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Author
Professor Fabio Botelho, Ph.D.
Department of Mathematics
Federal University of Pelotas
Pelotas, RS, BRAZIL
email: fabio.silva.botelho@gmail.com
Abstract
This work is a kind of revised and enlarged edition of the title Variational Convex Analysis, published by Lambert Academic Publishing. First we present the basic tools of analysis necessary to develop the core theory and applications. New results concerning duality principles for systems originally modeled by non-linear differential equations are shown in Chapters 10 to 18. A key aspect of this work is that although the original problems are non-linear with corresponding non-convex variational formulations, the dual formulations obtained are almost always concave and amenable to numerical computations. When the primal problem has no solution in the classical sense, the solution of dual problem is a weak limit of minimizing sequences, and the evaluation of such average behavior is important in many practical applications. Among the results we highlight the dual formulations for micro-magnetism, phase transition models, composites in elasticity and conductivity and others. To summarize, in the present work we introduce convex analysis as an interesting alternative approach for the understanding and computation of some important problems in the modern calculus of variations.
Finally, in Chapter 13 we develop a new numerical procedure, called the generalized method of lines. Through such a method the domain is discretized in lines (or more generally in curves) and the solution on them is written as functions of boundary conditions and boundary shape. In Chapter 17 we apply the method to an approximation of the incompressible Navier-Stokes system.
AMS Classification: 46N10, 49N15
Key words: Banach spaces, convex analysis, duality, calculus of variations, non-convex systems, generalized method of lines
Contents
| Introduction | ix | |
|---|---|---|
| Summary of Main Results | ix | |
| Duality Applied to a Plate Model | ix | |
| Duality Applied to Finite Elasticity | x | |
| Duality Applied to a Shell Model | x | |
| Duality Applied to Ginzburg-Landau Type Equations | x | |
| Duality Applied to Conductivity in Composites | xi | |
| Duality Applied to the Optimal Design in Elasticity | xi | |
| Duality Applied to Micro-Magnetism | xi | |
| Duality Applied to Fluid Mechanics | xi | |
| Duality Applied to a Beam Model | xi | |
| Acknowledgments | xii | |
| Part 1. Basic Functional Analysis | 1 | |
| Chapter 1. Topological Vector Spaces | 3 | |
| 1.1 | Introduction | 3 |
| 1.2 | Vector Spaces | 3 |
| 1.3 | Some Properties of Topological Vector Spaces | 8 |
| 1.4 | Compactness in Topological Vector Spaces | 11 |
| 1.5 | Normed and Metric Spaces | 12 |
| 1.6 | Compactness in Metric Spaces | 13 |
| 1.7 | The Arzela-Ascoli Theorem | 20 |
| 1.8 | Linear Mappings | 23 |
| 1.9 | Linearity and Continuity | 24 |
| 1.10 | Continuity of Operators on Banach Spaces | 25 |
| 1.11 | Some Classical Results on Banach Spaces | 26 |
| 1.12 | Hilbert Spaces | 32 |
| Chapter 2. The Hahn-Banach Theorems and Weak Topologies | 39 | |
| 2.1 | Introduction | 39 |
| 2.2 | The Hahn-Banach Theorem | 39 |
| 2.3 | Weak Topologies | 45 |
| 2.4 | The Weak-Star Topology | 47 |
| 2.5 | Weak-Star Compactness | 48 |
| 2.6 | Separable Sets | 52 |
