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Library | Item Barcode | Call Number | Material Type | Item Category 1 | Status |
|---|---|---|---|---|---|
Searching... | 30000010082237 | QA377 O93 2005 | Open Access Book | Book | Searching... |
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Summary
Summary
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, Schrödinger, 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.
Author Notes
El-Maati Ouhabaz is Professor of Analysis and Geometry at Université Bordeaux 1
Table of Contents
| Preface | p. ix |
| Notation | p. xiii |
| Chapter 1 Sesquilinear Forms, Associated Operators, And Semigroups | p. 1 |
| 1.1 Bounded sesquilinear forms | p. 1 |
| 1.2 Unbounded sesquilinear forms and their associated operators | p. 3 |
| 1.3 Semigroups and unbounded operators | p. 18 |
| 1.4 Semigroups associated with sesquilinear forms | p. 29 |
| 1.5 Correspondence between forms, operators, and semigroups | p. 38 |
| Chapter 2 Contractivity Properties | p. 43 |
| 2.1 Invariance of closed convex sets | p. 44 |
| 2.2 Positive andL p -contractive semigroups | p. 49 |
| 2.3 Domination of semigroups | p. 58 |
| 2.4 Operations on the form-domain | p. 64 |
| 2.5 Semigroups acting on vector-valued functions | p. 68 |
| 2.6 Sesquilinear forms with nondense domains | p. 74 |
| Chapter 3 Inequalities For Sub-Markovian Semigroups | p. 79 |
| 3.1 Sub-Markovian semigroups and Kato type inequalities | p. 79 |
| 3.2 Further inequalities and the corresponding domain inL p | p. 88 |
| 3.3 L p -holomorphy of sub-Markovian semigroups | p. 95 |
| Chapter 4 Uniformly Elliptic Operators On Domains | p. 99 |
| 4.1 Examples of boundary conditions | p. 99 |
| 4.2 Positivity and irreducibility | p. 103 |
| 4.3 L 1 -contractivity | p. 107 |
| 4.4 The conservation property | p. 120 |
| 4.5 Domination | p. 125 |
| 4.6 L p -contractivity for 1 | p. 134 |
| 4.7 Operators with unbounded coefficients | p. 137 |
| Chapter 5 Degenerate-Elliptic Operators | p. 143 |
| 5.1 Symmetric degenerate-elliptic operators | p. 144 |
| 5.2 Operators with terms of order 1 | p. 145 |
| Chapter 6 Gaussian Upper Bounds For Heat Kernels | p. 155 |
| 6.1 Heat kernel bounds, Sobolev, Nash, and Gagliardo-Nirenberg inequalities | p. 155 |
| 6.2 Houml;lder-continuity estimates of the heat kernel | p. 160 |
| 6.3 Gaussian upper bounds | p. 163 |
| 6.4 Sharper Gaussian upper bounds | p. 174 |
| 6.5 Gaussian bounds for complex time andL p -analyticity | p. 180 |
| 6.6 Weighted gradient estimates | p. 185 |
| Chapter 7 Gaussian Upper Bounds Andl P -Spectral Theory | p. 193 |
| 7.1 L p -bounds and holomorphy | p. 196 |
| 7.2 L p -spectral independence | p. 204 |
| 7.3 Riesz means and regularization of the Schrouml;dinger group | p. 208 |
| 7.4 L p -estimates for wave equations | p. 214 |
| 7.5 Singular integral operators on irregular domains | p. 228 |
| 7.6 Spectral multipliers | p. 235 |
| 7.7 Riesz transforms associated with uniformly elliptic operators | p. 240 |
| 7.8 Gaussian lower bounds | p. 245 |
| Chapter 8 A Review Of The Kato Square Root Problem | p. 253 |
| 8.1 The problem in the abstract setting | p. 253 |
| 8.2 The Kato square root problem for elliptic operators | p. 257 |
| 8.3 Some consequences | p. 261 |
| Bibliography | p. 265 |
| Index | p. 283 |
