List of Figures |
Preface |
Introduction / 1: |
Geometry as a Variable |
Shape Analysis / 2: |
Geometric Measure Theory / 3: |
Distance Functions, Smoothness, Curvatures / 4: |
Shape Optimization / 5: |
Shape Derivatives / 6: |
Shape Calculus and Tangential Differential Calculus / 7: |
Shape Analysis in This Book / 8: |
Overview of the Book / 9: |
Analysis, Shape Spaces, and Optimization / 9.1: |
Continuity, Derivatives, Shape Calculus, and Tangential Differential Calculus / 9.2: |
Classical Descriptions and Properties of Domains |
Notation and Definitions |
Basic Notation / 2.1: |
Continuous and C[superscript k] Functions / 2.2: |
Holder and Lipschitz Continuous Functions / 2.3: |
Smoothness of Domains, Boundary Integral, Boundary Curvatures |
C[superscript k] and Holderian Sets / 3.1: |
Boundary Integral, Canonical Density, and Hausdorff Measures / 3.2: |
Fundamental Forms and Principal Curvatures / 3.3: |
Domains as Level Sets of a Function |
Domains as Local Epigraphs |
Sets That Are Locally Lipschitzian Epigraphs / 5.1: |
Local Epigraphs and Sets of Class C[superscript k,e] / 5.2: |
Boundary Integral from the Graph of a Function / 5.3: |
Volume and Surface Measures of the Boundary / 5.4: |
Convex Sets / 5.5: |
Nonhomogeneous Neumann and Dirichlet Problems / 5.6: |
Uniform Cone Property for Lispchitzian Domains |
Segment Property and C[superscript 0]-Epigraphs |
Courant Metric Topology on Images of a Domain |
The Complete Metric Group of Diffeomorphisms F[superscript k subscript 0] / 8.1: |
The Courant Metric on the Images / 8.2: |
Perturbations of the Identity / 8.3: |
The Generic Framework of Micheletti |
B[superscript k] (R[superscript N], R[superscript N])-Mappings |
Lipschitzian Mappings |
The Murat-Simon Approach / 9.2.1: |
The Micheletti Approach / 9.2.2: |
B[superscript k] (D, R[superscript N])- and C[superscript k subscript 0] (D, R[superscript N])-Mappings / 9.3: |
Domains in a Closed Submanifold of R[superscript N] / 10: |
Relaxation to Measurable Domains |
Characteristic Functions in L[superscript p]-Topologies |
Strong Topologies and C[superscript [infinity]-Approximations |
Weak Topologies and Microstructures |
Nice or Measure Theoretic Representative |
The Family of Convex Sets / 2.4: |
Sobolev Spaces for Measurable Domains / 2.5: |
Some Compliance Problems with Two Materials |
Transmission Problem and Compliance |
The Original Problem of Cea and Malanowski |
Relaxation and Homogenization |
Buckling of Columns |
Caccioppoli or Finite Perimeter Sets |
Finite Perimeter Sets |
Decomposition of the Integral along Level Sets |
Domains of Class W[superscript [varepsilon],p] (D), 0 [less than or equal] [varepsilon] [ 1/p, p [greater than or equal] 1 |
Compactness and Uniform Cone Property |
Existence for the Bernoulli Free Boundary Problem |
An Example: Elementary Modeling of the Water Wave / 6.1: |
Existence for a Class of Free Boundary Problems / 6.2: |
Weak Solutions of Some Generic Free Boundary Problems / 6.3: |
Problem without Constraint / 6.3.1: |
Constraint on the Measure of the Domain [Omega] / 6.3.2: |
Weak Existence with Surface Tension / 6.4: |
Continuity of the Dirichlet Boundary Value Problem |
Continuity of Transmission Problems / 7.1: |
Approximation by Transmission Problems / 7.2: |
Continuity of the Dirichlet Problem / 7.3: |
Topologies Generated by Distance Functions |
Hausdorff Metric Topologies |
The Family of Distance Functions C[subscript d](D) |
Hausdorff Metric Topology |
Hausdorff Complementary Metric Topology and C[superscript c subscript d](D) |
Projection, Skeleton, Crack, and Differentiability |
W[superscript 1,p]-Topology and Characteristic Functions |
Sets of Bounded and Locally Bounded Curvature |
Definitions and Properties |
Examples |
Characterization of Convex Sets |
Federer's Sets of Positive Reach |
Compactness Theorems |
Global Conditions on D |
Local Conditions in Tubular Neighborhoods |
Oriented Distance Function and Smoothness of Sets |
Uniform Metric Topology |
The Family of Oriented Distance Functions C[subscript b](D) |
Projection, Skeleton, and Differentiability of b[subscript A] |
Boundary Smoothness and Smoothness of b[subscript A] |
W[superscript 1,p](D)-Topology and the Family C[superscript 0 subscript b](D) |
