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E-raamat: Maximum Principles and Sharp Constants for Solutions of Elliptic and Parabolic Systems

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Maximum Principles and Sharp Constants for Solutions of Elliptic and Parabolic SystemsSuurem pilt
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The main goal of this book is to present results pertaining to various versions of the maximum principle for elliptic and parabolic systems of arbitrary order. In particular, the authors present necessary and sufficient conditions for validity of the classical maximum modulus principles for systems of second order and obtain sharp constants in inequalities of Miranda-Agmon type and in many other inequalities of a similar nature. Somewhat related to this topic are explicit formulas for the norms and the essential norms of boundary integral operators. The proofs are based on a unified approach using, on one hand, representations of the norms of matrix-valued integral operators whose target spaces are linear and finite dimensional, and, on the other hand, on solving certain finite dimensional optimization problems. This book reflects results obtained by the authors, and can be useful to research mathematicians and graduate students interested in partial differential equations.
Introduction
Part 1 Elliptic Equations and Systems
7(194)
Chapter 1 Prerequisites on Operators Acting into Finite Dimensional Spaces
9(12)
1.1 Introduction
9(1)
1.2 Linear bounded operators defined on spaces of continuous vector-valued functions and acting into Rm or Cm
10(7)
1.3 Linear bounded operators defined on Lebesgue spaces of vector-valued functions and acting into Rm or Cm
17(3)
1.4 Comments to
Chapter 1
20(1)
Chapter 2 Maximum Modulus Principle for Second Order Strongly Elliptic Systems
21(34)
2.1 Introduction
21(2)
2.2 Systems with constant coefficients without lower order terms
23(10)
2.3 General second order strongly elliptic systems
33(19)
2.4 Comments to
Chapter 2
52(3)
Chapter 3 Sharp Constants in the Miranda-Agmon Inequalities for Solutions of Certain Systems of Mathematical Physics
55(22)
3.1 Introduction
55(3)
3.2 Best constants in the Miranda-Agmon inequalities for solutions of strongly elliptic systems in a half-space
58(6)
3.3 The Lame and Stokes systems in a half-space
64(5)
3.4 Planar deformed state
69(2)
3.5 The system of quasistatic viscoelasticity
71(4)
3.6 Comments to
Chapter 3
75(2)
Chapter 4 Sharp Pointwise Estimates for Solutions of Elliptic Systems with Boundary Data from LP
77(16)
4.1 Introduction
77(2)
4.2 Best constants in pointwise estimates for solutions of strongly elliptic systems with boundary data from LP
79(4)
4.3 The Stokes system in a half-space
83(2)
4.4 The Stokes system in a ball
85(2)
4.5 The Lame system in a half-space
87(4)
4.6 The Lame system in a ball
91(1)
4.7 Comments to
Chapter 4
92(1)
Chapter 5 Sharp Constant in the Miranda-Agmon Type Inequality for Derivatives of Solutions to Higher Order Elliptic Equations
93(12)
5.1 Introduction
93(1)
5.2 Weak form of the Miranda-Agmon inequality with the sharp constant
94(4)
5.3 Sharp constants for biharmonic functions
98(6)
5.4 Comments to
Chapter 5
104(1)
Chapter 6 Sharp Pointwise Estimates for Directional Derivatives and Khavinson's Type Extremal Problems for Harmonic Functions
105(46)
6.1 Introduction
105(5)
6.2 Khavinson's type extremal problem for bounded or semibounded harmonic functions in a ball and a half-space
110(7)
6.3 Sharp estimates for directional derivatives and Khavinson's type extremal problem in a half-space with boundary data from LP
117(14)
6.4 Sharp estimates for directional derivatives and Khavinson's type extremal problem in a ball with boundary data from LP
131(14)
6.5 Sharp estimates for the gradient of a solution of the Neumann problem in a half-space
145(3)
6.6 Comments to
Chapter 6
148(3)
Chapter 7 The Norm and the Essential Norm for Double Layer Vector-Valued Potentials
151(50)
7.1 Introduction
151(3)
7.2 Definition and certain properties of a solid angle
154(7)
7.3 Matrix-valued integral operators of the double layer potential type
161(12)
7.4 Boundary integral operators of elasticity and hydrodynamics
173(24)
7.5 Comments to
Chapter 7
197(4)
Part 2 Parabolic Systems
201(96)
Chapter 8 Maximum Modulus Principle for Parabolic Systems
203(34)
8.1 Introduction
203(2)
8.2 The Cauchy problem for systems of order 2l
205(12)
8.3 Second order systems
217(13)
8.4 The parabolic Lame system
230(5)
8.5 Comments to
Chapter 8
235(2)
Chapter 9 Maximum Modulus Principle for Parabolic Systems with Zero Boundary Data
237(14)
9.1 Introduction
237(1)
9.2 The case of real coefficients
238(8)
9.3 The case of complex coefficients
246(3)
9.4 Comments to
Chapter 9
249(2)
Chapter 10 Maximum Norm Principle for Parabolic Systems without Lower Order Terms
251(26)
10.1 Introduction
251(4)
10.2 Some notation
255(1)
10.3 Representation of the constant K(Rn,T)
256(3)
10.4 Necessary condition for validity of the maximum norm principle for the system δu/δt - 2l0(x, t, Dx)u = 0
259(3)
10.5 Sufficient condition for validity of the maximum norm principle for the system δu/δt - 2l0(x, t, Dx)u = 0
262(2)
10.6 Necessary and sufficient condition for validity of the maximum norm principle for the system δu/δt - 2l0(x, Dx)u = 0
264(5)
10.7 Certain particular cases and examples
269(6)
10.8 Comments to
Chapter 10
275(2)
Chapter 11 Maximum Norm Principle with Respect to Smooth Norms for Parabolic Systems
277(20)
11.1 Introduction
277(3)
11.2 Representation for the constant K(Rn,T)
280(4)
11.3 Necessary condition for validity of the maximum norm principle for the system δu/δt - 2l(x, t, Dx)u = 0
284(4)
11.4 Sufficient condition for validity of the maximum norm principle for the system δu/δt - 2l(x, t, Dx)u = 0 with scalar principal part
288(3)
11.5 Criteria for validity of the maximum norm principle for the system δu/δt - 2l(x, Dx)u =
0. Certain particular cases
291(3)
11.6 Example: criterion for validity of the maximum p-norm principle, 2 < p < ∞
294(2)
11.7 Comments to
Chapter 11
296(1)
Bibliography 297(10)
List of Symbols 307(6)
Index 313
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