| 2.7 | Uniformly Convex Spaces | 53 |
| Chapter 3. Topics on Linear Operators | 55 | |
| 3.1 | Topologies for Bounded Operators | 55 |
| 3.2 | Adjoint Operators | 56 |
| 3.3 | Compact Operators | 59 |
| 3.4 | The Square Root of a Positive Operator | 61 |
| 3.5 | About the Spectrum of a Linear Operator | 66 |
| 3.6 | The Spectral Theorem for Bounded Self-Adjoint Operators | 70 |
| 3.6.1 | The Spectral Theorem | 75 |
| 3.7 | The Spectral Decomposition of Unitary Transformations | 78 |
| 3.8 | Unbounded Operators | 81 |
| 3.8.1 | Introduction | 81 |
| 3.9 | Symmetric and Self-Adjoint Operators | 84 |
| 3.9.1 | The Spectral Theorem Using Cayley Transform | 86 |
| Chapter 4. Measure and Integration | 91 | |
| 4.1 | Basic Concepts | 91 |
| 4.2 | Simple Functions | 93 |
| 4.3 | Measures | 94 |
| 4.4 | Integration of Simple Functions | 95 |
| 4.5 | The Fubini Theorem | 99 |
| 4.5.1 | Product Measures | 99 |
| 4.6 | The Lebesgue Measure in Rn | 105 |
| 4.6.1 | Properties of the Outer Measure | 106 |
| 4.6.2 | The Lebesgue Measure | 109 |
| 4.6.3 | Properties of Measurable Sets | 110 |
| 4.7 | Lebesgue Measurable Functions | 115 |
| Chapter 5. Distributions | 125 | |
| 5.1 | Basic Definitions and Results | 125 |
| 5.2 | Differentiation of Distributions | 129 |
| Chapter 6. The Lebesgue and Sobolev Spaces | 131 | |
| 6.1 | Definition and Properties of $L^p$ Spaces | 131 |
| 6.1.1 | Spaces of Continuous Functions | 137 |
| 6.2 | The Sobolev Spaces | 139 |
| 6.3 | The Sobolev Imbedding Theorem | 144 |
| 6.3.1 | The Statement of Sobolev Imbedding Theorem | 144 |
| 6.4 | The Proof of the Sobolev Imbedding Theorem | 145 |
| 6.4.1 | Relatively Compact Sets in Lp(Ω) | 149 |
| 6.4.2 | Some Approximation Results | 153 |
| 6.4.3 | Extensions | 156 |
| 6.4.4 | The Main Results | 158 |
| 6.5 | Compact Imbeddings | 166 |
| Part 2. Variational Convex Analysis | 171 | |
| Chapter 7. Basic Concepts on the Calculus of Variations | 173 | |
| 7.1 | Introduction to the Calculus of Variations | 173 |
| 7.2 | Evaluating the Gateaux variations | 175 |
| 7.3 | The Gateaux Variation in W1,2(Ω) | 177 |
| 7.4 | Elementary Convexity | 179 |
| 7.5 | The Legendre-Hadamard Condition | 182 |
| 7.6 | The Weierstrass Necessary Condition | 184 |
| 7.6.1 | The Weierstrass Condition for $n=1$ | 184 |
| 7.7 | The du Bois-Reymond Lemma | 187 |
| 7.8 | The Weierstrass-Erdmann Conditions | 189 |
| 7.9 | Natural Boundary Conditions | 191 |
| Chapter 8. Basic Concepts on Convex Analysis | 195 | |
| 8.1 | Convex Sets and Convex Functions | 195 |
| 8.2 | Duality in Convex Optimization | 204 |
| 8.3 | Relaxation for the Scalar Case | 208 |
| 8.4 | Duality Suitable for the Vectorial Case | 217 |
| Chapter 9. Constrained Variational Optimization | 223 | |
| 9.1 | Basic Concepts | 223 |
| 9.2 | Duality | 227 |
| 9.3 | The Lagrange Multiplier Theorem | 229 |
| Part 3. Applications | 233 | |
| Chapter 10. Duality Applied to a Plate Model | 235 | |
| 10.1 | Introduction | 235 |
| 10.2 | The Primal Variational Formulation | 240 |
| 10.3 | The Legendre Transform | 242 |
| 10.4 | The Classical Dual Formulation | 244 |
| 10.4.1 | The Polar Functional Related to F | 248 |
| 10.4.2 | The First Duality Principle | 249 |
| 10.5 | The Second Duality Principle | 250 |
| 10.6 | The Third Duality Principle | 255 |
| 10.7 | A Convex Dual Formulation | 258 |
| 10.8 | A Final Result, other Sufficient Conditions of Optimality | 261 |
