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= Ø".- 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 = Ø.- 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 = Ø".- 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 = Ø".- 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.

1. We begin by giving a simple example of a partial differential inequality that occurs in an elementary physics problem. We consider a fluid with pressure u(x, t) at the point x at the instant t that 3 occupies a region Q oflR bounded by a membrane r of negligible thickness that, however, is semi-permeable, i. e., a membrane that permits the fluid to enter Q freely but that prevents all outflow of fluid. One can prove then (cf. the details in Chapter 1, Section 2.2.1) that au (aZu azu aZu) (1) in Q, t>o, -a - du = g du = -a z + -a z + -a z t Xl X X3 z l g a given function, with boundary conditions in the form of inequalities u(X,t»o => au(x,t)/an=O, XEr, (2) u(x,t)=o => au(x,t)/an?:O, XEr, to which is added the initial condition (3) u(x,O)=uo(x). We note that conditions (2) are non linear; they imply that, at each fixed instant t, there exist on r two regions r~ and n where u(x, t) =0 and au (x, t)/an = 0, respectively. These regions are not prescribed; thus we deal with a "free boundary" problem.