ж ш Л Fi I /V T I Jf"\ Ik 1 Л 1 ANALYSIS IN SOBOLEV AND BV SPACES APPLICATIONS TO PDES AND OPTIMIZATION Hedy Attouch Universite Montpellier I! Montpellier, France Giuseppe Buttazzo Universitä di Pisa Pisa, Italy Gerard Michatlle Universite Montpellier II Montpellier, France giajtl Society for Industrial and Applied Mathematics Philadelphia MPS Mathematical Programming Society Philadelphia
Preface xi 1 Introduction 1 Part I: Basic Variational Principles 5 2 Weak solution methods in variational analysis 7 2.1 The Dirichlet problem: Historical presentation 7 2.2 Test functions and distribution theory 15 2.2.1 Definition of distributions 15 2.2.2 Locally integrable functions as distributions: Regularization by convolution and mollifiers 18 2.2.3 Radon measures 24 2.2.4 Derivation of distributions, introduction to Sobolev spaces 24 2.2.5 Convergence of sequences of distributions 27 2.3 Weak solutions 31 2.3.1 Weak formulation of the model examples 31 2.3.2 Positive quadratic forms and convex minimization... 36 2.4 Weak topologies and weak convergences 41 2.4.1 Topologies induced by functions in general topological spaces 41 2.4.2 The weak topology a(v, V*) 44 2.4.3 Weak convergence and geometry of uniformly convex spaces 52 2.4.4 Weak compactness theorems in reflexive Banach spaces. 54 2.4.5 The Dunford-Pettis weak compactness theorem in L 1 (Q) 57 2.4.6 The weak* topology a(v*,v) 59 3 Abstract variational principles 67 3.1 The Lax-MiIgram theorem and the Galerkin method 67 3.1.1 The Lax-Milgram theorem 67 3.1.2 The Galerkin method 73 3.2 Minimization problems: The topological approach 76 v
3.2.1 Extended real-valued functions 77 3.2.2 The interplay between functions and sets: The role of the epigraph 78 3.2.3 Lower semicontinuous functions 80 3.2.4 The lower closure of a function and the relaxation problem 82 3.2.5 Inf-compactness functions, coercivity 86 3.2.6 Topological minimization theorems 87 3.2.7 Weak topologies and minimization of weakly lower semicontinuous functions 91 3.3 Convex minimization theorems 91 3.3.1 Extended real-valued convex functions and weak lower semicontinuity 91 3.3.2 Convex minimization in reflexive Banach spaces 93 3.4 Ekeland's e-variational principle 98 3.4.1 Ekeland's e-variational principle and the direct method.. 98 3.4.2 A dynamical approach and proof of Ekeland's e-variational principle 100 Complements on measure theory 109 4.1 Hausdorff measures and Hausdorff dimension 109 4.1.1 Outer Hausdorff measures and Hausdorff measures.... 109 4.1.2 Hausdorff measures: Scaling properties and Lipschitz transformations 117 4.1.3 Hausdorff dimension 120 4.2 Set functions and duality approach to Borel measures 124 4.2.1 Borel measures as set functions 124 4.2.2 Duality approach 129 4.3 Introduction to Young measures 138 4.3.1 Definition 138 4.3.2 Slicing Young measures 139 4.3.3 Prokhorov's compactness theorem 142 4.3.4 Young measures associated with functions and generated by functions 142 4.3.5 Semicontinuity and continuity properties 143 4.3.6 Young measures capture oscillations 146 4.3.7 Young measures do not capture concentrations 149 Sobolev spaces 151 5.1 Sobolev spaces: Definition, density results 152 5.2 The topological dual of tfj ( 2). The space Я _1 (^) 165 5.3 Poincare inequality and Rellich-Kondrakov theorem in W J 0 '(Q)... 168 5.4 Extension operators from W l - p (Q) into W l,p (R N ). Poincare inequalities and the Rellich-Kondrakov theorem in W lp ( 2) 174 5.5 The Fourier approach to Sobolev spaces. The space H S (Q), ser.. 180 5.6 Trace theory for W''''( 2) spaces 186 5.7 Sobolev embedding theorems 192
VII 5.7.1 Case 1 < p < N 194 5.7.2 Case p > N 200 5.7.3 Case p = N 202 5.8 Capacity theory and elements of potential theory 206 5.8.1 Contractions operate on W l - p (Q.) 206 5.8.2 Capacity 212 6 Variational problems: Some classical examples 217 6.1 The Dirichlet problem 218 6.1.1 The homogenous Dirichlet problem 218 6.1.2 The nonhomogenous Dirichlet problem 222 6.2 The Neumann problem 225 6.2.1 The coercive homogenous Neumann problem 225 6.2.2 The coercive nonhomogenous Neumann problem 229 6.2.3 The semicoercive homogenous Neumann problem... 230 6.2.4 The semicoercive nonhomogenous Neumann problem.. 