly in English.
Special thanks are due to G. M. Hayes for her corrections of our poor En
glish. Last but not least and with great pleasure we thank W. Beiglbock who
managed the realisation of this book in spite of many complications.
Bielefeld and Cape Town, Ph. Blanchard
January 1992 E. Bruning
Contents
Some Remarks on the History and Objectives
of the Calculus of Variations .................................... 1
1. Direct Methods of the Calculus of Variations ........ . 15
1.1 The Fundamental Theorem of the Calculus of Variations ... . 15
1.2 Applying the Fundamental Theorem in Banach Spaces ..... . 20
1.2.1 Sequentially Lower Semi continuous Functionals ............ . 22
1.3 Minimising Special Classes of Functions .................... . 25
1.3.1 Quadratic Functionals ..................................... . 28
1.4 Some Remarks on Linear Optimisation ..................... . 30
1.5 Ritz's Approximation Method ............................. . 31
2. Differential Calculus in Banach Spaces ............... . 35
2.1 General Remarks .......................................... . 35
2.2 The Fn§chet Derivative .................................... . 36
2.2.1 Higher Derivatives ......................................... . 43
2.2.2 Some Properties of Frechet Derivatives ..................... . 44
2.3 The Gateaux Derivative ................................... . 46
2.4 nth Variation .............................................. . 49
2.5 The Assumptions of the Fundamental Theorem
of Variational Calculus ..................................... . 51
2.6 Convexity of f and Monotonicity of l' ..................... . 52
3. Extrema of Differentiable Functions .................. . 54
3.1 Extrema and Critical Values ............................... . 54
3.2 Necessary Conditions for an Extremum .................... . 55
3.3 Sufficient Conditions for an Extremum ..................... . 60
4. Constrained Minimisation Problems
(Method of Lagrange Multipliers) ..................... . 63
4.1 Geometrical Interpretation
of Constrained Minimisation Problems 63
4.2 Ljusternik's Theorems ..................................... . 66
X Contents
4.3 Necessary and Sufficient Conditions
for Extrema Subject to Constraints ........................ . 72
4.4 A Special Case ............................................ . 75
5. Classical Variational Problems ........................ . 77
5.1 General Remarks .......................................... . 77
5.2 Hamilton's Principle in Classical Mechanics ................ . 80
5.2.1 Systems with One Degree of Freedom ...................... . 81
5.2.2 Systems with Several Degrees of Freedom .................. . 95
5.2.3 An Example from Classical Mechanics ..................... . 105
5.3 Symmetries and Conservation Laws in Classical Mechanics .. 107
5.3.1 Hamiltonian Formulation of Classical Mechanics ........... . 107
5.3.2 Coordinate Transformations and Integrals of Motion ....... . 109
5.4 The Brachystochrone Problem ............................. . 113
5.5 Systems with Infinitely Many Degrees of Freedom: Field Theory 116
5.5.1 Hamilton's Principle in Local Field Theory ................. . 117
5.5.2 Examples of Local Classical Field Theories ................. . 122
5.6 Noether's Theorem in Classical Field Theory ............... . 124
5.7 The Principle of Symmetric Criticality ..................... . 130
6. The Variational Approach to Linear Boundary
and Eigenvalue Problems .............................. . 142
6.1 The Spectral Theorem for Compact Self-Adjoint Operators.
Courant's Classical Minimax Principle. Projection Theorem. 142
6.2 Differential Operators and Forms .......................... . 148
6.3 The Theorem of Lax-Milgram and Some Generalisations ... . 152
6.4 The Spectrum of Elliptic Differential Operators in a Bounded
Domain. Some Problems from Classical Potential Theory .... 156
6.5 Variational Solution of Parabolic Differential Equations.
The Heat Conduction Equation. The Stokes Equations ..... . 159
6.5.1 A General Framework for the Variational Solution
of Parabolic Problems ..................................... . 161
6.5.2 The Heat Conduction Equation ........................... .. 166
6.5.3 The Stokes Equations in Hydrodynamics ................... . 167
7. Nonlinear Elliptic Boundary Value Problems
and Monotonic Operators .............................. . 171
7.1 Forms and Operators - Boundary Value Problems .......... . 171
7.2 Surjectivity of Coercive Monotonic Operators.
Theorems of Browder and Minty ........................... . 173
7.3 Nonlinear Elliptic Boundary Value Problems.
A Variational Solution ..................................... . 178
Contents XI
8. Nonlinear Elliptic Eigenvalue Problems .............. . 192
8.1 Introduction ............................................... . 192
8.2 Determination of the Ground State
in Nonlinear Elliptic Eigenvalue Problems .................. . 195
8.2.1 Abstract Versions of Some Existence Theorems ............. . 195
