ANALYSIS IN SOBOLEV AND BV SPACES

Transcription

1 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

2 Preface xi 1 Introduction 1 Part I: Basic Variational Principles 5 2 Weak solution methods in variational analysis The Dirichlet problem: Historical presentation Test functions and distribution theory Definition of distributions Locally integrable functions as distributions: Regularization by convolution and mollifiers Radon measures Derivation of distributions, introduction to Sobolev spaces Convergence of sequences of distributions Weak solutions Weak formulation of the model examples Positive quadratic forms and convex minimization Weak topologies and weak convergences Topologies induced by functions in general topological spaces The weak topology a(v, V*) Weak convergence and geometry of uniformly convex spaces Weak compactness theorems in reflexive Banach spaces The Dunford-Pettis weak compactness theorem in L 1 (Q) The weak* topology a(v*,v) 59 3 Abstract variational principles The Lax-MiIgram theorem and the Galerkin method The Lax-Milgram theorem The Galerkin method Minimization problems: The topological approach 76 v

3 3.2.1 Extended real-valued functions The interplay between functions and sets: The role of the epigraph Lower semicontinuous functions The lower closure of a function and the relaxation problem Inf-compactness functions, coercivity Topological minimization theorems Weak topologies and minimization of weakly lower semicontinuous functions Convex minimization theorems Extended real-valued convex functions and weak lower semicontinuity Convex minimization in reflexive Banach spaces Ekeland's e-variational principle Ekeland's e-variational principle and the direct method A dynamical approach and proof of Ekeland's e-variational principle 100 Complements on measure theory Hausdorff measures and Hausdorff dimension Outer Hausdorff measures and Hausdorff measures Hausdorff measures: Scaling properties and Lipschitz transformations Hausdorff dimension Set functions and duality approach to Borel measures Borel measures as set functions Duality approach Introduction to Young measures Definition Slicing Young measures Prokhorov's compactness theorem Young measures associated with functions and generated by functions Semicontinuity and continuity properties Young measures capture oscillations Young measures do not capture concentrations 149 Sobolev spaces Sobolev spaces: Definition, density results The topological dual of tfj ( 2). The space Я _1 (^) Poincare inequality and Rellich-Kondrakov theorem in W J 0 '(Q) Extension operators from W l - p (Q) into W l,p (R N ). Poincare inequalities and the Rellich-Kondrakov theorem in W lp ( 2) The Fourier approach to Sobolev spaces. The space H S (Q), ser Trace theory for W''''( 2) spaces Sobolev embedding theorems 192

4 VII Case 1 < p < N Case p > N Case p = N Capacity theory and elements of potential theory Contractions operate on W l - p (Q.) Capacity Variational problems: Some classical examples The Dirichlet problem The homogenous Dirichlet problem The nonhomogenous Dirichlet problem The Neumann problem The coercive homogenous Neumann problem The coercive nonhomogenous Neumann problem The semicoercive homogenous Neumann problem The semicoercive nonhomogenous Neumann problem Mixed Dirichlet-Neumann problems The Dirichlet-Neumann problem Mixed Dirichlet-Neumann boundary conditions Heterogenous media: Transmission conditions Linear elliptic operators The nonlinear Laplacian Д р The Stokes system The finite element method The Galerkin method: Further results Description of finite element methods An example Convergence of the finite element method Complements Flat triangles H 2 ( i) regularity of the solution of the Dirichlet problem on a convex polygon Finite element methods of type Po Spectral Analysis of the Laplacian Introduction The Laplace-Dirichlet operator: Functional setting Existence of a Hilbertian basis of eigenvectors of the Laplace-Dirichlet operator The Couranl-Fisher min-max and max-min formulas Multiplicity and asymptotic properties of the eigenvalues of the Laplace-Dirichlet operator A general abstract theory for spectral analysis of elliptic boundary value problems 303

5 VIII Contents 9 Convex duality and optimization Dual representation of convex sets Passing from sets to functions: Elements of epigraphical calculus Legendre-Fenchel transform Legendre-Fenchel calculus Subdifferential calculus for convex functions Mathematical programming: Multipliers and duality Karush-Kuhn-Tucker optimality conditions The marginal approach to multipliers The Lagrangian approach to duality Duality for linear programming A general approach to duality in convex optimization Duality in the calculus of variations: First examples 365 Part II: Advanced Variational Analysis Spaces В V and SB V The space В V(Q): Definition, convergences, and approximation The trace operator, the Green's formula, and its consequences The coarea formula and the structure of В V functions Notion of density and regular points Sets of finite perimeter, structure of simple BV functions Structure of В V functions Structure of the gradient of В V functions The space SB V(Q) Definition Properties Relaxation in Sobolev, В V, and Young measures spaces Relaxation in abstract metrizable spaces Relaxation of integral functionals with domain W lp ( 2, R m ), p > Relaxation of integral functionals with domain W IJ (Q, R m ) Relaxation in the space of Young measures in nonlinear elasticity Young measures generated by gradients Relaxation of classical integral functionals in y(q; E) Г-convergence and applications Г-convergence in abstract metrizable spaces Application to the nonlinear membrane model Application to homogenization of composite media The quadratic case in one dimension Periodic homogenization in the general case Application to image segmentation and phase transitions The Mumford-Shah model 482

6 ix Variational approximation of a more elementary problem: A phase transitions model Variational approximation of the Mumford-Shah functional energy Integral functionals of the calculus of variations Lower semicontinuity in the scalar case Lower semicontinuity in the vectorial case Lower semicontinuity for funclionals defined on the space of measures Functionals with linear growth: Lower semicontinuity in BV and SBV Lower semicontinuity and relaxation in В V Compactness and lower semicontinuity in SBV Application in mechanics and computer vision Problems in pseudoplasticity Introduction The Hencky model The spaces ß >(ß), M(div), and /(Q) Relaxation of the Hencky model Some variational models in fracture mechanics A few considerations in fracture mechanics A first model in one dimension A second model in one dimension The Mumford-Shah model Variational problems with a lack of coercivity Convex minimization problems and recession functions Nonconvex minimization problems and topological recession Some examples Limit analysis problems An introduction to shape optimization problems The isoperimetric problem The Newton problem Optimal Dirichlet free boundary problems Optimal distribution of two conductors 609 Bibliography 615 Index 631