| 10.9 | Final Remarks | 264 |
| Chapter 11. Duality Applied to Elasticity | 267 | |
| 11.1 | Introduction and Primal Formulation | 267 |
| 11.2 | The First Duality Principle | 269 |
| 11.3 | The Second Duality Principle | 274 |
| 11.4 | Conclusion | 276 |
| Chapter 12. Duality Applied to a Membrane Shell Model | 277 | |
| 12.1 | Introduction and Primal Formulation | 277 |
| 12.2 | The Legendre Transform | 279 |
| 12.3 | The Polar Functional Related to F | 280 |
| 12.4 | The Final Format of First Duality Principle | 280 |
| 12.5 | The Second Duality Principle | 281 |
| 12.6 | Conclusion | 285 |
| Chapter 13. Duality Applied to Ginzburg-Landau Type Equations | 287 | |
| 13.1 | Introduction | 287 |
| 13.1.1 | Existence of Solution for the Ginzburg-Landau Equation | 288 |
| 13.2 | A Concave Dual Variational Formulation | 289 |
| 13.3 | Applications to Phase Transition in Polymers | 293 |
| 13.3.1 | Another Two Phase Model in Polymers | 295 |
| 13.4 | A Numerical Example | 298 |
| 13.5 | A New Path for Relaxation | 301 |
| 13.6 | A New Numerical Procedure, the Generalized Method of Lines | 306 |
| 13.7 | A Simple Numerical Example | 314 |
| 13.8 | Conclusion | 315 |
| Chapter 14 Duality Applied to Conductivity in Composites | 317 | |
| 14.1 | Introduction | 317 |
| 14.2 | The Primal Formulation | 318 |
| 14.3 | The Duality Principle | 318 |
| 14.4 | Conclusion | 320 |
| Chapter 15. Duality Applied to the Optimal Design in Elasticity | 323 | |
| 15.1 | Introduction | 323 |
| 15.2 | The Main Duality Principles | 324 |
| 15.3 | The First Applied Duality Principle | 329 |
| 15.4 | A Concave Dual Formulation | 332 |
| 15.5 | Duality for a Two-Phase Problem in Elasticity | 335 |
| 15.6 | A Numerical Example | 338 |
| 15.7 | Conclusion | 339 |
| Chapter 16. Duality Applied to Micro-Magnetism | 341 | |
| 16.1 | Introduction | 341 |
| 16.2 | The Primal formulations and the Duality Principles | 342 |
| 16.2.1 | Summary of Results for the Hard Uniaxial Case | 342 |
| 16.2.2 | The Results for the Full Semi-linear Case | 343 |
| 16.3 | A Preliminary Result | 344 |
| 16.4 | The Duality Principle for the Hard Case | 345 |
| 16.5 | An Alternative Dual Formulation for the Hard Case | 349 |
| 16.5.1 | The Primal Formulation | 349 |
| 16.5.2 | The Duality Principle | 350 |
| 16.6 | The Full Semi-linear Case | 354 |
| 16.7 | The Cubic Case in Micro-magnetism | 360 |
| 16.7.1 | The Primal Formulation | 361 |
| 16.7.2 | The Duality Principles | 362 |
| 16.8 | Final Results, Other Duality Principles | 364 |
| 16.9 | Conclusion | 372 |
| Chapter 17. Duality Applied to Fluid Mechanics | 373 | |
| 17.1 | Introduction and Primal Formulation | 373 |
| 17.2 | The Legendre Transform | 375 |
| 17.3 | Linear Systems which the Solutions Solve the Navier-Stokes One | 376 |
| 17.4 | The Method of Lines for the Navier-Stokes System | 379 |
| 17.5 | Conclusion | 385 |
| Chapter 18. Duality Applied to a Beam Model | 387 | |
| 18.1 | Introduction and Statement of the Primal Formulation | 387 |
| 18.2 | Existence and Regularity Results for Problem P | 388 |
| 18.3 | A Convex Dual Formulation for the Beam Model | 390 |
| 18.4 | A Final Result, Another Duality Principle | 392 |
| 18.5 | Conclusion | 394 |
| Bibliography | 395 | |
Bibliography (BibTex Format)
\bibitem{1} R.A. Adams, {\it Sobolev Spaces,} Academic Press, New York (1975).