Definitions and Main Properties |
Examples and Limits of the Tubular Norms as h Goes to Zero |
(m,p)-Sobolev Domains (or W[superscript m,p]-Domains) |
Characterization of Convex and Semiconvex Sets |
Compactness Theorems for Sets of Bounded Curvature |
Compactness and Uniform Cusp Property / 11: |
Optimization of Shape Functions |
Introduction and Generic Examples |
Embedding of H[superscript 1 subscript 0]([Omega]) into H[superscript 1 subscript 0](D) and First Eigenvalue |
Hausdorff Complementary Topology |
Continuity of u[subscript [Omega] with Respect to [Omega] for the Dirichlet Problem |
Continuity of the Neumann Problem |
Optimization over Families of Lipschitzian or Convex Domains |
Minimization of an Objective Function under State Equation Constraint |
Optimization of the First Eigenvalue |
Elements of Capacity Theory |
Definition and Basic Properties |
Quasi-Continuous Representative and H[superscript 1]-Functions |
Transport of Sets of Zero Capacity |
Continuity under Capacity Constraints |
Flat Cone Condition and the Compact Family O[subscript c,r](D) |
Examples with a Constraint on the Gradient |
Transformations versus Flows of Velocities |
Shape Functions and Choice of Shape Derivatives |
Families of Transformations of Domains |
C[infinity]-Domains |
C[superscript k]-Domains |
Cartesian Graphs |
Polar Coordinates / 3.4: |
Level Sets / 3.5: |
Unconstrained Families of Domains |
Equivalence between Velocities and Transformations / 4.1: |
Equivalence for Special Families of Velocities / 4.2: |
Constrained Families of Domains |
Transformation of Condition (V2[subscript D]) into a Linear Constraint |
Continuity of Shape Functions |
Courant Metrics and Flows of Velocities |
Shape Continuity and Velocity Method |
Shape Derivatives and Calculus, and Tangential Differential Calculus |
Review of Differentiation in Banach Spaces |
Definitions of Semiderivatives and Derivatives |
Locally Lipschitz Functions |
Chain Rule for Semiderivatives |
Semiderivatives of Convex Functions |
Conditions for Frechet Differentiability |
Hadamard Semiderivative and Velocity Method / 2.6: |
First-Order Semiderivatives and Shape Gradient |
Perturbations of the Identity and Frechet Derivative |
Shape Gradient and Structure Theorem |
Elements of Shape Calculus |
Basic Formula for Domain Integrals |
Basic Formula for Boundary Integrals |
Examples of Shape Derivative |
Volume of [Omega] and Area of [Gamma] / 4.3.1: |
H[superscript 1]([Omega])-Norm / 4.3.2: |
Normal Derivative / 4.3.3: |
Elements of Tangential Calculus |
Intrinsic Definition of the Tangential Gradient |
First-Order Derivatives |
Second-Order Derivatives |
A Few Useful Formulae and the Chain Rule |
The Stokes and Green Formulae |
Relation between Tangential and Covariant Derivatives |
Back to the Example of Section 4.3.3 / 5.7: |
Second-Order Semiderivative and Shape Hessian |
Second-Order Derivative of the Domain Integral |
Nonautonomous Case |
Autonomous Case |
Decomposition of d[superscript 2] J([Omega]; V(0), W(0)) / 6.5: |
Shape Gradients under a State Equation Constraint |
Min Formulation |
An Illustrative Example and a Shape Variational Principle |
Function Space Parametrization |
Differentiability of a Minimum with Respect to a Parameter |
Application of the Theorem |
Domain and Boundary Integral Expressions of the Shape Gradient |
Eigenvalue Problems |
Transport of H[superscript k subscript 0]([Omega]) by W[superscript k,[infinity]-Transformations of R[superscript N] |
Laplacian and Bi-Laplacian |
Linear Elasticity |
Saddle Point Formulation and Function Space Parametrization |
An Illustrative Example |
Saddle Point Formulation |
Differentiability of a Saddle Point with Respect to a Parameter |
Domain and Boundary Expressions for the Shape Gradient |
Multipliers and Function Space Embedding |
The Nonhomogeneous Dirichlet Problem |
A Saddle Point Formulation of the State Equation |
Saddle Point Expression of the Objective Function |
Verification of the Assumptions of Theorem 5.1 |
Elements of Bibliography |
Index of Notation |
Index |
List of Figures |
Preface |
Introduction / 1: |
Geometry as a Variable |
Shape Analysis / 2: |
Geometric Measure Theory / 3: |