Applied Mathematical Sciences Volume 78 Editors S.S Antman J.E Marsden L Sirovich Advisors Si nh Vi en Zo ne C om J.K Hale P Holmes J Keener J Keller B.J Matkowsky A Mielke C.S Peskin K.R Sreenivasan SinhVienZone.com https://fb.com/sinhvienzonevn Applied Mathematical Sciences ne C om 33 Grenander: Regular Structures: Lectures in Pattern Theory, Vol III 34 Kevorkian/Cole: Perturbation Methods in Applied Mathematics 35 Carr: Applications of Centre Manifold Theory 36 Bengtsson/Ghil/Källén: Dynamic Meteorology: Data Assimilation Methods 37 Saperstone: Semidynamical Systems in Infinite Dimensional Spaces 38 Lichtenberg/Lieberman: Regular and Chaotic Dynamics, 2nd ed 39 Piccini/Stampacchia/Vidossich: Ordinary Differential Equations in Rn 40 Naylor/Sell: Linear Operator Theory in Engineering and Science 41 Sparrow: The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors 42 Guckenheimer/Holmes: Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields 43 Ockendon/Taylor: Inviscid Fluid Flows 44 Pazy: Semigroups of Linear Operators and Applications to Partial Differential Equations 45 Glashoff/Gustafson: Linear Operations and Approximation: An Introduction to the Theoretical Analysis and Numerical Treatment of Semi-Infinite Programs 46 Wilcox: Scattering Theory for Diffraction Gratings 47 Hale: Dynamics in Infinite Dimensions/Magalhāes/ Oliva, 2nd ed 48 Murray: Asymptotic Analysis 49 Ladyzhenskaya: The Boundary-Value Problems of Mathematical Physics 50 Wilcox: Sound Propagation in Stratified Fluids 51 Golubitsky/Schaeffer: Bifurcation and Groups in Bifurcation Theory, Vol I 52 Chipot: Variational Inequalities and Flow in Porous Media 53 Majda: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables 54 Wasow: Linear Turning Point Theory 55 Yosida: Operational Calculus: A Theory of Hyperfunctions 56 Chang/Howes: Nonlinear Singular Perturbation Phenomena: Theory and Applications 57 Reinhardt: Analysis of Approximation Methods for Differential and Integral Equations 58 Dwoyer/Hussaini/Voigt (eds): Theoretical Approaches to Turbulence 59 Sanders/Verhulst: Averaging Methods in Nonlinear Dynamical Systems 60 Ghil/Childress: Topics in Geophysical Dynamics: Atmospheric Dynamics, Dynamo Theory and Climate Dynamics Si nh Vi en Zo John: Partial Differential Equations, 4th ed Sirovich: Techniques of Asymptotic Analysis Hale: Theory of Functional Differential Equations, 2nd ed Percus: Combinatorial Methods von Mises/Friedrichs: Fluid Dynamics Freiberger/Grenander: A Short Course in Computational Probability and Statistics Pipkin: Lectures on Viscoelasticity Theory Giacaglia: Perturbation Methods in Non-linear Systems Friedrichs: Spectral Theory of Operators in Hilbert Space 10 Stroud: Numerical Quadrature and Solution of Ordinary Differential Equations 11 Wolovich: Linear Multivariable Systems 12 Berkovitz: Optimal Control Theory 13 Bluman/Cole: Similarity Methods for Differential Equations 14 Yoshizawa: Stability Theory and the Existence of Periodic Solution and Almost Periodic Solutions 15 Braun: Differential Equations and Their Applications, 3rd ed 16 Lefschetz: Applications of Algebraic Topology 17 Collatz/Wetterling: Optimization Problems 4th ed 18 Grenander: Pattern Synthesis: Lectures in Pattern