# Inequalities in Mechanics and Physics

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

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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.

## Authors and Affiliations

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

Georges Duvaut

• ### Collège de France, Paris, France

Jacques Louis Lions

## Bibliographic Information

• Book Title: Inequalities in Mechanics and Physics

• Authors: Georges Duvaut, Jacques Louis Lions

• Series Title:

• DOI: https://doi.org/10.1007/978-3-642-66165-5

• Publisher: Springer Berlin, Heidelberg

• eBook Packages:

• Copyright Information: Springer-Verlag Berlin Heidelberg 1976

• Softcover ISBN: 978-3-642-66167-9Published: 15 November 2011

• eBook ISBN: 978-3-642-66165-5Published: 06 December 2012

• Series ISSN: 0072-7830

• Series E-ISSN: 2196-9701

• Edition Number: 1

• Number of Pages: XVI, 400

• Topics: