E-raamat: Inequalities in Mechanics and Physics

I. Problems of Semi-Permeable Media and of Temperature Control.-
1.
Review of Continuum Mechanics.- 1.1. Stress Tensor.- 1.2. Conservation Laws.-
1.3. Strain Tensor.- 1.4. Constituent Laws.-
2. Problems of Semi-Permeable
Membranes and of Temperature Control.- 2.1. Formulation of Equations.- 2.1.1.
Equations of Thermics.- 2.1.2. Equations of Mechanics of Fluids in Porous
Media.- 2.1.3. Equations of Electricity.- 2.2. Semi-Permeable Walls.- 2.2.1.
Wall of Negligible Thickness.- 2.2.2. Semi-Permeable Wall of Finite
Thickness.- 2.2.3. Semi-Permeable Partition in the Interior of ?.- 2.2.4.
Volume Injection Through a Semi-Permeable Wall.- 2.3. Temperature Control.-
2.3.1. Temperature Control Through the Boundary, Regulated by the Temperature
at the Boundary.- 2.3.2. Temperature Control Through the Interior, Regulated
by the Temperature in the Interior.-
3. Variational Formulation of Problems
of Temperature Control and of Semi-Permeable Walls.- 3.1. Notation.- 3.2.
Variational Inequalities.- 3.3. Examples. Equivalence with the Problems of
Section 2.- 3.3.1. Functions ? of Type 1.- 3.3.2. Functions ? of Type 2.-
3.3.3. Functions ? of Type 3.- 3.4. Some Extensions.- 3.5. Stationary Cases.-
3.5.1. The Function ? Is of Type 1.- 3.5.2. The Function ? Is of Type 2.-
3.5.3. The Function ? Is of Type 3.- 3.5.4. Stationary Case and Problems of
the Calculus of Variations.-
4. Some Tools from Functional Analysis.- 4.1.
Sobolev Spaces.- 4.2. Applications: The Convex Sets K.- 4.3. Spaces of
Vector-Valued Functions.-
5. Solution of the Variational Inequalities of
Evolution of Section 3.- 5.1. Definitive Formulation of the Problems.- 5.1.1.
Data V, H, V' and a(u, v).- 5.1.2. The Functional ?.- 5.1.3. Formulation of
the Problem.- 5.2. Statement of the Principal Results.- 5.3. Verification of
the Assumptions.- 5.4. Other Methods of Approximation.- 5.5. Uniqueness Proof
in Theorem 5.1 (and 5.2).- 5.6. Proof of Theorems 5.1 and 5.2.- 5.6.1.
Solution of (5.14).- 5.6.2. Estimates for uj and u'j.- 5.6.3. Proof of
(5.7).-
6. Properties of Positivity and of Comparison of Solutions.- 6.1.
Positivity of Solutions.- 6.2. Comparison of Solutions (I).- 6.3. Comparison
of Solutions (II).-
7. Stationary Problems.- 7.1. The Strictly Coercive
Case.- 7.2. Approximation of the Stationary Condition by the Solution of
Problems of Evolution when t ? + ?.- 7.3. The Not Strictly Coercive Case.-
7.3.1. Necessary Conditions for the Existence of Solutions.- 7.3.2.
Sufficient Conditions for the Existence of a Solution.- 7.3.3. The Problem of
Uniqueness under Assumption (7.48).- 7.3.4. The Limiting Cases in (7.48).-
8.
Comments.- II. Problems of Heat Control.-
1. Heat Control.- 1.1.
Instantaneous Control.- 1.1.1. Temperature Control at the Boundary.- 1.1.2.
Temperature Control in the Interior.- 1.1.3. Properties of the Solutions.-
1.1.4. Other Controls.- 1.2. Delayed Control.-
2. Variational Formulation of
Control Problems.- 2.1. Notation.- 2.2. Variational Inequalities.- 2.2.1.