8.2.2 Determining the Ground State Solution
for Nonlinear Elliptic Eigenvalue Problems ................. . 203
8.3 Ljusternik-Schnirelman Theory for Compact Manifolds ..... . 205
8.3.1 The Topological Basis of the Generalised Minimax Principle . 205
8.3.2 The Deformation Theorem ................................. . 207
8.3.3 The Ljusternik-Schnirelman Category and the Genus of a Set 210
8.3.4 Minimax Characterisation of Critical Values
of Ljusternik-Schnirelman .................................. . 215
8.4 The Existence of Infinitely Many Solutions
of Nonlinear Elliptic Eigenvalue Problems .................. . 217
8.4.1 Sphere-Like Constraints ................................... . 217
8.4.2 Galerkin Approximation for Nonlinear Eigenvalue Problems
in Separable Banach Spaces ................................ . 220
8.4.3 The Existence of Infinitely Many Critical Points as Solutions
of Abstract Eigenvalue Problems in Separable Banach Spaces 225
8.4.4 The Existence of Infinitely Many Solutions
of Nonlinear Eigenvalue Problems .......................... . 228
9. Semilinear Elliptic Differential Equations.
Some Recent Results on Global Solutions 241
9.1 Introduction ............................................... . 241
9.2 Technical Preliminaries .................................... . 247
9.2.1 Some Function Spaces and Their Properties ................ . 247
9.2.2 Some Continuity Results for Niemytski Operators .......... . 252
9.2.3 Some Results on Concentration of Function Sequences ...... . 256
9.2.4. A One-dimensional Variational Problem .................... . 262
9.3 Some Properties of Weak Solutions
of Semilinear Elliptic Equations ............................ . 266
9.3.1 Regularity of Weak Solutions .............................. . 266
9.3.2 Pohozaev's Identities ...................................... . 278
9.4 Best Constant in Sobolev Inequality ....................... . 283
9.5 The Local Case with Critical Sobolev Exponent ............ . 287
9.6 The Constrained Minimisation Method Under Scale Covariance 294
9.7 Existence of a Minimiser I: Some General Results .......... . 302
9.7.1 Symmetries ................................................ . 302
9.7.2. Necessary and Sufficient Conditions ........................ . 304
9.7.3 The Concentration Condition .............................. . 305
9.7.4 Minimising Subsets ........................................ . 308
XII Contents
9.7.5 Growth Restrictions on the Potential ....................... . 310
9.8 Existence of a Minimiser II: Some Examples ............... . 312
9.8.1 Some Non-translation-invariant Cases ...................... . 313
9.8.2 Spherically Symmetric Cases ............................... . 316
9.8.3 The Translation-invariant Case Without Spherical Symmetry 319
9.9 Nonlinear Field Equations in Two Dimensions .............. . 322
9.9.1 Some Properties of Niemytski Operators on Eq ............. . 323
9.9.2 Solution of Some Two-Dimensional Vector Field Equations .. 326
9.10 Conclusion and Comments ................................. . 332
9.10.1 Conclusion ................................................ . 332
9.10.2 Generalisations ............................................ . 334
9.10.3 Comments ................................................. . 335
9.11 Complementary Remarks .................................. . 337
10. Thomas-Fermi Theory .................................. . 340
10.1 General Remarks .......................................... . 340
10.2 Some Results from the Theory of LP Spaces (1 :::; P :::; 00) ... . 342
10.3 Minimisation of the Thomas-Fermi Energy Functional ...... . 344
lOA Thomas-Fermi Equations and the Minimisation Problem
for the TF Functional ...................................... . 351
10.5 Solution of TF Equations for Potentials
of the Form Vex) = E~=l Ix~jxjl ........................... . 357
10.6 Remarks on Recent Developments in Thomas-Fermi
and Related Theories ...................................... . 361
Appendix A. Banach Spaces .................................... 363
Appendix B. Continuity and Semicontinuity .................. 371
Appendix C. Compactness in Banach Spaces ................. 373
Appendix D. The Sobolev Spaces Wm,P(O) .................... 380
D.1 Definition and Properties. . . .. . .. . . ... .. ..... .. ... .. .. . . 380
D.2 Poincare's Inequality ................................... 385
D.3 Continuous Embeddings of Sobolev Spaces.............. 386
DA Compact Embeddings of Sobolev Spaces................ 388
Appendix E ...................................................... 391
E.1 Bessel Potentials. ... .. .. .... .. . .. . . ... ....... ..... .. .... 391
E.2 Some Properties of Weakly Differentiable Functions ..... 392
E.3 Proof of Theorem 9.2.3 ................................. 393
References ........................................................ 395
Index of Names .................................................. 405
Subject Index .................................................... 407
