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Analysis of Heat Equations on Domains. (LMS-31) : London Mathematical Society Monographs - El-Maati Ouhabaz

Analysis of Heat Equations on Domains. (LMS-31)

London Mathematical Society Monographs

Hardcover Published: 31st October 2004
ISBN: 9780691120164
Number Of Pages: 296

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This is the first comprehensive reference published on heat equations associated with non self-adjoint uniformly elliptic operators. The author provides introductory materials for those unfamiliar with the underlying mathematics and background needed to understand the properties of heat equations. He then treats "Lp" properties of solutions to a wide class of heat equations that have been developed over the last fifteen years. These primarily concern the interplay of heat equations in functional analysis, spectral theory and mathematical physics.

This book addresses new developments and applications of Gaussian upper bounds to spectral theory. In particular, it shows how such bounds can be used in order to prove "Lp" estimates for heat, Schrodinger, and wave type equations. A significant part of the results have been proved during the last decade.

The book will appeal to researchers in applied mathematics and functional analysis, and to graduate students who require an introductory text to sesquilinear form techniques, semigroups generated by second order elliptic operators in divergence form, heat kernel bounds, and their applications. It will also be of value to mathematical physicists. The author supplies readers with several references for the few standard results that are stated without proofs.

"This book is both an excellent introduction for those learning about heat operators for the first time, and a reference work for the mathematician searching for information. The author has presented an especially lucid exposition of the subject." - Alan McIntosh, Australian National University; "This book contains very interesting material, starting with the basics and progressing to lively trends of current research." - Thierry Coulhon, Cergy-Pontoise University"

Prefacep. ix
Notationp. xiii
Sesquilinear Forms, Associated Operators, And Semigroupsp. 1
Bounded sesquilinear formsp. 1
Unbounded sesquilinear forms and their associated operatorsp. 3
Semigroups and unbounded operatorsp. 18
Semigroups associated with sesquilinear formsp. 29
Correspondence between forms, operators, and semigroupsp. 38
Contractivity Propertiesp. 43
Invariance of closed convex setsp. 44
Positive andL p -contractive semigroupsp. 49
Domination of semigroupsp. 58
Operations on the form-domainp. 64
Semigroups acting on vector-valued functionsp. 68
Sesquilinear forms with nondense domainsp. 74
Inequalities For Sub-Markovian Semigroupsp. 79
Sub-Markovian semigroups and Kato type inequalitiesp. 79
Further inequalities and the corresponding domain inL pp. 88
L p -holomorphy of sub-Markovian semigroupsp. 95
Uniformly Elliptic Operators On Domainsp. 99
Examples of boundary conditionsp. 99
Positivity and irreducibilityp. 103
L 1 -contractivityp. 107
The conservation propertyp. 120
Dominationp. 125
L p -contractivity for 1p. 134
Operators with unbounded coefficientsp. 137
Degenerate-Elliptic Operatorsp. 143
Symmetric degenerate-elliptic operatorsp. 144
Operators with terms of order 1p. 145
Gaussian Upper Bounds For Heat Kernelsp. 155
Heat kernel bounds, Sobolev, Nash, and Gagliardo-Nirenberg inequalitiesp. 155
Houml;lder-continuity estimates of the heat kernelp. 160
Gaussian upper boundsp. 163
Sharper Gaussian upper boundsp. 174
Gaussian bounds for complex time andL p -analyticityp. 180
Weighted gradient estimatesp. 185
Gaussian Upper Bounds Andl P -Spectral Theoryp. 193
L p -bounds and holomorphyp. 196
L p -spectral independencep. 204
Riesz means and regularization of the Schrouml;dinger groupp. 208
L p -estimates for wave equationsp. 214
Singular integral operators on irregular domainsp. 228
Spectral multipliersp. 235
Riesz transforms associated with uniformly elliptic operatorsp. 240
Gaussian lower boundsp. 245
A Review Of The Kato Square Root Problemp. 253
The problem in the abstract settingp. 253
The Kato square root problem for elliptic operatorsp. 257
Some consequencesp. 261
Bibliographyp. 265
Indexp. 283
Table of Contents provided by Publisher. All Rights Reserved.

ISBN: 9780691120164
ISBN-10: 0691120161
Series: London Mathematical Society Monographs
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 296
Published: 31st October 2004
Publisher: Princeton University Press
Country of Publication: US
Dimensions (cm): 23.5 x 15.2  x 1.85
Weight (kg): 0.54