Instantaneous Control.- 2.2.2. Delayed Control.- 2.3. Examples.- 2.3.1. The
Function ? of Type 1.- 2.3.2. The Function ? of Type 2.- 2.3.3. The Function
? of Type 3.- 2.4. Orientation.-
3. Solution of the Problems of Instantaneous
Control.- 3.1. Statement of the Principal Results.- 3.2. Uniqueness Proof for
Theorem 3.1 (and 3.2).- 3.3. Proof of Theorems 3.1 and 3.2.- 3.3.1. Solution
of the Galerkin Approximation of (3.15).- 3.3.2. Solution of (3.15) and a
Priori Estimates for uj.- 3.3.3. Proof of the Statements of the Theorems.-
4.
A Property of the Solution of the Problem of Instantaneous Control at a Thin
Wall.-
5. Partial Results for Delayed Control.- 5.1. Statement of a Result.-
5.2. Proof of Existence in Theorem 5.1.- 5.3. Proof of Uniqueness in Theorem
5.1.-
6. Comments.- III. Classical Problems and Problems with Friction in
Elasticity and Visco-Elasticity.-
1. Introduction.-
2. Classical Linear
Elasticity.- 2.1. The Constituent Law.- 2.2. Classical Problems of Linear
Elasticity.- 2.2.1. Linearization of the Equation of Conservation of Mass and
of the Equations of Motion.- 2.2.2. Boundary Conditions.- 2.2.3. Summary.-
2.3. Variational Formulation of the Problem of Evolution.- 2.3.1. Green's
Formula.- 2.3.2. Variational Formulation.-
3. Static Problems.- 3.1.
Classical Formulation.- 3.2. Variational Formulation.- 3.3. Korn's Inequality
and its Consequences.- 3.4. Results.- 3.4.1. The Case "?U has Positive
Measure".- 3.4.2. The Case "?U is Empty".- 3.5. Dual Formulations.- 3.5.1.
Statically Admissible Fields and Potential Energy.- 3.5.2. Duality and
Lagrange Multipliers.-
4. Dynamic Problems.- 4.1. Statement of the Principal
Results.- 4.2. Proof of Theorem 4.1.- 4.3. Other Boundary Conditions.- 4.3.1.
Variant I (for Example, a Body on a Rigid Support).- 4.3.2. Variant II (a
Body Placed in an Elastic Envelope).-
5. Linear Elasticity with Friction or
Unilateral Constraints.- 5.1. First Laws of Friction. Dynamic Case.- 5.1.1.
Coulomb's Law.- 5.1.2. Problems under Consideration.- 5.2. Coulomb's Law.
Static Case.- 5.2.1. Problems under Consideration.- 5.2.2. Variational
Formulation.- 5.2.3. Results. The Case "?U with Positive Measure".- 5.2.4.
Results. The Case "?U= O".- 5.3. Dual Variational Formulation.- 5.3.1.
Statically Admissible Fields and Potential Energy.- 5.3.2. Duality and
Lagrange Multipliers.- 5.4. Other Boundary Conditions and Open Questions.-
5.4.1. Normal Displacement with Friction.- 5.4.2. Signorini's Problem as
Limit Case of Problems with Friction.- 5.4.3. Another Condition for Friction
with Imposed Normal Displacement.- 5.4.4. Coulomb Friction with Imposed
Normal Displacement.- 5.4.5. Signorini's Problem with Friction.- 5.5. The
Dynamic Cases.- 5.5.1. Variational Formulation.- 5.5.2. Statement of
Results.- 5.5.3. Uniqueness Proof.- 5.5.4. Existence Proof.-
6. Linear
Visco-Elasticity. Material with Short Memory.- 6.1. Constituent Law and
General Remarks.- 6.2. Dynamic Case. Formulation of the Problem.- 6.3.
