© 1976

Inequalities in Mechanics and Physics


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

  1. 1.Mécanique ThéoretiqueUniversité de Paris VIParisFrance
  2. 2.Collège de FranceParisFrance

Bibliographic information

  • Book Title Inequalities in Mechanics and Physics
  • Authors G. Duvant
    J. L. Lions
  • Series Title Grundlehren der mathematischen Wissenschaften
  • DOI
  • Copyright Information Springer-Verlag Berlin Heidelberg 1976
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Hardcover ISBN 978-3-540-07327-7
  • Softcover ISBN 978-3-642-66167-9
  • eBook ISBN 978-3-642-66165-5
  • Series ISSN 0072-7830
  • Edition Number 1
  • Number of Pages XVI, 400
  • Number of Illustrations 0 b/w illustrations, 0 illustrations in colour
  • Topics Mathematics, general
  • Buy this book on publisher's site
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