\bibitem{1.1} R.A. Adams and J.F. Fournier, {\it Sobolev Spaces, second edition} , Elsevier (2003).
\bibitem{100} J.F. Annet, {\it Superconductivity, Superfluids and Condensates}, { Oxford Master Series in Condensed Matter Physics, Oxford University Press, Reprint (2010)}
\bibitem{400} G. Bachman and L. Narici, {\it Functional Analysis}, {Dover Publications,} Reprint (2000).
\bibitem{13} J.M. Ball and R.D. James, {\it Fine mixtures as minimizers of energy }, Archive for Rational Mechanics and Analysis, 100, 15-52 (1987).
\bibitem{15} F. Botelho, {\it Dual Variational Formulations for a Non-linear Model of Plates}, Journal of Convex Analysis, 17 , No. 1, 131-158 (2010).
\bibitem{870} F. Botelho, {\it Duality for Ginzburg-Landau Type Equations and the Generalized Method of Lines}, submitted.
\bibitem{36} F. Botelho, {\it Dual Variational Formulations for Models in Micro-magnetism - The Uniaxial and Cubic Cases}, submitted.
\bibitem{27} H.Brezis, {\it Analyse Fonctionnelle}, Masson (1987).
\bibitem{4} I.V. Chenchiah and K. Bhattacharya, {\it The Relaxation of Two-well Energies with Possibly Unequal Moduli}, Arch. Rational Mech. Anal. 187 (2008), 409-479.
\bibitem{17} M. Chipot, {\it Approximation and oscillations}, Microstructure and Phase Transition, the IMA volumes in mathematics and applications, {\bf 54}, 27-38 (1993).
\bibitem{14} R. Choksi, M.A. Peletier, and J.F. Williams, {\it On the Phase Diagram for Microphase Separation of Diblock Copolymers: an Approach via a Nonlocal Cahn-Hilliard Functional}, to appear in SIAM J. Appl. Math. (2009).
\bibitem{2} P.Ciarlet, {\it Mathematical Elasticity}, {Vol. I -- Three Dimensional Elasticity}, North Holland Elsivier (1988).
\bibitem{3} P.Ciarlet, {\it Mathematical Elasticity}, {Vol. II -- Theory of Plates}, North Holland Elsevier (1997).
\bibitem{4} P.Ciarlet, {\it Mathematical Elasticity}, {Vol. III -- Theory of Shells}, North Holland Elsevier (2000).
\bibitem{5} B. Dacorogna, {\it Direct methods in the Calculus of Variations}, Springer-Verlag (1989).
\bibitem{6} I.Ekeland and R.Temam, {\it Convex Analysis and Variational Problems.} North Holland (1976).
\bibitem{4000} L. C. Evans, {\it Partial Differential Equations}, Graduate Studies in Mathematics, {\bf 19}, AMS (1998).
\bibitem{40} U.Fidalgo and P.Pedregal, {\it A General Lower Bound for the Relaxation of an Optimal Design Problem in Conductivity with a Quadratic Cost Functional}, Pre-print (2007).
\bibitem{18} N.B. Firoozye and R.V. Khon, {\it Geometric Parameters and the Relaxation for Multiwell Energies}, Microstructure and Phase Transition, the IMA volumes in mathematics and applications, {\bf 54}, 85-110 (1993).
\bibitem{8} D.Y.Gao and G.Strang, {\it Geometric Nonlinearity: Potential Energy, Complementary Energy and the Gap Function}, Quartely Journal of Applied Mathematics, {\bf 47}, 487-504 (1989a).