Theory, Vol I 19 Marsden/McCracken: Hopf Bifurcation and Its Applications 20 Driver: Ordinary and Delay Differential Equations 21 Courant/Friedrichs: Supersonic Flow and Shock Waves 22 Rouche/Habets/Laloy: Stability Theory by Liapunov’s Direct Method 23 Lamperti: Stochastic Processes: A Survey of the Mathematical Theory 24 Grenander: Pattern Analysis: Lectures in Pattern Theory, Vol II 25 Davies: Integral Transforms and Their Applications, 2nd ed 26 Kushner/Clark: Stochastic Approximation Methods for Constrained and Unconstrained Systems 27 de Boor: A Practical Guide to Splines: Revised Edition 28 Keilson: Markov Chain Models–Rarity and Exponentiality 29 de Veubeke: A Course in Elasticity 30 Sniatycki: Geometric Quantization and Quantum Mechanics 31 Reid: Sturmian Theory for Ordinary Differential Equations 32 Meis/Markowitz: Numerical Solution of Partial Differential Equations (continued after index) SinhVienZone.com https://fb.com/sinhvienzonevn Bernard Dacorogna om Direct Methods in the Calculus of Variations Si nh Vi en Zo ne C Second Edition ABC SinhVienZone.com https://fb.com/sinhvienzonevn Bernard Dacorogna ´ ´ Departement de Mathematiques ´ ´ ´ Ecole Polytechnique Federale de Lausanne CH-1015 Lausanne, Switzerland Editors: L Sirovich Division of Applied Mathematics Brown University Providence, RI 02912 USA chico@camelot.mssm.edu Si nh Vi en Zo ne C om J.E Marsden Control and Dynamical Systems, 107-81 California Institute of Technology Pasadena, CA 91125 USA marsden@cds.caltech.edu S.S Antman Department of Mathematics and Institute for Physical Science and Technology University of Maryland College Park, MD 20742-4015 USA ssa@math.umd.edu ISBN: 978-0-387-35779-9 e-ISBN: 978-0-387-55249-1 Library of Congress Control Number: 2007938908 Mathematics Subject Classification (2000): 74S05 © 2008 Springer Science+Business Media, LLC All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights Printed on acid-free paper springer.com SinhVienZone.com https://fb.com/sinhvienzonevn om Contents C Preface Si nh Vi en Zo ne Introduction 1.1 The direct methods of the calculus of variations 1.2 Convex analysis and the scalar case 1.2.1 Convex analysis 1.2.2 Lower semicontinuity and existence results 1.2.3 The one dimensional case 1.3 Quasiconvex analysis and the vectorial case 1.3.1 Quasiconvex functions 1.3.2 Quasiconvex envelopes 1.3.3 Quasiconvex sets 1.3.4 Lower semicontinuity and existence theorems 1.4 Relaxation and non-convex problems 1.4.1 Relaxation theorems 1.4.2 Some existence theorems for differential inclusions 1.4.3 Some existence results for non-quasiconvex integrands 1.5 Miscellaneous 1.5.1 Hăolder and Sobolev spaces 1.5.2 Singular values 1.5.3 Some underdetermined partial differential equations 1.5.4 Extension of Lipschitz maps I xi 1 9 12 13 15 17 18 19 20 23 23 23 24 25 Convex analysis and the scalar case Convex sets and convex functions 2.1 Introduction 2.2 Convex sets 2.2.1 Basic definitions and properties 2.2.2 Separation theorems SinhVienZone.com https://fb.com/sinhvienzonevn 29 31 31 32 32 34 vi CONTENTS 38 42 44 44 46 52 56 57 61 68 70 C om 2.3 2.2.3 Convex hull and Carath´eodory theorem 2.2.4 Extreme points and Minkowski theorem Convex functions 2.3.1 Basic definitions and