Author: Fabio Botelho Title: TOPICS ON FUNCTIONAL ANALYSIS, CALCULUS OF VARIATIONS AND DUALITY ISBN: 978-954-2940-08-1 Publisher: Academic Publications, Ltd. Date: September, 2011 Pages: 397
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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 Summary of Main Results Part 1. Basic Functional Analysis ix 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 1 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 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 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 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 115 125 5.1 Basic Definitions and Results 125 5.2 Differentiation of Distributions 129 131 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 166 171 173 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 191 195 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 217 223 9.1 Basic Concepts 223 9.2 Duality 227 9.3 The Lagrange Multiplier Theorem 229 233 235 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 264 267 11.1 Introduction and Primal Formulation 267 11.2 The First Duality Principle 269 11.3 The Second Duality Principle 274 11.4 Conclusion 276 277 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 285 287 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 315 317 14.1 Introduction 317 14.2 The Primal Formulation 318 14.3 The Duality Principle 318 14.4 Conclusion 320 323 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 339 341 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 372 373 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 385 387 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 394 395

## Bibliography (BibTex Format)

\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).