# Inequalities in Mechanics and Physics

• Georges Duvaut
• Jacques Louis Lions
Book

Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 219)

1. Front Matter
Pages I-XVI
2. Georges Duvaut, Jacques Louis Lions
Pages 1-76
3. Georges Duvaut, Jacques Louis Lions
Pages 77-101
4. Georges Duvaut, Jacques Louis Lions
Pages 102-196
5. Georges Duvaut, Jacques Louis Lions
Pages 197-227
6. Georges Duvaut, Jacques Louis Lions
Pages 228-277
7. Georges Duvaut, Jacques Louis Lions
Pages 278-327
8. Georges Duvaut, Jacques Louis Lions
Pages 328-381
9. Back Matter
Pages 382-400

### Introduction

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.

### Keywords

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

#### Authors and affiliations

• Georges Duvaut
• 1
• Jacques Louis Lions
• 2
1. 1.Mécanique ThéoretiqueUniversité de Paris VIParisFrance
2. 2.Collège de FranceParisFrance

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-642-66165-5
• Copyright Information Springer-Verlag Berlin Heidelberg 1976
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Print ISBN 978-3-642-66167-9
• Online ISBN 978-3-642-66165-5
• Series Print ISSN 0072-7830