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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-p...

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Bibliographic Details
Main Authors: Duvant, G., Lions, J. L. (Author)
Format: eBook
Language:English
Published: Berlin, Heidelberg Springer Berlin Heidelberg 1976, 1976
Edition:1st ed. 1976
Series:Grundlehren der mathematischen Wissenschaften, A Series of Comprehensive Studies in Mathematics
Subjects:
Mathematics
Online Access:
Collection: Springer Book Archives -2004 - Collection details see MPG.ReNa
Table of Contents:
  • 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
  • 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

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