Elliptic Equations: An Introductory Course
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
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chel Chipot
Elliptic Equations: An Introductory Course
Birkhäuser Basel · Boston · Berlin
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Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
Part I Basic Techniques 1 Hilbert Space Techniques 1.1 The projection on a closed convex set . 1.2 The Riesz representation theorem . . . 1.3 The Lax–Milgram theorem . . . . . . . 1.4 Convergence techniques . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . .
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3 6 8 10 11
2 A Survey of Essential Analysis 2.1 Lp -techniques . . . . . . . . 2.2 Introduction to distributions 2.3 Sobolev Spaces . . . . . . . Exercises . . . . . . . . . . . . . .
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13 18 22 32
3 Weak Formulation of Elliptic Problems 3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The weak formulation . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35 38 41
4 Elliptic Problems in Divergence Form 4.1 Weak formulation . . . . . . . . 4.2 The weak maximum principle . 4.3 Inhomogeneous problems . . . . Exercises . . . . . . . . . . . . . . . .
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43 49 53 54
5 Singular Perturbation Problems 5.1 A prototype of a singular perturbation problem . . . . . . . . . . 5.2 Anisotropic singular perturbation problems . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57 61 69
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