\bibitem{7} D.Y.Gao, {\it On the extreme variational principles for non-linear elastic plates.} Quarterly of Applied Mathematics, {\bf XLVIII}, No. 2 (June 1990), 361-370.
\bibitem{41} D.Y.Gao, {\it Finite Deformation Beam Models and Triality Theory in Dynamical Post-Buckling Analysis.} International Journal of Non-linear Mechanics {\bf 35} (2000), 103-131.
\bibitem{9} D.Y.Gao, {\it Duality Principles in Nonconvex Systems, Theory, Methods and Applications,} {Kluwer, Dordrecht},(2000).
\bibitem{21 }R.D.James and D. Kinderlehrer {\it Frustration in ferromagnetic materials}, Continuum Mech. Thermodyn. 2 (1990) 215-239.
\bibitem{24} H. Kronmuller and M. Fahnle, {\it Micromagnetism and the Microstructure of Ferromagnetic Solids}, Cambridge University Press (2003).
\bibitem{28} D.G. Luenberger, {\it Optimization by Vector Space Methods}, John Wiley and Sons, Inc. (1969).
\bibitem{22} G.W. Milton, {\it Theory of composites,} Cambridge Monographs on Applied and Computational Mathematics. Cambridge University Press, Cambridge (2002).
\bibitem{19} P.Pedregal, {\it Parametrized measures and variational principles}, Progress in Nonlinear Differential Equations and Their Applications, {\bf 30}, Birkhauser (1997).
\bibitem{23} P.Pedregal and B.Yan, {\it On Two Dimensional Ferromagnetism}, pre-print (2007).
\bibitem{87} M. Reed and B. Simon, {\it Methods of Modern Mathematical Physics, Volume I, Functional Analysis}, Reprint Elsevier (Singapore, 2003).
\bibitem{29} R. T. Rockafellar, {\it Convex Analysis}, Princeton Univ. Press, (1970).
\bibitem{80} R.C. Rogers, {\it A nonlocal model of the exchange energy in ferromagnet materials}, Journal of Integral Equations and Applications, {\bf 3}, No. 1 (Winter 1991).
\bibitem{81} R.C. Rogers, {\it Nonlocal variational problems in nonlinear electromagneto-elastostatics}, SIAM J. Math, Anal. {\bf 19}, No. 6 (November 1988).
\bibitem{51} W. Rudin, {\it Functional Analysis}, second edition, McGraw-Hill (1991).
\bibitem{5000} W. Rudin, {\it Real and Complex Analysis}, third edition, McGraw-Hill (1987).
\bibitem{20} D.R.S.Talbot and J.R.Willis, {\it Bounds for the effective contitutive relation of a nonlinear composite,} Proc. R. Soc. Lond. (2004), {\bf 460}, 2705-2723.
\bibitem{6000} E. M. Stein and R. Shakarchi, {\it Real Analysis}, Princeton Lectures in Analysis III, Princeton University Press (2005).
\bibitem{890}R. Temam, {\em Navier-Stokes Equations,} { AMS Chelsea, reprint (2001).}
\bibitem{10} J.J. Telega, {\it On the complementary energy principle in non-linear elasticity. Part I: Von Karman plates and three dimensional solids}, C.R. Acad. Sci. Paris, Serie II, 308, 1193-1198; Part II: Linear elastic solid and non-convex boundary condition. Minimax approach, ibid, pp. 1313-1317 (1989)
\bibitem{11} A.Galka and J.J.Telega {\it Duality and the complementary energy principle for a class of geometrically non-linear structures. Part I. Five parameter shell model; Part II. Anomalous dual variational priciples for compressed elastic beams}, Arch. Mech. 47 (1995) 677-698, 699-724.
\bibitem{12} J.F. Toland, {\it A duality principle for non-convex optimisation and the calculus of variations}, Arch. Rath. Mech. Anal., {\bf 71}, No. 1 (1979), 41-61.
\bibitem{120} J. L. Troutman, {\it Variational Calculus and Optimal Control}, second edition, Springer (1996).