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

By: El-Maati Ouhabaz

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Hardcover | 10 January 2005

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296 Pages

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23.4 x 15.2 x 1.27

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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.

Industry Reviews

"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"

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Preface	p. ix
Notation	p. xiii
Sesquilinear Forms, Associated Operators, And Semigroups	p. 1
Bounded sesquilinear forms	p. 1
Unbounded sesquilinear forms and their associated operators	p. 3
Semigroups and unbounded operators	p. 18
Semigroups associated with sesquilinear forms	p. 29
Correspondence between forms, operators, and semigroups	p. 38
Contractivity Properties	p. 43
Invariance of closed convex sets	p. 44
Positive andL p -contractive semigroups	p. 49
Domination of semigroups	p. 58
Operations on the form-domain	p. 64
Semigroups acting on vector-valued functions	p. 68
Sesquilinear forms with nondense domains	p. 74
Inequalities For Sub-Markovian Semigroups	p. 79
Sub-Markovian semigroups and Kato type inequalities	p. 79
Further inequalities and the corresponding domain inL p	p. 88
L p -holomorphy of sub-Markovian semigroups	p. 95
Uniformly Elliptic Operators On Domains	p. 99
Examples of boundary conditions	p. 99
Positivity and irreducibility	p. 103
L 1 -contractivity	p. 107
The conservation property	p. 120
Domination	p. 125
L p -contractivity for 1	p. 134
Operators with unbounded coefficients	p. 137
Degenerate-Elliptic Operators	p. 143
Symmetric degenerate-elliptic operators	p. 144
Operators with terms of order 1	p. 145
Gaussian Upper Bounds For Heat Kernels	p. 155
Heat kernel bounds, Sobolev, Nash, and Gagliardo-Nirenberg inequalities	p. 155
Houml;lder-continuity estimates of the heat kernel	p. 160
Gaussian upper bounds	p. 163
Sharper Gaussian upper bounds	p. 174
Gaussian bounds for complex time andL p -analyticity	p. 180
Weighted gradient estimates	p. 185
Gaussian Upper Bounds Andl P -Spectral Theory	p. 193
L p -bounds and holomorphy	p. 196
L p -spectral independence	p. 204
Riesz means and regularization of the Schrouml;dinger group	p. 208
L p -estimates for wave equations	p. 214
Singular integral operators on irregular domains	p. 228
Spectral multipliers	p. 235
Riesz transforms associated with uniformly elliptic operators	p. 240
Gaussian lower bounds	p. 245
A Review Of The Kato Square Root Problem	p. 253
The problem in the abstract setting	p. 253
The Kato square root problem for elliptic operators	p. 257
Some consequences	p. 261
Bibliography	p. 265
Index	p. 283
Table of Contents provided by Publisher. All Rights Reserved.

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

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Hardcover

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