Contents

Introduction 1

Part 1. Elliptic Equations and Systems 7

Chapter 1. Prerequisites on Operators Acting into Finite Dimensional

Spaces 9

1.1. Introduction 9

1.2. Linear bounded operators defined on spaces of continuous vector-

valued functions and acting into

Rm

or

Cm

10

1.3. Linear bounded operators defined on Lebesgue spaces of vector-valued

functions and acting into

Rm

or

Cm

17

1.4. Comments to Chapter 1 20

Chapter 2. Maximum Modulus Principle for Second Order Strongly

Elliptic Systems 21

2.1. Introduction 21

2.2. Systems with constant coeﬃcients without lower order terms 23

2.3. General second order strongly elliptic systems 33

2.4. Comments to Chapter 2 52

Chapter 3. Sharp Constants in the Miranda-Agmon Inequalities for

Solutions of Certain Systems of Mathematical Physics 55

3.1. Introduction 55

3.2. Best constants in the Miranda-Agmon inequalities for solutions of

strongly elliptic systems in a half-space 58

3.3. The Lam´ e and Stokes systems in a half-space 64

3.4. Planar deformed state 69

3.5. The system of quasistatic viscoelasticity 71

3.6. Comments to Chapter 3 75

Chapter 4. Sharp Pointwise Estimates for Solutions of Elliptic

Systems with Boundary Data from

Lp

77

4.1. Introduction 77

4.2. Best constants in pointwise estimates for solutions of strongly elliptic

systems with boundary data from

Lp

79

4.3. The Stokes system in a half-space 83

4.4. The Stokes system in a ball 85

4.5. The Lam´ e system in a half-space 87

4.6. The Lam´ e system in a ball 91

4.7. Comments to Chapter 4 92

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