Analysis of Heat Equations on Domains. (LMS-30)
El-Maati Ouhabaz

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Chapter 1 [in PDF format]

TABLE OF CONTENTS:

Preface ix
Notation xiii Chapter 1. SESQUILINEAR FORMS, ASSOCIATED OPERATORS, AND SEMIGROUPS 1
1.1 Bounded sesquilinear forms 1
1.2 Unbounded sesquilinear forms and their associated operators 3
1.3 Semigroups and unbounded operators 18
1.4 Semigroups associated with sesquilinear forms 29
1.5 Correspondence between forms, operators, and semigroups 38
Chapter 2. CONTRACTIVITY PROPERTIES 43
2.1 Invariance of closed convex sets 44
2.2 Positive and L^p-contractive semigroups 49
2.3 Domination of semigroups 58
2.4 Operations on the form-domain 64
2.5 Semigroups acting on vector-valued functions 68
2.6 Sesquilinear forms with nondense domains 74
Chapter 3. INEQUALITIES FOR SUB-MARKOVIAN SEMIGROUPS 79
3.1 Sub-Markovian semigroups and Kato type inequalities 79
3.2 Further inequalities and the corresponding domain in L^p 88
3.3 L^p-holomorphy of sub-Markovian semigroups 95
Chapter 4. UNIFORMLY ELLIPTIC OPERATORS ON DOMAINS 99
4.1 Examples of boundary conditions 99
4.2 Positivity and irreducibility 103
4.3 L¹-contractivity 107
4.4 The conservation property 120
4.5 Domination 125
4.6 L^p-contractivity for 1 < p < 134
4.7 Operators with unbounded coefficients 137
Chapter 5. DEGENERATE-ELLIPTIC OPERATORS 143
5.1 Symmetric degenerate-elliptic operators 144
5.2 Operators with terms of order 1 145
Chapter 6. GAUSSIAN UPPER BOUNDS FOR HEAT KERNELS 155
6.1 Heat kernel bounds, Sobolev, Nash, and Gagliardo-Nirenberg inequalities 155
6.2 Hölder-continuity estimates of the heat kernel 160
6.3 Gaussian upper bounds 163
6.4 Sharper Gaussian upper bounds 174
6.5 Gaussian bounds for complex time and L^p-analyticity 180
6.6 Weighted gradient estimates 185
Chapter 7. GAUSSIAN UPPER BOUNDS AND L^p-SPECTRAL THEORY 193
7.1 L^p-bounds and holomorphy 196
7.2 L^p-spectral independence 204
7.3 Riesz means and regularization of the Schrödinger group 208
7.4 L^p-estimates for wave equations 214
7.5 Singular integral operators on irregular domains 228
7.6 Spectral multipliers 235
7.7 Riesz transforms associated with uniformly elliptic operators 240
7.8 Gaussian lower bounds 245
Chapter 8. A REVIEW OF THE KATO SQUARE ROOT PROBLEM 253
8.1 The problem in the abstract setting 253
8.2 The Kato square root problem for elliptic operators 257
8.3 Some consequences 261
Bibliography 265
Index 283

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