Inequalities in Mechanics and Physics

1. Front Matter

   Pages I-XVI

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2. Problems of Semi-Permeable Media and of Temperature Control

   • Georges Duvaut, Jacques Louis Lions

   Pages 1-76

3. Problems of Heat Control

   • Georges Duvaut, Jacques Louis Lions

   Pages 77-101

4. Classical Problems and Problems with Friction in Elasticity and Visco-Elasticity

   • Georges Duvaut, Jacques Louis Lions

   Pages 102-196

5. Unilateral Phenomena in the Theory of Flat Plates

   • Georges Duvaut, Jacques Louis Lions

   Pages 197-227

6. Introduction to Plasticity

   • Georges Duvaut, Jacques Louis Lions

   Pages 228-277

7. Rigid Visco-Plastic Bingham Fluid

   • Georges Duvaut, Jacques Louis Lions

   Pages 278-327

8. Maxwell’s Equations. Antenna Problems

   • Georges Duvaut, Jacques Louis Lions

   Pages 328-381

9. Back Matter

   Pages 382-400

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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.

• Finite
• Ungleichung
• approximation
• calculus
• continuum mechanics
• duality
• elasticity
• equation
• function
• mathematical physics
• mechanics
• plasticity
• proof
• theorem
• transfinite induction

• Mécanique Théoretique, Université de Paris VI, Paris, France

  Georges Duvaut

• Collège de France, Paris, France

  Jacques Louis Lions

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