â" Goes to Plus Infinity

By: Michel Chipot

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Hardcover | 1 December 2001

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196 Pages

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23.5 x 17.15 x 1.27

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This graduate textbook accompanies the reader starting from the basics in the theory of elliptic problems to recent research topics concerning the asymptotic theory in cylindrical domains where many physical problems are settled. It is designed as a resource for graduates and researchers in applied mathematics and for engineers. Many physical problems are meaningfully formulated in a cylindrical domain. When the size of the cylinder goes to infinity, the solutions, under certain symmetry conditions, are expected to be identical in every cross-section of the domain. The proof of this, however, is sometimes difficult and almost never given in the literature. This book aims to partially fill this gap by providing proofs of the asymptotic behaviour of solutions to various important cases of linear and non-linear problems in the theory of elliptic and parabolic partial differential equations. Many results presented here are original and have not been published elsewhere. They are included with the aim of motivating and enabling the reader to apply the theory to other problems in partial differential equations.

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Preface
Introduction to Linear Elliptic Problems	p. 1
The Lax-Milgram theorem	p. 1
Elementary notions on Sobolev spaces	p. 3
Applications to linear elliptic problems	p. 8
Some Model Techniques	p. 17
The case of lateral Dirichlet boundary conditions on a rectangle	p. 17
The case of lateral Neumann boundary conditions on a rectangle	p. 21
The case of lateral Dirichlet boundary conditions revisited	p. 29
A different point of view	p. 33
A General Asymptotic Theory for Linear Elliptic Problems	p. 37
A general convergence result in H[superscript 1](actual symbol not reproducible])	p. 40
A sharper rate of convergence	p. 45
Convergence in higher Sobolev spaces	p. 52
Nonlinear Elliptic Problems	p. 59
Variational inequalities	p. 59
Quasilinear elliptic problems	p. 66
Strongly nonlinear problems	p. 74
Asymptotic Behaviour of some Nonlinear Elliptic Problems	p. 77
The case of variational inequalities	p. 77
The case of quasilinear problems	p. 84
Elliptic Systems	p. 101
Some inequalities	p. 101
Existence results for linear elliptic systems	p. 116
Nonlinear elliptic systems	p. 125
Asymptotic Behaviour of Elliptic Systems	p. 127
The case of linear elliptic systems satisfying the Legendre condition	p. 127
The system of elasticity	p. 131
Parabolic Equations	p. 137
Functional spaces for parabolic problems	p. 137
Linear parabolic problems	p. 142
Nonlinear parabolic problems	p. 146
Asymptotic Behaviour of Parabolic Problems	p. 153
The linear case	p. 153
A nonlinear case	p. 162
Concluding Remark	p. 177
Bibliography	p. 179
Index	p. 181
Table of Contents provided by Blackwell. All Rights Reserved.

â" Goes to Plus Infinity

At a Glance

Hardcover

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