Inequalities in Mechanics and Physics

  • Georges Duvaut
  • Jacques Louis Lions

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

Table of contents

  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 197-227
  5. Georges Duvaut, Jacques Louis Lions
    Pages 228-277
  6. Georges Duvaut, Jacques Louis Lions
    Pages 278-327
  7. Georges Duvaut, Jacques Louis Lions
    Pages 328-381
  8. Back Matter
    Pages 382-400

About this book


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

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
  • Copyright Information Springer-Verlag Berlin Heidelberg 1976
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-66167-9
  • Online ISBN 978-3-642-66165-5
  • Series Print ISSN 0072-7830
  • Buy this book on publisher's site
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