properties 2.3.2 Continuity of convex functions 2.3.3 Convex envelope 2.3.4 Lower semicontinuous envelope 2.3.5 Legendre transform and duality 2.3.6 Subgradients and differentiability of convex functions 2.3.7 Gauges and their polars 2.3.8 Choquet function Si nh Vi en Zo ne Lower semicontinuity and existence theorems 3.1 Introduction 3.2 Weak lower semicontinuity 3.2.1 Preliminaries 3.2.2 Some approximation lemmas 3.2.3 Necessary condition: the case without lower order terms 3.2.4 Necessary condition: the general case 3.2.5 Sufficient condition: a particular case 3.2.6 Sufficient condition: the general case 3.3 Weak continuity and invariant integrals 3.3.1 Weak continuity 3.3.2 Invariant integrals 3.4 Existence theorems and Euler-Lagrange equations 3.4.1 Existence theorems 3.4.2 Euler-Lagrange equations 3.4.3 Some regularity results The 4.1 4.2 4.3 4.4 4.5 4.6 4.7 SinhVienZone.com 73 73 74 74 77 82 84 94 96 101 101 103 105 105 108 116 119 119 120 125 125 129 132 132 132 137 143 148 one dimensional case Introduction An existence theorem The Euler-Lagrange equation 4.3.1 The classical and the weak forms 4.3.2 Second form of the Euler-Lagrange equation Some inequalities 4.4.1 Poincar´e-Wirtinger inequality 4.4.2 Wirtinger inequality Hamiltonian formulation Regularity Lavrentiev phenomenon https://fb.com/sinhvienzonevn vii CONTENTS II Quasiconvex analysis and the vectorial case 153 155 155 156 156 158 163 171 174 178 179 191 Polyconvex, quasiconvex and rank one convex envelopes 6.1 Introduction 6.2 The polyconvex envelope 6.2.1 Duality for polyconvex functions 6.2.2 Another representation formula 6.3 The quasiconvex envelope 6.4 The rank one convex envelope 6.5 Some more properties of the envelopes 6.5.1 Envelopes and sums of functions 6.5.2 Envelopes and invariances 6.6 Examples 6.6.1 Duality for SO (n) × SO (n) and O (N ) × O (n) invariant functions 6.6.2 The case of singular values 6.6.3 Functions depending on a quasiaffine function 6.6.4 The area type case 6.6.5 The Kohn-Strang example 6.6.6 The Saint Venant-Kirchhoff energy function 6.6.7 The case of a norm Si nh Vi en Zo ne C om Polyconvex, quasiconvex and rank one convex functions 5.1 Introduction 5.2 Definitions and main properties 5.2.1 Definitions and notations 5.2.2 Main properties 5.2.3 Further properties of polyconvex functions 5.2.4 Further properties of quasiconvex functions 5.2.5 Further properties of rank one convex functions 5.3 Examples 5.3.1 Quasiaffine functions 5.3.2 Quadratic case 5.3.3 Convexity of SO (n) × SO (n) and O (N ) × O (n) invariant functions 5.3.4 Polyconvexity and rank one convexity of SO (n) × SO (n) and O (N ) × O (n) invariant functions 5.3.5 Functions depending on a quasiaffine function 5.3.6 The area type case 5.3.7 The example of Sverak 5.3.8 The example of Alibert-Dacorogna-Marcellini 5.3.9 Quasiconvex functions with subquadratic growth 5.3.10 The case of homogeneous functions of degree one 5.3.11 Some more examples 5.4 Appendix: some basic properties of determinants SinhVienZone.com https://fb.com/sinhvienzonevn 197 202 212 215 219 221 237 239 245 249 265 265 266 266 269 271 277 280 280 282 285 285 291 296 298 300 305 309 viii CONTENTS Zo ne C om Polyconvex, quasiconvex and rank one convex sets 7.1 