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9780898716009

Variational Analysis in Sobolev And Bv Spaces

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  • ISBN13:

    9780898716009

  • ISBN10:

    0898716004

  • Format: Paperback
  • Copyright: 2005-12-30
  • Publisher: Society for Industrial & Applied
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Summary

This self-contained book is excellent for graduate-level courses devoted to variational analysis, optimization, and partial differential equations (PDEs). It provides readers with a complete guide to problems in these fields as well as a detailed presentation of the most important tools and methods of variational analysis. New trends in variational analysis are also presented, along with recent developments and applications in this area. It contains several applications to problems in geometry, mechanics, elasticity, and computer vision, along with a complete list of references. The book is divided into two parts. In Part I, classical Sobolev spaces are introduced and the reader is provided with the basic tools and methods of variational analysis and optimization in infinite dimensional spaces, with applications to classical PDE problems. In Part II, BV spaces are introduced and new trends in variational analysis are presented.

Author Biography

Hedy Attouch is Professor of Mathematics at Université Montpellier II, France. He is Director of Laboratoire d'Analyse Convexe and of ACSIOM (the analysis, computational, and optimization component of the Institute of Mathematics and Modelization of Montpellier). He has published 100 articles and has supervised 23 theses in the fields of variational analysis and optimization.Giuseppe Buttazzo is Professor of Mathematics at Università di Pisa. He is an editor of several international journals and author of more than 130 scientific articles and 16 books. He has supervised 13 theses in the fields of calculus of variations, nonlinear PDEs, control theory, and related topics.Gérard Michaille is Professor at Centre Universitaire de Formation et de Recherche at Nîmes (CUFR) and a member of the UMR-CNRS I3S of the Mathematical Department at Université Montpellier II, France. He works in the fields of variational analysis and homogenization and their applications in mechanics.

