Functional Analysis, Calculus of Variations and Numerical Methods for Models in Physics and Engineering

589 Pages·2020·5.489 MB·English

by  Fabio Silva Botelho

Functional Analysis, Calculus
of Variations and Numerical
Methods for Models in
Physics and Engineering
Fabio Silva Botelho, Adjunct Professor
Department of Mathematics
Universidade Federal de Santa Catarina
Florianópolis, SC, Brazil
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A SCIENCE PUBpL,ISHERS BOOK
A SCIENCE PUBLISHERS BOOK
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Preface
The first part of this book develops in details the basic tools of functional analysis, measure and integration
and Sobolev spaces theory, which will be used in the development of the subsequent chapters. The results are
established in a rigorous and precise fashion, however we believe the proofs presented are relatively easy to
follow, since all steps are developed in a very transparent and clear way.
Of course, in general the results in this first part are not new and may be found in similar format in many
other excellent books on functional analysis, such as those by Bachman and Narici [6], Brezis [24], Rudin
[67] and others.
In many cases we give our own version of some proofs we believe the original ones are not clear enough,
always aiming to improve their clarity and preciseness.
In the second book part, we develop the basic concepts on the calculus of variations, convex analysis,
duality theory and constrained optimization in a Banach spaces context.
We start with a more basic approach on the calculus of variations which gradually evolves to the
development of more advanced results on convex analysis and optimization. The text includes a formal proof
of the Lagrange multiplier theorem, in a first step, for the equality-restriction case and in the subsequent
sections we address the general case of constrained optimization also in a Banach spaces context.
Finally, in the third part we address the development of several important examples which apply the
previous theoretical approaches exposed.
Applications include duality principles for non-linear plate and shell models, duality principles for
problems in non-linear elasticity, duality for shape optimization models, existence and duality for the
Ginzburg-Landau system in superconductivity, duality for a semi-linear model in micro-magnetism and
others.
About such applications we highlight the results on the Generalized Method of Lines, which is a
numerical method for which the domain of the partial differential equation in question is discretized in lines,
and the concerning solution is written on these lines as function of the boundary conditions and boundary
shape. Some important models are addressed and among them we highlight the applications of such a method
to the Navier-Stokes system.
Also as an application of the Generalized Method of Lines, we finish the book with a chapter which
presents a numerical procedure for the solution of an inverse optimization problem.
At this point, it is worth emphasizing that the Chapter 21 is co-authored by myself and my colleague
Alexandre Molter and Chapter 23 is co-authored by myself and my graduate student Eduardo Pandini Barros,
to whom I would like to express my gratitude for their contributions.
As a final note we highlight that the content of the present book overlaps something with the ones of my
previous books ‘‘Topics on Functional Analysis, Calculus of Variations and Duality” published by Academic
Publications, Sofia, 2011 and, ‘‘Functional Analysis and Applied Optimization in Banach Spaces” published
by Springer in 2014, but only relating a part on basic standard mathematics, so that, among some other new
chapters, the applications presented from Chapters 17 and 30 are almost all new developments.
Acknowledgements
I would like to thank some people of Virginia Tech—USA, where I got my Ph.D. degree in Mathematics in
2009. I am especially grateful to Professor Robert C Rogers for his excellent work as advisor. I would like to
thank the Department of Mathematics for its constant support and this opportunity of studying mathematics
in advanced level. Among many other Professors, I particularly thank Martin Day (Calculus of Variations),
William Floyd and Peter Haskell (Elementary Real Analysis), James Thomson (Real Analysis), Peter
Linnell (Abstract Algebra) and George Hagedorn (Functional Analysis) for the excellent lectured courses.
Finally, special thanks to all my Professors at I.T.A. (Instituto Tecnológico de Aeronáutica, SP-Brasil) my
undergraduate and masters school. Specifically about I.T.A., among many others I would like to express
my gratitude to Professors Leo H Amaral, Tânia Rabelo and my master thesis advisor Antônio Marmo de
Oliveira, also for their valuable work.
