Semi-Bounded Differential Operators, Contractive Semigroups and Beyond

ISBN-13: 9783319045573 / Angielski / Twarda / 2014 / 252 str.

Alberto Cialdea; Vladimir Maz'ya

Opis

​In the present book the conditions are studied for the semi-boundedness of partial differential operators which is interpreted in different ways. Nowadays one knows rather much about L²-semibounded differential and pseudo-differential operators, although their complete characterization in analytic terms causes difficulties even for rather simple operators. Until recently almost nothing was known about analytic characterizations of semi-boundedness for differential operators in other Hilbert function spaces and in Banach function spaces. The goal of the present book is to partially fill this gap. Various types of semi-boundedness are considered and some relevant conditions which are either necessary and sufficient or best possible in a certain sense are given. Most of the results reported in this book are due to the authors.

Recenzje

"This book is valuable; it contains a lot of new information and deep, complicated proofs. ... it is a very good book, and every serious research university library should get it. I expect it to inspire new research." (Jerome A. Goldstein, Bulletin of the American Mathematical Society, Vol. 55 (1), January, 2018)

"The book is logically ordered and clearly written ... . It will be of interest to specialists and graduate students working on partial differential operators in function spaces." (Hector O. Fattorini, zbMATH 1317.47002, 2015)

Spis treści

Introduction.- 1 Preliminary facts on semi-boundedness of forms and operators.- 2 Lp-dissipativity of scalar second order operators with complex coefficients.- 3 Elasticity system.- 4 Lp-dissipativity for systems of partial differential operators.- 5 The angle of Lp-dissipativity.- 6 Higher order differential operators in Lp.- 7 Weighted positivity and other related results.- References.

Blurb

This book examines the conditions for the semi-boundedness of partial differential operators, which are interpreted in different ways. For example, today we know a great deal about L2-semibounded differential and pseudodifferential operators, although their complete characterization in analytic terms still poses difficulties, even for fairly simple operators. In contrast, until recently almost nothing was known about analytic characterizations of semi-boundedness for differential operators in other Hilbert function spaces and in Banach function spaces. This book works to address that gap. As such, various types of semi-boundedness are considered and a number of relevant conditions which are either necessary and sufficient or best possible in a certain sense are presented. The majority of the results reported on are the authors’ own contributions.