Elliptic Equations

The aim of this book is to introduce the reader to different topics of the theory of elliptic partial differential equations by avoiding technicalities and refinements. Apart from the basic theory of equations in divergence form it includes subjects such as singular perturbation problems, homogenization, computations, asymptotic behaviour of problems in cylinders, elliptic systems, nonlinear problems, regularity theory, Navier-Stokes system, p-Laplace equation. Just a minimum on Sobolev spaces has been introduced, and work or integration on the boundary has been carefully avoided to keep the reader's attention on the beauty and variety of these issues. The chapters are relatively independent of each other and can be read or taught separately. Numerous results presented here are original and have not been published elsewhere. The book will be of interest to graduate students and faculty members specializing in partial differential equations.

Preface	p. ix
Basic Techniques
Hilbert Space Techniques
The projection on a closed convex set	p. 3
The Riesz representation theorem	p. 6
The Lax-Milgram theorem	p. 8
Convergence techniques	p. 10
Exercises	p. 11
A Survey of Essential Analysis
Lp-techniques	p. 13
Introduction to distributions	p. 18
Sobolev Spaces	p. 22
Exercises	p. 32
Weak Formulation of Elliptic Problems
Motivation	p. 35
The weak formulation	p. 38
Exercises	p. 41
Elliptic Problems in Divergence Form
Weak formulation	p. 43
The weak maximum principle	p. 49
Inhomogeneous problems	p. 53
Exercises	p. 54
Singular Perturbation Problems
A prototype of a singular perturbation problem	p. 57
Anisotropic singular perturbation problems	p. 61
Exercises	p. 69
Problems in Large Cylinders
A model problem	p. 73
Another type of convergence	p. 79
The general case	p. 82
An application	p. 86
Exercises	p. 89
Periodic Problems
A general theory	p. 93
Some additional remarks	p. 101
Exercises	p. 103
Homogenization
More on periodic functions	p. 106
Homogenization of elliptic equations	p. 109
The one-dimensional case	p. 109
The n-dimensional case	p. 112
Exercises	p. 119
Eigenvalues
The one-dimensional case	p. 121
The higher-dimensional case	p. 123
An application	p. 127
Exercises	p. 128
Numerical Computations
The finite difference method	p. 129
The finite element method	p. 135
Exercises	p. 147
More Advanced Theory
Nonlinear Problems
Monotone methods	p. 153
Quasilinear equations	p. 160
Nonlocal problems	p. 166
Variational inequalities	p. 170
Exercises	p. 174
L&infinitive;-estimates
Some simple cases	p. 177
A more involved estimate	p. 180
The Sobolev-Gagliardo-Nirenberg inequality	p. 183
The maximum principle on small domains	p. 187
Exercises	p. 188
Linear Elliptic Systems
The general framework	p. 191
Some examples	p. 197
Exercises	p. 202
The Stationary Navier-Stokes System
Introduction	p. 203
Existence and uniqueness result	p. 205
Exercises	p. 208
Some More Spaces
Motivation	p. 211
Essential features of the Sobolev spaces Wk,p	p. 212
An application	p. 215
Exercises	p. 216
Regularity Theory
Introduction	p. 217
The translation method	p. 221
Regularity of functions in Sobolev spaces	p. 224
The bootstrap technique	p. 227
Exercises	p. 229
The p-Laplace Equation
A minimization technique	p. 231
A weak maximum principle and its consequences	p. 237
A generalization of the Lax-Milgram theorem	p. 239
Exercises	p. 245
The Strong Maximum Principle
A first version of the maximum principle	p. 247
The Hopf maximum principle	p. 252
Application: the moving plane technique	p. 255
Exercises	p. 259
Problems in the Whole Space
The harmonic functions, Liouville theorem	p. 261
The Schröet;dinger equation	p. 267
Exercises	p. 273
Appendix: Fixed Point Theorems
The Brouwer fixed point theorem	p. 275
The Schauder fixed point theorem	p. 279
Exercises	p. 280
Bibliography	p. 281
Index	p. 287
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