234 6.3 Mixed Dirichlet-Neumann problems 236 6.3.1 The Dirichlet-Neumann problem 236 6.3.2 Mixed Dirichlet-Neumann boundary conditions 238 6.4 Heterogenous media: Transmission conditions 240 6.5 Linear elliptic operators 245 6.6 The nonlinear Laplacian Д р 249 6.7 The Stokes system 253 7 The finite element method 257 7.1 The Galerkin method: Further results 257 7.2 Description of finite element methods 260 7.3 An example 262 7.4 Convergence of the finite element method 263 7.5 Complements 276 7.5.1 Flat triangles 276 7.5.2 H 2 ( i) regularity of the solution of the Dirichlet problem on a convex polygon 277 7.5.3 Finite element methods of type Po 277 8 Spectral Analysis of the Laplacian 279 8.1 Introduction 279 8.2 The Laplace-Dirichlet operator: Functional setting 281 8.3 Existence of a Hilbertian basis of eigenvectors of the Laplace-Dirichlet operator 286 8.4 The Couranl-Fisher min-max and max-min formulas 289 8.5 Multiplicity and asymptotic properties of the eigenvalues of the Laplace-Dirichlet operator 297 8.6 A general abstract theory for spectral analysis of elliptic boundary value problems 303
VIII Contents 9 Convex duality and optimization 307 9.1 Dual representation of convex sets 307 9.2 Passing from sets to functions: Elements of epigraphical calculus...312 9.3 Legendre-Fenchel transform 318 9.4 Legendre-Fenchel calculus 328 9.5 Subdifferential calculus for convex functions 331 9.6 Mathematical programming: Multipliers and duality 340 9.6.1 Karush-Kuhn-Tucker optimality conditions 341 9.6.2 The marginal approach to multipliers 345 9.6.3 The Lagrangian approach to duality 353 9.6.4 Duality for linear programming 356 9.7 A general approach to duality in convex optimization 358 9.8 Duality in the calculus of variations: First examples 365 Part II: Advanced Variational Analysis 369 10 Spaces В V and SB V 371 10.1 The space В V(Q): Definition, convergences, and approximation...371 10.2 The trace operator, the Green's formula, and its consequences 378 10.3 The coarea formula and the structure of В V functions 387 10.3.1 Notion of density and regular points 388 10.3.2 Sets of finite perimeter, structure of simple BV functions 395 10.3.3 Structure of В V functions 402 10.4 Structure of the gradient of В V functions 406 10.5 The space SB V(Q) 408 10.5.1 Definition 409 10.5.2 Properties 410 11 Relaxation in Sobolev, В V, and Young measures spaces 417 11.1 Relaxation in abstract metrizable spaces 417 11.2 Relaxation of integral functionals with domain W lp ( 2, R m ), p > 1.421 11.3 Relaxation of integral functionals with domain W IJ (Q, R m ) 437 11.4 Relaxation in the space of Young measures in nonlinear elasticity...449 11.4.1 Young measures generated by gradients 450 11.4.2 Relaxation of classical integral functionals in y(q; E).. 457 12 Г-convergence and applications 463 12.1 Г-convergence in abstract metrizable spaces 463 12.2 Application to the nonlinear membrane model 467 12.3 Application to homogenization of composite media 472 12.3.1 The quadratic case in one dimension 472 12.3.2 Periodic homogenization in the general case 475 12.4 Application to image segmentation and phase transitions 482 12.4.1 The Mumford-Shah model 482
ix 12.4.2 Variational approximation of a more elementary problem: A phase transitions model 483 12.4.3 Variational approximation of the Mumford-Shah functional energy 487 13 Integral functionals of the calculus of variations 497 13.1 Lower semicontinuity in the scalar case 497 13.2 Lower semicontinuity in the vectorial case 503 13.3 Lower semicontinuity for funclionals defined on the space of measures 510 13.4 Functionals with linear growth: Lower semicontinuity in BV and SBV.513 13.4.1 Lower semicontinuity and relaxation in В V 513 13.4.2 Compactness and lower semicontinuity in SBV 515 14 Application in mechanics and computer vision 521 14.1 Problems in pseudoplasticity 521 14.1.1 Introduction 521 14.1.2 The Hencky model 523 14.1.3 The spaces ß >(ß), M(div), and /(Q) 524 14.1.4 Relaxation of the Hencky model 528 14.2 Some variational models in fracture mechanics 529 14.2.1 A few considerations in fracture mechanics 529 14.2.2 A first model in one dimension 532 14.2.3 A second model in one dimension 543 14.3 The Mumford-Shah model 549 15 Variational problems with a lack of coercivity 553 15.1 Convex minimization problems and recession functions 553 15.2 Nonconvex minimization problems and topological recession 572 15.3 Some examples 581 15.4 Limit analysis problems 588 16 An introduction to shape optimization problems 601 16.1 The isoperimetric problem 602 16.2 The Newton problem 604 16.3 Optimal Dirichlet free boundary problems 605 16.4 Optimal distribution of two conductors 609 Bibliography 615 Index 631