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  m or  m .......... 10
1.3 Linear bounded operators defined on Lebesgue spaces of
vector-valued functions and acting into m or m .......... 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 Lame 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 Lame system in a half-space ........................... 87
4.6 The Lame system in a ball ................................. 91
4.7 Comments to Chapter 4 ..................................... 92
Chapter 5 Sharp Constant in the Miranda-Agmon Type
Inequality for Derivatives of Solutions to Higher Order
Elliptic Equations ............................................. 93
5.1 Introduction .............................................. 93
5.2 Weak form of the Miranda-Agmon inequality with the sharp
constant .................................................. 94
5.3 Sharp constants for biharmonic functions .................. 98
5.4 Comments to Chapter 5 .................................... 104
Chapter 6 Sharp Pointwise Estimates for Directional
Derivatives and Khavinson's Type Extremal Problems for
Harmonic Functions ............................................ 105
6.1 Introduction ............................................. 105
6.2 Khavinson's type extremal problem for bounded or
semibounded harmonic functions in a ball and
a half-space ............................................. 110
6.3 Sharp estimates for directional derivatives and
Khavinson's type extremal problem in a half-space with
boundary data from LP .................................... 117
6.4 Sharp estimates for directional derivatives and
Khavinson's type extremal problem in a ball with
boundary data from LP .................................... 131
6.5 Sharp estimates for the gradient of a solution of the
Neumann problem in a half-space .......................... 145
6.6 Comments to Chapter 6 .................................... 148
Chapter 7 The Norm and the Essential Norm for Double
Layer Vector-Valued Potentials ................................ 151
7.1 Introduction ............................................. 151
7.2 Definition and certain properties of a solid angle ....... 154
7.3 Matrix-valued integral operators of the double layer
potential type ........................................... 161
7.4 Boundary integral operators of elasticity and
hydrodynamics ............................................ 173
7.5 Comments to Chapter 7 .................................... 197
Part 2 Parabolic Systems ..................................... 201
Chapter 8 Maximum Modulus Principle for Parabolic Systems .... 203
8.1 Introduction ............................................. 203
8.2 The Cauchy problem for systems of order 2ℓ ............... 205
8.3 Second order systems ..................................... 217
8.4 The parabolic Lamé system ................................ 230
8.5 Comments to Chapter 8 .................................... 235
Chapter 9 Maximum Modulus Principle for Parabolic Systems
with Zero Boundary Data ....................................... 237
9.1 Introduction ............................................. 237
9.2 The case of real coefficients ............................ 238
9.3 The case of complex coefficients ......................... 246
9.4 Comments to Chapter 9 .................................... 249
Chapter 10 Maximum Norm Principle for Parabolic Systems
without Lower Order Terms ..................................... 251
10.1 Introduction ............................................. 251
10.2 Some notation ............................................ 255
10.3 Representation of the constant  (n, T) ................. 256
10.4 Necessary condition for validity of the maximum norm
principle for the system ∂u/∂t —  0(x, t, Dx)u = 0 ...... 259
10.5 Sufficient condition for validity of the maximum norm
principle for the system ∂u/∂t — 0(x, t, Dx)u = 0 ...... 262
10.6 Necessary and sufficient condition for validity of the
maximum norm principle for the system ∂u/∂t —
0(х, Dx)u = 0 .......................................... 264
10.7 Certain particular cases and examples .................... 269
10.8 Comments to Chapter 10 ................................... 275
Chapter 11 Maximum Norm Principle with Respect to Smooth
Norms for Parabolic Systems ................................... 277
11.1 Introduction ............................................. 277
11.2 Representation for the constant (n, T) ................ 280
11.3 Necessary condition for validity of the maximum norm
principle for the system ∂u/∂t — 0(x, t, Dx)u = 0 ...... 284
11.4 Sufficient condition for validity of the maximum norm
principle for the system ∂u/∂t — 0(x, t, Dx)u = 0 with
scalar principal part .................................... 288
11.5 Criteria for validity of the maximum norm principle for
the system ∂u/∂t — 0(x, t, Dx)u = 0. Certain particular
cases .................................................... 291
11.6 Example: criterion for validity of the maximum p-norm
principle, 2 < p < ∞ ..................................... 294
11.7 Comments to Chapter 11 ................................... 296
Bibliography .................................................. 297
List of Symbols ............................................... 307
Index ......................................................... 313
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