**Mathematical Surveys and Monographs**

Volume: 183;
2012;
317 pp;
Hardcover

MSC: Primary 35;
Secondary 31

Print ISBN: 978-0-8218-8981-7

Product Code: SURV/183

List Price: $96.00

Individual Member Price: $76.80

**Electronic ISBN: 978-0-8218-9169-8
Product Code: SURV/183.E**

List Price: $96.00

Individual Member Price: $76.80

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#### Supplemental Materials

# Maximum Principles and Sharp Constants for Solutions of Elliptic and Parabolic Systems

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*Gershon Kresin; Vladimir Maz’ya*

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.

#### Table of Contents

# Table of Contents

## Maximum Principles and Sharp Constants for Solutions of Elliptic and Parabolic Systems

- Cover Cover11 free
- Title page iii4 free
- Contents v6 free
- Introduction 110 free
- Part I. Elliptic equations and systems 716 free
- Prerequisites on operators acting into finite dimensional spaces 918
- Maximum modulus principle for second order strongly elliptic systems 2130
- Sharp constants in the Miranda-Agmon inequalities for solutions of certain systems of mathematical physics 5564
- Sharp pointwise estimates for solutions of elliptic systems with boundary data from 𝐿^{𝑝} 7786
- Sharp constant in the Miranda-Agmon type inequality for derivatives of solutions to higher order elliptic equation 93102
- Sharp pointwise estimates for directional derivatives and Khavinson’s type extremal problems for harmonic functions 105114
- The norm and the essential norm for double layer vector-valued potentials 151160
- Part II. Parabolic systems 201210
- Maximum modulus principle for parabolic systems 203212
- Maximum modulus principle for parabolic systems with zero boundary data 237246
- Maximum norm principle for parabolic systems without lower order terms 251260
- Maximum norm principle with respect to smooth norms for parabolic systems 277286
- Bibliography 297306
- List of symbols 307316
- Index 313322 free
- Back Cover Back Cover1330

#### Readership

Graduate students and research mathematicians interested in partial differential equations.