Maximum Principles and Sharp Constants for Solutions of Elliptic and Parabolic Systems ( Mathematical Surveys and Monographs )

Publication series ：Mathematical Surveys and Monographs

Author： Gershon Kresin;Vladimir Maz’ya  

Publisher： American Mathematical Society‎

Publication year： 2012

E-ISBN: 9780821891698

P-ISBN(Paperback): 9780821889817

Subject： O178 Inequality and Its him.

Keyword： Differential Equations

Language： ENG

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

Description

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.

Chapter

Cover

Title page

Contents

Introduction

Part I. Elliptic equations and systems

Prerequisites on operators acting into finite dimensional spaces

Maximum modulus principle for second order strongly elliptic systems

Sharp constants in the Miranda-Agmon inequalities for solutions of certain systems of mathematical physics

Sharp pointwise estimates for solutions of elliptic systems with boundary data from 𝐿^{𝑝}

Sharp constant in the Miranda-Agmon type inequality for derivatives of solutions to higher order elliptic equation

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

The norm and the essential norm for double layer vector-valued potentials

Part II. Parabolic systems

Maximum modulus principle for parabolic systems

Maximum modulus principle for parabolic systems with zero boundary data

Maximum norm principle for parabolic systems without lower order terms

Maximum norm principle with respect to smooth norms for parabolic systems

Bibliography

List of symbols

Index

Back Cover

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