Elements of Nonlinear Analysis

by Chipot, Michel

• ISBN13: 9783764364069
• ISBN10: 3764364068
• Format: Hardcover
• Copyright: 2001-01-01
• Publisher: Birkhauser

Summary

This textbook explores the vast field of nonlinear analysis by emphasizing the underlying ideas rather than the sophisticated refinements of the theory. Two classical examples from physics, namely elasticity and diffusion, serve to motivate the theoretical parts that are then applied to various aspects of elliptic and parabolic problems. In particular, existence, uniqueness, regularity and approximation of solutions for quasilinear and monotone problems are studied, as well as some new aspects of the calculus of variations including Young measures or approximation of minimizing sequences. The book is reasonably self-contained. Wherever possible, original proofs are given that are not to be found elsewhere. The text is geared towards graduate students and nonspecialists in nonlinear analysis who wish to become acquainted with the basic ideas of the subject. The study of this book will enable the reader to access the many ramifications of the field.

Table of Contents

Preface	vii
Some Physical Motivations	1	(14)
An elementary theory of elasticity	1	(8)
A problem in biology	9	(5)
Exercises	14	(1)
A Short Background in Functional Analysis	15	(24)
An introduction to distributions	15	(3)
Integration on boundaries	18	(3)
Introduction to Sobolev spaces	21	(16)
Exercises	37	(2)
Elliptic Linear Problems	39	(10)
The Dirichlet problem	39	(2)
The Lax-Milgram theorem and its applications	41	(7)
Exercises	48	(1)
Elliptic Variational Inequalities	49	(10)
A generalization of the Lax-Milgram theorem	49	(3)
Some applications	52	(6)
Exercises	58	(1)
Nonlinear Elliptic Problems	59	(26)
A compactness method	59	(3)
A monotonicity method	62	(4)
A generalization of variational inequalities	66	(5)
Some multivalued problems	71	(11)
Exercises	82	(3)
A Regularity Theory for Nonlocal Variational Inequalities	85	(10)
Some general results	85	(6)
Applications to second order variational inequalities	91	(3)
Exercises	94	(1)
Uniqueness and Nonuniqueness Issues	95	(10)
Uniqueness result for local nonlinear problems	95	(4)
Nonuniqueness issues	99	(3)
Exercises	102	(3)
Finite Element Methods for Elliptic Problems	105	(26)
An abstract setting	105	(1)
Some simple finite elements	106	(7)
Interpolation error	113	(3)
Convergence results	116	(6)
Approximation of nonlinear problems	122	(7)
Exercises	129	(2)
Minimizers	131	(14)
Introduction	131	(2)
The direct method	133	(2)
Applications	135	(7)
The Euler Equation	142	(1)
Exercises	143	(2)
Minimizing Sequences	145	(40)
Some model problems	145	(2)
Young measures	147	(3)
Construction of the minimizing sequences	150	(4)
A more elaborate issue	154	(12)
Numerical analysis of oscillations	166	(16)
Exercises	182	(3)
Linear Parabolic Equations	185	(22)
Introduction	185	(1)
Functional analysis for parabolic problems	185	(6)
The resolution of parabolic problems	191	(7)
Applications	198	(7)
Exercises	205	(2)
Nonlinear Parabolic Problems	207	(14)
Local problems	207	(8)
Nonlocal problems	215	(5)
Exercises	220	(1)
Asymptotic Analysis	221	(30)
The case of one stationary point	221	(1)
The case of several stationary points	222	(1)
A nonlinear case	223	(18)
Blow-up	241	(7)
Exercises	248	(3)
Bibliography	251	(4)
Index	255