Existence Theorem and Uniqueness in the Dynamic Case.- 6.4. Quasi-Static
Problems. Variational Formulation.- 6.5. Existence and Uniqueness Theorem for
the Case when ?U has Measure >0.- 6.6. Discussion of the Case when ?U = O.-
6.7. Justification of the Quasi-Static Case in the Problems without
Friction.- 6.7.1. Statement of the Problem.- 6.7.2. The Case "Measure ?U >
0".- 6.7.3. The Case "?U = O".- 6.8. The Case without Viscosity as Limit of
the Case with Viscosity.- 6.9. Interpretation of Viscous Problems as
Parabolic Systems.-
7. Linear Visco-Elasticity. Material with Long Memory.-
7.1. Constituent Law and General Remarks.- 7.2. Dynamic Problems with
Friction.- 7.3. Existence and Uniqueness Theorem in the Dynamic Case.- 7.4.
The Quasi-Static Case.- 7.4.1. Necessary Conditions for the Initial Data.-
7.4.2. Discussion of the Case "Measure ?U >0".- 7.4.3. Discussion of the Case
" ?U = O".- 7.5. Use of the Laplace Transformation in the Cases without
Friction.- 7.6. Elastic Case as Limit of the Case with Memory.-
8. Comments.-
IV. Unilateral Phenomena in the Theory of Flat Plates.-
1. Introduction.-
2.
General Theory of Plates.- 2.1. Definitions and Notation.- 2.2. Analysis of
Forces.- 2.3. Linearized Theory.- 2.3.1. Hypotheses.- 2.3.2. Formulation of
Equations. First Method.- 2.3.3. Formulation of Equations. Second Method (due
to Landau and Lifshitz).- 2.3.4. Summary.-
3. Problems to be Considered.-
3.1. Classical Problems.- 3.2. Unilateral Problems.-
4. Stationary Unilateral
Problems.- 4.1. Notation.- 4.2. Problems (Stationary).- 4.3. Solution of
Problem 4.1. Necessary Conditions for the Existence of a Solution.- 4.4.
Solution of Problem 4.1. Sufficient Conditions.- 4.5. The Question of
Uniqueness in Problems 4.1 and 4.3.- 4.6. Solution of Problem 4.1a.- 4.7.
Solution of Problem 4.2.-
5. Unilateral Problems of Evolution.- 5.1.
Formulation of the Problems.- 5.2. Solution of Unilateral Problems of
Evolution.-
6. Comments.- V. Introduction to Plasticity.-
1. Introduction.-
2. The Elastic Perfectly Plastic Case (Prandtl-Reuss Law) and the
Elasto-Visco-Plastic Case.- 2.1. Constituent Law of Prandtl-Reuss.- 2.1.1.
Preliminary Observation.- 2.1.2. Generalization.- 2.2. Elasto-Visco-Plastic
Constituent Law.- 2.3. Problems to be Discussed.-
3. Discussion of
Elasto-Visco-Plastic, Dynamic and Quasi-Static Problems.- 3.1. Variational
Formulation of the Problems.- 3.2. Statement of Results.- 3.3. Uniqueness
Proof in the Theorems.- 3.4. Existence Proof in the Dynamic Case.- 3.5.
Existence Proof in the Quasi-Static Case.-
4. Discussion of Elastic Perfectly
Plastic Problems.- 4.1. Statement of the Problems.- 4.2. Formulation of the
Results.- 4.3. Proof of the Uniqueness Results.- 4.4. Proof of Theorems 4.1
and 4.2.- 4.5. Proof of Theorems 4.3 and 4.4.-
5. Discussion of
Rigid-Visco-Plastic and Rigid Perfectly Plastic Problems.- 5.1.
Rigid-Visco-Plastic Problems.- 5.2. Rigid Perfectly Plastic Problems.-
6.
Hencky's Law. The Problem of Elasto-Plastic Torsion.- 6.1. Constituent Law.-
6.2. Problems to be Considered.- 6.3. Variational Formulation for the
Stresses.- 6.4. Determination of the Field of Displacements.- 6.5. Isotropic
Material with the Von Mises Condition.- 6.6. Torsion of a Cylindrical Tree
(Fig. 19).-
7. Locking Material.- 7.1. Constituent Law.- 7.2. Problem to be
Considered.- 7.3. Double Variational Formulation of the Problem.- 7.4.