Introduction 7.2 Polyconvex, quasiconvex and rank one convex sets 7.2.1 Definitions and main properties 7.2.2 Separation theorems for polyconvex sets 7.2.3 Appendix: functions with finitely many gradients 7.3 The different types of convex hulls 7.3.1 The different convex hulls 7.3.2 The different convex finite hulls 7.3.3 Extreme points and Minkowski type theorem for polyconvex, quasiconvex and rank one convex sets 7.3.4 Gauges for polyconvex sets 7.3.5 Choquet functions for polyconvex and rank one convex sets 7.4 Examples 7.4.1 The case of singular values 7.4.2 The case of potential wells 7.4.3 The case of a quasiaffine function 7.4.4 A problem of optimal design Si nh Vi en Lower semi continuity and existence theorems in the vectorial case 8.1 Introduction 8.2 Weak lower semicontinuity 8.2.1 Necessary condition 8.2.2 Lower semicontinuity for quasiconvex functions without lower order terms 8.2.3 Lower semicontinuity for general quasiconvex functions for p = ∞ 8.2.4 Lower semicontinuity for general quasiconvex functions for ≤ p < ∞ 8.2.5 Lower semicontinuity for polyconvex functions 8.3 Weak Continuity 8.3.1 Necessary condition 8.3.2 Sufficient condition 8.4 Existence theorems 8.4.1 Existence theorem for quasiconvex functions 8.4.2 Existence theorem for polyconvex functions 8.5 Appendix: some properties of Jacobians III Relaxation and non-convex problems 313 313 315 315 321 322 323 323 331 335 342 344 347 348 355 362 364 367 367 368 368 369 377 381 391 393 393 394 403 403 404 407 413 Relaxation theorems 415 9.1 Introduction 415 9.2 Relaxation Theorems 416 SinhVienZone.com https://fb.com/sinhvienzonevn CONTENTS ix 9.2.1 9.2.2 The case without lower order terms 416 The general case 424 439 439 440 440 444 449 451 451 459 461 462 463 465 465 467 472 483 483 485 487 488 490 492 493 494 498 ne C om 10 Implicit partial differential equations 10.1 Introduction 10.2 Existence theorems 10.2.1 An abstract theorem 10.2.2 A sufficient condition for the relaxation property 10.2.3 Appendix: Baire one functions 10.3 Examples 10.3.1 The scalar case 10.3.2 The case of singular values 10.3.3 The case of potential wells 10.3.4 The case of a quasiaffine function 10.3.5 A problem of optimal design Si nh Vi en Zo 11 Existence of minima for non-quasiconvex integrands 11.1 Introduction 11.2 Sufficient conditions 11.3 Necessary conditions 11.4 The scalar case 11.4.1 The case of single integrals 11.4.2 The case of multiple integrals 11.5 The vectorial case 11.5.1 The case of singular values 11.5.2 The case of quasiaffine functions 11.5.3 The Saint Venant-Kirchhoff energy 11.5.4 A problem of optimal design 11.5.5 The area type case 11.5.6 The case of potential wells IV Miscellaneous 501 12 Function spaces 12.1 Introduction 12.2 Main notation 12.3 Some properties of Hă older spaces 12.4 Some properties of Sobolev spaces 12.4.1 Definitions and notations 12.4.2 Imbeddings and compact imbeddings 12.4.3 Approximation by smooth and piecewise SinhVienZone.com https://fb.com/sinhvienzonevn affine functions 503 503 503 506 509 510 510 512 x CONTENTS 13 Singular values 515 13.1 Introduction 515 13.2 Definition and basic properties 515 13.3 Signed singular values and von Neumann type inequalities 519 529 529 529 529 531 533 