Inequalities in Mechanics and Physics

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.

I. Problems of Semi-Permeable Media and of Temperature Control.- 1. Review of Continuum Mechanics.- 2. Problems of Semi-Permeable Membranes and of Temperature Control.- 3. Variational Formulation of Problems of Temperature Control and of Semi-Permeable Walls.- 4. Some Tools from Functional Analysis.- 5. Solution of the Variational Inequalities of Evolution of Section 3.- 6. Properties of Positivity and of Comparison of Solutions.- 7. Stationary Problems.- 8. Comments.- II. Problems of Heat Control.- 1. Heat Control.- 2. Variational Formulation of Control Problems.- 3. Solution of the Problems of Instantaneous Control.- 4. A Property of the Solution of the Problem of Instantaneous Control at a Thin Wall.- 5. Partial Results for Delayed Control.- 6. Comments.- III. Classical Problems and Problems with Friction in Elasticity and Visco-Elasticity.- 1. Introduction.- 2. Classical Linear Elasticity.- 3. Static Problems.- 4. Dynamic Problems.- 5. Linear Elasticity with Friction or Unilateral Constraints.- 6. Linear Visco-Elasticity. Material with Short Memory.- 7. Linear Visco-Elasticity. Material with Long Memory.- 8. Comments.- IV. Unilateral Phenomena in the Theory of Flat Plates.- 1. Introduction.- 2. General Theory of Plates.- 3. Problems to be Considered.- 4. Stationary Unilateral Problems.- 5. 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.- 3. Discussion of Elasto-Visco-Plastic, Dynamic and Quasi-Static Problems.- 4. Discussion of Elastic Perfectly Plastic Problems.- 5. Discussion of Rigid-Visco-Plastic and Rigid Perfectly Plastic Problems.- 6. Hencky’s Law. The Problem of Elasto-Plastic Torsion.- 7. Locking Material.- 8.Comments.- VI. Rigid Visco-Plastic Bingham Fluid.- 1. Introduction and Problems to be Considered.- 2. Flow in the Interior of a Reservoir. Formulation in the Form of a Variational Inequality.- 3. Solution of the Variational Inequality, Characteristic for the Flow of a Bingham Fluid in the Interior of a Reservoir.- 4. A Regularity Theorem in Two Dimensions.- 5. Newtonian Fluids as Limits of Bingham Fluids.- 6. Stationary Problems.- 7. Exterior Problem.- 8. Laminar Flow in a Cylindrical Pipe.- 9. Interpretation of Inequalities with Multipliers.- 10. Comments.- VII. Maxwell’s Equations. Antenna Problems.- 1. Introduction.- 2. The Laws of Electromagnetism.- 3. Physical Problems to be Considered.- 4. Discussion of Stable Media. First Theorem of Existence and Uniqueness.- 5. Stable Media. Existence of “Strong” Solutions.- 6. Stable Media. Strong Solutions in Sobolev Spaces.- 7. Slotted Antennas. Non-Homogeneous Problems.- 8. Polarizable Media.- 9. Stable Media as Limits of Polarizable Media.- 10. Various Additions.- 11. Comments.- Additional Bibliography and Comments.- 1. Comments.- 2. Bibliography.