Author	Duvaut, Georges. author
Title	Inequalities in Mechanics and Physics [electronic resource] / by Georges Duvaut, Jacques Louis Lions
Imprint	Berlin, Heidelberg : Springer Berlin Heidelberg, 1976
Connect to	http://dx.doi.org/10.1007/978-3-642-66165-5
Descript	XVI, 400 p. online resource

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,tยปo => 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 r̃ 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. Henckyโ{128}{153}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. Maxwellโ{128}{153}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 โ{128}{156}Strongโ{128}{157} 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