535 535 539 541 543 15 Extension of Lipschitz functions on Banach 15.1 Introduction 15.2 Preliminaries and notation 15.3 Norms induced by an inner product 15.4 Extension from a general subset of E to E 15.5 Extension from a convex subset of E to E 549 549 549 551 558 565 Bibliography 611 Index SinhVienZone.com spaces 569 Si nh Vi Notation en Zo ne C om 14 Some underdetermined partial differential equations 14.1 Introduction 14.2 The equations div u = f and curl u = f 14.2.1 A preliminary lemma 14.2.2 The case div u = f 14.2.3 The case curl u = f 14.3 The equation det ∇u = f 14.3.1 The main theorem and some corollaries 14.3.2 A deformation argument 14.3.3 A proof under a smallness assumption 14.3.4 Two proofs of the main theorem 615 https://fb.com/sinhvienzonevn 228 Polyconvex, quasiconvex and rank one convex functions is polyconvex and the proof will be complete As we already mentioned, there are three proofs of the preceding fact: the original one of Alibert-Dacorogna, the one of Hartwig and that of Iwaniec-Lutoborski, which is in the same spirit as the one of Alibert-Dacorogna but slightly simpler, and we will follow here this last one We will show that, for every ξ, η ∈ R2×2 , 2 f1 (η) ≥ f1 (ξ) + 4(|ξ| − det ξ) ξ; η − ξ − |ξ| [det η − det ξ] This last inequality, combined with Theorem 5.6, gives that f1 is polyconvex om In order to show the inequality, it is sufficient (see Theorem 5.43 and the remark following it) to verify it on diagonal matrices, so we will set C ξ = diag (a, b) and η = diag (x, y) We therefore have to prove that 2 x2 + y ≥ (a − b) a2 + b2 + a2 + b2 − ab [a (x − a) + b (y − b)] − a2 + b2 (xy − ab) ne (x − y) Zo This can be rewritten, setting X = x − a and Y = y − b, as where en αX − 2βXY + γY ≥ γ 2 2 = (x − y + a) + a2 + (a − b) = (a − b) (x − y + a − b) Si nh Vi α β (5.76) = (x − y − b) + b2 + (a − b) The inequality (5.76), and thus the polyconvexity of f1 , follows from the fact that α, γ ≥ and from ∆ = = αγ − β [ a2 + b2 − (x − y) (a − b) ]2 ≥ 2 + (x − y + a − b) [ (x − y) + (a − b) ] This concludes the claim for the polyconvexity We finally show the statement on quasiconvexity It is clearly the most difficult to prove and we will first start with the following result, proved by Alibert-Dacorogna [14], which is a consequence of regularity results for Laplace equation We will use it twice: once when ξ = and p = 4, in the proof of Theorem 5.51, and the second time when ξ = and < p < in Theorem 5.54 The statement with ξ = and p = is just a curiosity SinhVienZone.com https://fb.com/sinhvienzonevn 229 Examples Theorem 5.52 Let < p < ∞ and Ω ⊂ R2 be a bounded open set Then there exists ǫ = ǫ (Ω, p) > such that p p/2 [ |∇ϕ (x)| ± det (∇ϕ (x)) ] dx ≥ ǫ |∇ϕ (x)| dx (5.77) Ω Ω 1,∞ Ω; R for every ϕ ∈ W0 Moreover, when p = 4, the inequality [ |ξ + ∇ϕ (x)| ± det (ξ + ∇ϕ (x)) ]2 dx Ω 2 ≥ (|ξ| ± det ξ) meas Ω + ǫ and every ϕ ∈ W01,∞ Ω; R Ω (5.78) |∇ϕ (x)| dx C holds for every ξ ∈ R 2×2 om Si nh Vi en Zo ne The result (5.77) is clearly non-trivial, except when p = (in this case we can take ǫ = and equality, instead of inequality, holds) Observe also that the inequality (5.77) shows that the functional on the left-hand side of (5.77) is coercive in W01,p Ω, R2 , even though the