Existence and Uniqueness of a Displacement Field Solution.- 7.5. The
Associated Field of Stresses.-
8. Comments.- VI. Rigid Visco-Plastic Bingham
Fluid.-
1. Introduction and Problems to be Considered.- 1.1. Constituent Law
of a Rigid Visco-Plastic, Incompressible Fluid.- 1.2. The Dissipation
Function.- 1.3. Problems to be Considered and Recapitulation of the
Equations.-
2. Flow in the Interior of a Reservoir. Formulation in the Form
of a Variational Inequality.- 2.1. Preliminary Notation.- 2.2. Variational
Inequality.-
3. Solution of the Variational Inequality, Characteristic for
the Flow of a Bingham Fluid in the Interior of a Reservoir.- 3.1. Tools from
Functional Analysis.- 3.2. Functional Formulation of the Variational
Inequalities.- 3.3. Proof of Theorem 3.2.- 3.4. Proof of Theorem 3.1.- 3.4.1.
Existence Proof.- 3.4.2. Uniqueness Proof.-
4. A Regularity Theorem in Two
Dimensions.-
5. Newtonian Fluids as Limits of Bingham Fluids.- 5.1. Statement
of the Result.- 5.2. Proof of Theorem 5.1.-
6. Stationary Problems.- 6.1.
Statement of the Results.- 6.2. Proof.-
7. Exterior Problem.- 7.1.
Formulation of the Problem as a Variational Inequality.- 7.2. Results.-
8.
Laminar Flow in a Cylindrical Pipe.- 8.1. Recapitulation of the Equations.-
8.2. Variational Formulation.- 8.3. Properties of the Solution.-
9.
Interpretation of Inequalities with Multipliers.-
10. Comments.- VII.
Maxwell's Equations. Antenna Problems.-
1. Introduction.-
2. The Laws of
Electromagnetism.- 2.1. Physical Quantities.- 2.2. Conservation of Electric
Charge.- 2.3. Faraday's Law.- 2.4. Recapitulation. Maxwell's Equations.- 2.5.
Constituent Laws.-
3. Physical Problems to be Considered.- 3.1. Stable Medium
with Supraconductive Boundary.- 3.2. Polarizable Medium with Supraconductive
Boundary.- 3.3. Bipolar Antenna.- 3.4. Slotted Antenna. Diffraction of an
Electromagnetic Wave by a Supraconductor.- 3.5. Recapitulation. Unified
Formulation of the Problems.-
4. Discussion of Stable Media. First Theorem of
Existence and Uniqueness.- 4.1. Tools from Functional Analysis for the "Weak"
Formulation of the Problem.- 4.2. The Operator A. "Weak" Formulation of the
Problem.- 4.3. Existence and Uniqueness of the Weak Solution.- 4.4.
Continuous Dependence of the Solution on the Dielectric Constants and on the
Magnetic Permeabilities.-
5. Stable Media. Existence of "Strong" Solutions.-
5.1. Strong Solutions in D(A).- 5.2. Solution of the Physical Problem.-
6.
Stable Media. Strong Solutions in Sobolev Spaces.- 6.1. Imbedding Theorem.-
6.2. B as Part of a Sobolev Space.- 6.3. D as Part of a Sobolev Space.-
7.
Slotted Antennas. Non-Homogeneous Problems.- 7.1. Statement of the Problem
(Cf. Sec. 3.4).- 7.2. Statement of the Result.- 7.3. Proof of Theorem 7.1.-
8. Polarizable Media.- 8.1. Existence and Uniqueness Result for a Variational
Inequality Associated with the Operators of Maxwell.- 8.2. Interpretation of
the Variational Inequality. Solution of the Problems for Polarizable Media.-
8.3. Proof of Theorem 8.1.- 8.3.1. Existence Proof.- 8.3.2. Uniqueness
Proof.-
9. Stable Media as Limits of Polarizable Media.- 9.1. Statement of
the Result.- 9.2. Proof of Theorem 9.1.-
10. Various Additions.-
11.
Comments.- Additional Bibliography and Comments.-
1. Comments.-
2.
Bibliography.