Table of Contents

Preface xi
Introduction
1(4)
Part I: Basic Variational Principles
5(364)
Weak solution methods in variational analysis
7(60)
The Dirichlet problem: Historical presentation
7(8)
Test functions and distribution theory
15(16)
Definition of distributions
15(3)
Locally integrable functions as distributions: Regularization by convolution and mollifiers
18(6)
Radon measures
24(1)
Derivation of distributions, introduction to Sobolev spaces
24(3)
Convergence of sequences of distributions
27(4)
Weak solutions
31(10)
Weak formulation of the model examples
31(5)
Positive quadratic forms and convex minimization
36(5)
Weak topologies and weak convergences
41(26)
Topologies induced by functions in general topological spaces
41(3)
The weak topology σ(V, V*)
44(8)
Weak convergence and geometry of uniformly convex spaces
52(2)
Weak compactness theorems in reflexive Banach spaces
54(3)
The Dunford--Pettis weak compactness theorem in L1(Ω)
57(2)
The weak* topology σ(V*, V)
59(8)
Abstract variational principles
67(42)
The Lax-Milgram theorem and the Galerkin method
67(9)
The Lax-Milgram theorem
67(6)
The Galerkin method
73(3)
Minimization problems: The topological approach
76(15)
Extended real-valued functions
77(1)
The interplay between functions and sets: The role of the epigraph
78(2)
Lower semicontinuous functions
80(2)
The lower closure of a function and the relaxation problem
82(4)
Inf-compactness functions, coercivity
86(1)
Topological minimization theorems
87(4)
Weak topologies and minimization of weakly lower semi-continuous functions
91(1)
Convex minimization theorems
91(7)
Extended real-valued convex functions and weak lower semicontinuity
91(2)
Convex minimization in reflexive Banach spaces
93(5)
Ekeland's ε-variational principle
98(11)
Ekeland's ε-variational principle and the direct method
98(2)
A dynamical approach and proof of Ekeland's ε-variational principle
100(9)
Complements on measure theory
109(42)
Hausdorff measures and Hausdorff dimension
109(15)
Outer Hausdorff measures and Hausdorff measures
109(8)
Hausdorff measures: Scaling properties and Lipschitz transformations
117(3)
Hausdorff dimension
120(4)
Set functions and duality approach to Borel measures
124(14)
Borel measures as set functions
124(5)
Duality approach
129(9)
Introduction to Young measures
138(13)
Definition
138(1)
Slicing Young measures
139(3)
Prokhorov's compactness theorem
142(1)
Young measures associated with functions and generated by functions
142(1)
Semicontinuity and continuity properties
143(3)
Young measures capture oscillations
146(3)
Young measures do not capture concentrations
149(2)
Sobolev spaces
151(66)
Sobolev spaces: Definition, density results
152(13)
The topological dual of H1/0 (Ω). The space H-1(Ω)
165(3)
Poincare inequality and Rellich-Kondrakov theorem in W1/0,p (Ω)
168(6)
Extension operators from W1,p (Ω) into W1,p (RN). Poincare inequalities and the Rellich--Kondrakov theorem in W1,p (Ω)
174(6)
The Fourier approach to Sobolev spaces. The space Hs(Ω), s ε R
180(6)
Trace theory for W1,p (Ω) spaces
186(6)
Sobolev embedding theorems
192(14)
Case 1 ≤ p < N
194(6)
Case p > N
200(2)
Case p = N
202(4)
Capacity theory and elements of potential theory
206(11)
Contractions operate on W1,p (Ω)
206(6)
Capacity
212(5)
Variational problems: Some classical examples
217(40)
The Dirichlet problem
218(7)
The homogeneous Dirichlet problem
218(4)
The nonhomogeneous Dirichlet problem
222(3)
The Neumann problem
225(11)
The coercive homogenous Neumann problem
225(4)
The coercive nonhomogenous Neumann problem
229(1)
The semicoercive homogenous Neumann problem
230(4)
The semicoercive nonhomogenous Neumann problem
234(2)
Mixed Dirichlet-Neumann problems
236(4)
The Dirichlet-Neumann problem
236(2)
Mixed Dirichlet-Neumann boundary conditions
238(2)
Heterogenous media: Transmission conditions
240(5)
Linear elliptic operators
245(4)
The nonlinear Laplacian Δp
249(4)
The Stokes system
253(4)
The finite element method
257(22)
The Galerkin method: Further results
257(3)
Description of finite element methods
260(2)
An example
262(1)
Convergence of the finite element method
263(13)
Complements
276(3)
Flat triangles
276(1)
H2(Ω) regularity of the solution of the Dirichlet problem on a convex polygon
277(1)
Finite element methods of type P2
277(2)
Spectral Analysis of the Laplacian
279(28)
Introduction
279(2)
The Laplace--Dirichlet operator: Functional setting
281(5)
Existence of a Hilbertian basis of eigenvectors of the Laplace--Dirichlet operator
286(3)
The Courant--Fisher min-max and max-min formulas
289(8)
Multiplicity and asymptotic properties of the eigenvalues of the Laplace--Dirichlet operator
297(6)
A general abstract theory for spectral analysis of elliptic boundary value problems
303(4)
Convex duality and optimization
307(62)
Dual representation of convex sets
307(5)
Passing from sets to functions: Elements of epigraphical calculus
312(6)
Legendre--Fenchel transform
318(10)
Legendre--Fenchel calculus
328(3)
Subdifferential calculus for convex functions
331(9)
Mathematical programming: Multipliers and duality
340(18)
Karush--Kuhn--Tucker optimality conditions
341(4)
The marginal approach to multipliers
345(8)
The Lagrangian approach to duality
353(3)
Duality for linear programming
356(2)
A general approach to duality in convex optimization
358(7)
Duality in the calculus of variations: First examples
365(4)
Part II: Advanced Variational Analysis
369(246)
Spaces BV and SBV
371(46)
The space BV(Ω): Definition, convergences, and approximation
371(7)
The trace operator, the Green's formula, and its consequences
378(9)
The coarea formula and the structure of BV functions
387(19)
Notion of density and regular points
388(7)
Sets of finite perimeter, structure of simple BV functions
395(7)
Structure of BV functions
402(4)
Structure of the gradient of BV functions
406(2)
The space SBV(Ω)
408(9)
Definition
409(1)
Properties
410(7)
Relaxation in Sobolev, BV, and Young measures spaces
417(46)
Relaxation in abstract metrizable spaces
417(4)
Relaxation of integral functionals with domain W1,p (Ω, Rm), p > 1
421(16)
Relaxation of integral functionals with domain W1,1 (Ω, Rm)
437(12)
Relaxation in the space of Young measures in nonlinear elasticity
449(14)
Young measures generated by gradients
450(7)
Relaxation of classical integral functionals in γ(Ω; E)
457(6)
Γ-convergence and applications
463(34)
Γ-convergence in abstract metrizable spaces
463(4)
Application to the nonlinear membrane model
467(5)
Application to homogenization of composite media
472(10)
The quadratic case in one dimension
472(3)
Periodic homogenization in the general case
475(7)
Application to image segmentation and phase transitions
482(15)
The Mumford--Shah model
482(1)
Variational approximation of a more elementary problem: A phase transitions model
483(4)
Variational approximation of the Mumford--Shah functional energy
487(10)
Integral functionals of the calculus of variations
497(24)
Lower semicontinuity in the scalar case
497(6)
Lower semicontinuity in the vectorial case
503(7)
Lower semicontinuity for functionals defined on the space of measures
510(3)
Functionals with linear growth: Lower semicontinuity in BV and SBV
513(8)
Lower semicontinuity and relaxation in BV
513(2)
Compactness and lower semicontinuity in SBV
515(6)
Application in mechanics and computer vision
521(32)
Problems in pseudoplasticity
521(8)
Introduction
521(2)
The Hencky model
523(1)
The spaces BD(Ω), M(div), and U(Ω)
524(4)
Relaxation of the Hencky model
528(1)
Some variational models in fracture mechanics
529(20)
A few considerations in fracture mechanics
529(3)
A first model in one dimension
532(11)
A second model in one dimension
543(6)
The Mumford--Shah model
549(4)
Variational problems with a lack of coercivity
553(48)
Convex minimization problems and recession functions
553(19)
Nonconvex minimization problems and topological recession
572(9)
Some examples
581(7)
Limit analysis problems
588(13)
An introduction to shape optimization problems
601(14)
The isoperimetric problem
602(2)
The Newton problem
604(1)
Optimal Dirichlet free boundary problems
605(4)
Optimal distribution of two conductors
609(6)
Bibliography 615(16)
Index 631

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