January 2020 Fabio Silva Botelho
Florian´opolis, SC, Brazil
Contents
Preface iii
Acknowledgements iv
SECTION I: FUNCTIONAL ANALYSIS
1. Metric Spaces 1
1.1 Introduction 1
1.2 The main definitions 1
1.2.1 The space l∞ 2
1.2.2 Discrete metric 2
1.2.3 The metric space s 3
1.2.4 The space B(A) 3
1.2.5 The space lp 3
1.2.6 Some fundamental definitions 6
1.2.7 Properties of open and closed sets in a metric space 7
1.2.8 Compact sets 10
1.2.9 Separable metric spaces 13
1.2.10 Complete metric spaces 14
1.3 Completion of a metric space 15
1.4 Advanced topics on compactness in metric spaces 19
1.5 The Arzela-Ascoli theorem 24
2. Topological Vector Spaces 26
2.1 Introduction 26
2.2 Vector spaces 26
2.3 Some properties of topological vector spaces 29
2.3.1 Nets and convergence 31
2.4 Compactness in topological vector spaces 34
2.4.1 A note on convexity in topological vector spaces 36
2.5 Normed and metric spaces 38
2.6 Linear mappings 40
2.7 Linearity and continuity 41
2.8 Continuity of operators in Banach spaces 42
vi < Functional Analysis, Calculus of Variations and Numerical Methods for Models in Physics and Engineering
2.9 Some classical results on Banach spaces 43
2.9.1 The Baire Category theorem 43
2.9.2 The Principle of Uniform Boundedness 45
2.9.3 The Open Mapping theorem 46
2.9.4 The Closed Graph theorem 48
2.10 A note on finite dimensional normed spaces 48
3. Hilbert Spaces 52
3.1 Introduction 52
3.2 The main definitions and results 52
3.3 Orthonormal basis 57
3.3.1 The Gram-Schmidt orthonormalization 59
3.4 Projection on a convex set 60
3.5 The theorems of Stampacchia and Lax-Milgram 63
4. The Hahn-Banach Theorems and the Weak Topologies 66
4.1 Introduction 66
4.2 The Hahn-Banach theorems 66
4.3 The weak topologies 72
4.4 The weak-star topology 76
4.5 Weak-star compactness 77
4.6 Separable sets 81
4.7 Uniformly convex spaces 84
5. Topics on Linear Operators 86
5.1 Topologies for bounded operators 86
5.2 Adjoint operators 87
5.3 Compact operators 92
5.4 The square root of a positive operator 93
6. Spectral Analysis, a General Approach in Normed Spaces 100
6.1 Introduction 100
6.2 Sesquilinear functionals 103
6.3 About the spectrum of a linear operator defined on a banach space 113
6.4 The spectral theorem for bounded self-adjoint operators 117
6.4.1 The spectral theorem 122
6.5 The spectral decomposition of unitary transformations 125
6.6 Unbounded operators 127
6.6.1 Introduction 127
6.7 Symmetric and self-adjoint operators 130
6.7.1 The spectral theorem using Cayley transform 132
7. Basic Results on Measure and Integration 136
7.1 Basic concepts 136
7.2 Simple functions 137
7.3 Measures 138
7.4 Integration of simple functions 139
7.5 Signed measures 142
7.6 The Radon-Nikodym theorem 145
Contents <  vii
7.7 Outer measure and measurability 149
7.8 Fubini’s theorem 156
7.8.1 Product measures 156
8. The Lebesgue Measure in Rn 161
8.1 Introduction 161
8.2 Properties of the outer measure 161
8.3 The Lebesgue measure 165
8.4 Outer and inner approximations of Lebesgue measurable sets 166
8.5 Some other properties of measurable sets 169
8.6 Lebesgue measurable functions 171
9. Other Topics in Measure and Integration 178
9.1 Some preliminary results 178
9.2 The Riesz representation theorem 183
9.3 The Lebesgue points 191
9.3.1 The main result on Lebesgue points 193
10. Distributions 195
10.1 Basic definitions and results 195
10.2 Differentiation of distributions 198
10.3 Examples of distributions 199
10.3.1 First example 199
10.3.2 Second example 199
10.3.3 Third example 200
11. The Lebesgue and Sobolev Spaces 201
11.1 Definition and properties of Lp spaces 201
11.1.1 Spaces of continuous functions 206
11.2 The Sobolev spaces 208
11.3 The Sobolev imbedding theorem 210
11.3.1 The statement of Sobolev imbedding theorem 210
11.4 The proof of the Sobolev imbedding theorem 211
11.4.1 Relatively compact sets in Lp(Ω) 215
11.4.2 Some approximation results 218
11.4.3 Extensions 222
11.4.4 The main results 227
11.5 The trace theorem 235
11.6 Compact imbeddings 238
11.7 On the regularity of Laplace equation solutions 245
11.7.1 Regularity, a more general result 249
SECTION II: CALCULUS OF VARIATIONS, CONVEX ANALYSIS
AND RESTRICTED OPTIMIZATION
12. Basic Topics on the Calculus of Variations 252
