Variational Methods for Problems from Plasticity Theory and for Generalized Newtonian Fluids
Variational methods are applied to prove the existence of weak solutions for boundary value problems from the deformation theory of plasticity as well as for the slow, steady state flow of generalized Newtonian fluids including the Bingham and Prandtl-Eyr
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1749
Springer Berlin Heidelberg New York Barcelona Hong Kong London Milan Paris Singapore Tokyo
Martin Fuchs Gregory Seregin
Variational Methods for Problems from Plasticity Theory and for Generalized Newtonian Fluids
Springer
Authors Martin Fuchs Universitat des Saarlandes Fachrichtung 6.1 Mathematik Postfach 151150 66041 Saarbriicken, Germany e-mail: [email protected] Gregory Seregin V.A. Steklov Mathematical Institute St. Petersburg Branch Fontanka 27 1910 11 St. Petersburg, Russia e-mail: [email protected]
Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Fuchs, Martin: Variational methods for problems from plasticity theory and for generalized Newtonian fluids I Martin Fuchs; Gregory Seregin. Berlin; Heidelberg; New York; Barcelona; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 2000 (Lecture notes in mathematics ; 1749) ISBN 3-540-41397-9
Mathematics Subject Classification (2000): 74, 74G40, 74G65, 76A05, 76M30, 49N15, 49N60, 35Q ISSN 0075- 8434 ISBN 3-540-41397-9 Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science-Business Media GmbH © Springer-Verlag Berlin Heidelberg 2000 Printed in Germany Typesetting: Camera-ready TEX output by the author SPIN: 10734279 41/3142-543210 - Printed on acid-free paper
Contents Introduction
1
1 Weak solutions to boundary value problems in the deformation theory of perfect elastoplasticity 5 1.0
Preliminaries . . . . . . . . . . .
5
1.1
The classical boundary value problem for the equilibrium state of a perfect elastoplastic body and its primary functional formulation
6
1.2 1.3
Relaxation of convex variational problems in non reflexive spaces. General construction . . . . . . . . . . . . . . . . . . . . . . . ..
15
Weak solutions to variational problems of perfect elastoplasticity
27
2 Differentiability properties of weak solutions to boundary value problems in the deformation theory of plasticity 40 2.0
Preliminaries
.
40
2.1
Formulation of the main results
42
2.2
Approximation and proof of Lemma 2.1.1
52
2.3
Proof of Theorem 2.1.1 and a local estimate of Caccioppoli-type for the stress tensor
57
Estimates for solutions of certain systems of PDE's with constant coefficients. . . . . . . . . . . . . .
71
2.5
The main lemma and its iteration.
76
2.6
Proof of Theorem 2.1.2 .
89
2.7
Open Problems . . . . .
98
2.8
Rema
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