integrand is not coercive (not even up to a quasiaffine function, which here can be at most quadratic) Proof (Theorem 5.52) We prove (5.77) and (5.78) only for the minus sign, the proof being identical for the plus sign For this purpose we adapt an idea of Sverak [552] Step We first prove the result for ξ = and < p < ∞ We start with an algebraic relation We clearly have that there exists a constant α = α (p) such that for every ξ ∈ R2×2 $ $ %p/2 2 2 %p/2 |ξ| − det ξ ξ11 − ξ22 + ξ21 + ξ12 = $& &p & &p % ≥ α &ξ11 − ξ22 & + &ξ21 + ξ12 & We now turn to theclaim and note that it is sufficient to prove the claim for ϕ = ϕ1 , ϕ2 ∈ C0∞ Ω, R2 , the general result being obtained by density We also extend the function outside Ω by setting ϕ ≡ there Then denoting ∂ϕj /∂xi by ∂i ϕj , i, j ∈ {1, 2}, we have from the above algebraic relation [ |∇ϕ (x)| − det (∇ϕ (x)) ]p/2 dx Ω & &p & &p ≥ α [ &∂1 ϕ1 (x) − ∂2 ϕ2 (x)& + &∂2 ϕ1 (x) + ∂1 ϕ2 (x)& ]dx Ω The classical regularity results for Cauchy-Riemann equations (see, for example, Proposition on page 60 in Stein [543]) leads to the existence of a constant β > such that &p &p & & ∇ϕpLp ≤ β [ &∂1 ϕ1 (x) − ∂2 ϕ2 (x)& + &∂2 ϕ1 (x) + ∂1 ϕ2 (x)& ]dx Ω SinhVienZone.com https://fb.com/sinhvienzonevn 230 Polyconvex, quasiconvex and rank one convex functions Choosing ǫ ≤ α/β, we have (5.77) Step We now prove the general case, where ξ is not necessarily but p = We start with the following algebraic observation η ]2 ≤ | ξ − ξ |2 |η|2 = 2[ |ξ|2 − det ξ ] |η|2 [ ξ − ξ; (5.79) We next compute = om = [ |ξ + ∇ϕ|2 − det (ξ + ∇ϕ) ]2 ∇ϕ + |∇ϕ|2 − det (∇ϕ) ]2 [ |ξ| − det ξ + ξ − ξ; ∇ϕ ]2 + [ |∇ϕ| − det (∇ϕ) ]2 [ |ξ| − det ξ ]2 + 4[ ξ − ξ; ∇ϕ − det (∇ϕ) ] +4[ |ξ| − det ξ ][ ξ − ξ; 2 ∇ϕ +2[ |ξ| − det ξ ] |∇ϕ| + 4[ |∇ϕ| − det (∇ϕ) ] ξ − ξ; 2 C Using (5.79), we obtain en Noticing that ne ∇ϕ − det (∇ϕ) ] +4[ |ξ| − det ξ ][ ξ − ξ; ∇ϕ +4[ |∇ϕ| − det (∇ϕ) ] ξ − ξ; Zo ≥ [ |ξ + ∇ϕ| − det (ξ + ∇ϕ) ]2 ∇ϕ ]2 + [ |∇ϕ|2 − det (∇ϕ) ]2 [ |ξ| − det ξ ]2 + 5[ ξ − ξ; ∇ϕ ]2 ≤ 5[ ξ − ξ; Si nh Vi ∇ϕ + [ |∇ϕ|2 − det (∇ϕ) ]2 +4[ |∇ϕ|2 − det (∇ϕ) ] ξ − ξ; we deduce that [ |ξ + ∇ϕ|2 − det (ξ + ∇ϕ) ]2 ≥ [ |ξ|2 − det ξ ]2 + 51 [ |∇ϕ|2 − det (∇ϕ) ]2 ∇ϕ − det (∇ϕ) ] +4[ |ξ| − det ξ ][ ξ − ξ; We then integrate the above inequality, bearing in mind that ϕ = on ∂Ω, and we find 2 [ |ξ + ∇ϕ| − det (ξ + ∇ϕ) ]2 dx ≥ [ |ξ| − det ξ ]2 meas Ω Ω [ |∇ϕ|2 − det (∇ϕ) ]2 dx + Ω Using Step 1, with p = 4, we find that α 2 [ |ξ + ∇ϕ| − det (ξ + ∇ϕ) ]2 dx ≥ [ |ξ| − det ξ ]2 meas Ω + |∇ϕ| dx 5β Ω Ω Choosing ǫ = α/5β, we have indeed established (5.78) and thus the theorem is proved SinhVienZone.com https://fb.com/sinhvienzonevn ... https://fb .com/ sinhvienzonevn Chapter C The direct methods of the calculus of variations ne 1.1 om Introduction Zo The main problem that we will be investigating throughout the present monograph is the. .. Numerical Solution of Partial Differential Equations (continued after index) SinhVienZone. com https://fb .com/ sinhvienzonevn Bernard Dacorogna om Direct Methods in the Calculus of Variations Si nh... with, in particular from the point of view of regularity It is discussed in the general SinhVienZone. com https://fb .com/ sinhvienzonevn Introduction framework of the scalar case in Chapter but also