12.1 Banach spaces 252
12.2 The Gâteaux variation 258
12.3 Minimization of convex functionals 260
viii < Functional Analysis, Calculus of Variations and Numerical Methods for Models in Physics and Engineering
12.4 Sufficient conditions of optimality for the convex case 263
12.5 Natural conditions, problems with free extremals 265
12.6 The du Bois-Reymond lemma 271
12.7 Calculus of variations, the case of scalar functions on Rn 274
12.8 The second Gâteaux variation 276
12.9 First order necessary conditions for a local minimum 277
12.10 Continuous functionals 278
12.11 The Gâteaux variation, the formal proof of its formula 279
13. More Topics on the Calculus of Variations 282
13.1 Introductory remarks 282
13.2 The Gâteaux variation, a more general case 283
13.3 Fréchet differentiability 284
13.4 The Legendre-Hadamard condition 284
13.5 The Weierstrass condition for n = 1 286
13.6 The Weierstrass condition, the general case 288
13.7 The Weierstrass-Erdmann conditions 292
13.8 Natural boundary conditions 294
14. Convex Analysis and Duality Theory 296
14.1 Convex sets and functions 296
14.2 Weak lower semi-continuity 296
14.3 Polar functionals and related topics on convex analysis 298
14.4 The Legendre transform and the Legendre functional 301
14.5 Duality in convex optimization 304
14.6 The min-max theorem 307
14.7 Relaxation for the scalar case 313
14.8 Duality suitable for the vectorial case 320
14.8.1 The Ekeland variational principle 320
14.9 Some examples of duality theory in convex and non-convex analysis 323
15. Constrained Variational Optimization 328
15.1 Basic concepts 328
15.2 Duality 331
15.3 The Lagrange multiplier theorem 332
15.4 Some examples concerning inequality constraints 338
15.5 The Lagrange multiplier theorem for equality and inequality constraints 338
15.6 Second order necessary conditions 342
15.7 On the Banach fixed point theorem 344
15.8 Sensitivity analysis 346
15.8.1 Introduction 346
15.9 The implicit function theorem 346
15.9.1 The main results about Gâteaux differentiability 350
16. On Central Fields in the Calculus of Variations 356
16.1 Introduction 356
16.2 Central fields for the scalar case in the calculus of variations 356
16.3 Central fields and the vectorial case in the calculus of variations 360
Contents <  ix
SECTION III: APPLICATIONS TO MODELS IN
PHYSICS AND ENGINEERING
17. Global Existence Results and Duality for Non-Linear Models of Plates and Shells 367
17.1 Introduction 367
17.2 On the existence of a global minimizer 369
17.3 Existence of a minimizer for the plate model for a more general case 372
17.4 The main duality principle 375
17.5 Existence and duality for a non-linear shell model 381
17.6 Existence of a minimizer 384
17.7 The duality principle for the shell model 386
17.8 Conclusion 390
18. A Primal Dual Formulation and a Multi-Duality Principle for a Non-Linear Model of Plates 391
18.1 Introduction 391
18.2 The first duality principle 393
18.3 The primal dual formulation and related duality principle 399
18.4 A multi-duality principle for non-convex optimization 402
18.5 Conclusion 408
19. On Duality Principles for One and Three-Dimensional Non-Linear Models in Elasticity 409
19.1 Introduction 409
19.2 The main duality principle for the one-dimensional model 410
19.3 The primal variational formulation for the three-dimensional model 414
19.4 The main duality principle for the three-dimensional model 415
19.5 Conclusion 419
20. A Primal Dual Variational Formulation Suitable for a Large Class of Non-Convex 420
Problems in Optimization
20.1 Introduction 420
20.2 The main duality principle 424
20.3 Conclusion 426
21. A Duality Principle and Concerning Computational Method for a Class of Optimal 427
Design Problems in Elasticity
Fabio Silva Botelho and Alexandre Molter
21.1 Introduction 427
21.2 Mathematical formulation of the topology optimization problem 428
21.3 About the computational method 431
21.4 Computational simulations and results 432
21.5 Final remarks and conclusions 435
22. Existence and Duality Principles for the Ginzburg-Landau System in Superconductivity 437
22.1 Introduction 437
22.2 The main result 440
22.3 A second multi-duality principle 446
22.4 Another duality principle for global optimization 452