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Hardcover ISBN:  9780821889817 
Product Code:  SURV/183 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9780821891698 
Product Code:  SURV/183.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9780821889817 
eBook ISBN:  9780821891698 
Product Code:  SURV/183.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 183; 2012; 317 ppMSC: Primary 35; Secondary 31
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 MirandaAgmon 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 matrixvalued 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.
ReadershipGraduate students and research mathematicians interested in partial differential equations.

Table of Contents

Chapters

Introduction

Part 1. Elliptic equations and systems

1. Prerequisites on operators acting into finite dimensional spaces

2. Maximum modulus principle for second order strongly elliptic systems

3. Sharp constants in the MirandaAgmon inequalities for solutions of certain systems of mathematical physics

4. Sharp pointwise estimates for solutions of elliptic systems with boundary data from $L^p$

5. Sharp constant in the MirandaAgmon type inequality for derivatives of solutions to higher order elliptic equations

6. Sharp pointwise estimates for directional derivatives and Khavinson’s type extremal problems for harmonic functions

7. The norm and the essential norm for double layer vectorvalued potentials

Part 2. Parabolic systems

8. Maximum modulus principle for parabolic systems

9. Maximum modulus principle for parabolic systems with zero boundary data

10. Maximum norm principle for parabolic systems without lower order terms

11. Maximum norm principle with respect to smooth norms for parabolic systems


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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 MirandaAgmon 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 matrixvalued 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.
Graduate students and research mathematicians interested in partial differential equations.

Chapters

Introduction

Part 1. Elliptic equations and systems

1. Prerequisites on operators acting into finite dimensional spaces

2. Maximum modulus principle for second order strongly elliptic systems

3. Sharp constants in the MirandaAgmon inequalities for solutions of certain systems of mathematical physics

4. Sharp pointwise estimates for solutions of elliptic systems with boundary data from $L^p$

5. Sharp constant in the MirandaAgmon type inequality for derivatives of solutions to higher order elliptic equations

6. Sharp pointwise estimates for directional derivatives and Khavinson’s type extremal problems for harmonic functions

7. The norm and the essential norm for double layer vectorvalued potentials

Part 2. Parabolic systems

8. Maximum modulus principle for parabolic systems

9. Maximum modulus principle for parabolic systems with zero boundary data

10. Maximum norm principle for parabolic systems without lower order terms

11. Maximum norm principle with respect to smooth norms for parabolic systems