Grundlehren der mathematischen Wissenschaften

# Inequalities in Mechanics and Physics

Authors: Duvant, G., Lions, J. L.

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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.

• Problems of Semi-Permeable Media and of Temperature Control

Duvaut, Georges (et al.)

Pages 1-76

• Problems of Heat Control

Duvaut, Georges (et al.)

Pages 77-101

• Classical Problems and Problems with Friction in Elasticity and Visco-Elasticity

Duvaut, Georges (et al.)

Pages 102-196

• Unilateral Phenomena in the Theory of Flat Plates

Duvaut, Georges (et al.)

Pages 197-227

• Introduction to Plasticity

Duvaut, Georges (et al.)

Pages 228-277

### Buy this book

eBook $89.00 price for USA in USD (gross) • ISBN 978-3-642-66165-5 • Digitally watermarked, DRM-free • Included format: PDF • ebooks can be used on all reading devices • Immediate eBook download after purchase Softcover$119.99
price for USA in USD
• ISBN 978-3-642-66167-9
• Free shipping for individuals worldwide
• Usually dispatched within 3 to 5 business days.

## Bibliographic Information

Bibliographic Information
Book Title
Inequalities in Mechanics and Physics
Authors
Translated by
John, C.W.
Series Title
Grundlehren der mathematischen Wissenschaften
Series Volume
219
1976
Publisher
Springer-Verlag Berlin Heidelberg
Springer-Verlag Berlin Heidelberg
eBook ISBN
978-3-642-66165-5
DOI
10.1007/978-3-642-66165-5
Softcover ISBN
978-3-642-66167-9
Series ISSN
0072-7830
Edition Number
1
Number of Pages
XVI, 400
Topics