Analysis of heat equations on domains

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. H...

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Hlavní autor:	Ouhabaz, El Maati
Médium:	E-kniha Kniha
Jazyk:	angličtina
Vydáno:	Princeton, N.J Princeton University Press 2005
Vydání:	1
Edice:	London Mathematical Society monographs
Témata:	Adjoint Analytic semigroup Approximation Boundary value problem Bounded operator Boundedness Cauchy problem Characterization (mathematics) Coefficient Complex number Convex set Differential operator Dirichlet boundary condition Ellipse Elliptic operator Equation Estimation Euclidean space Existential quantification Functional analysis Heat Heat > Transmission > Measurement Heat equation Heat kernel Hermitian adjoint Hilbert space Inverse trigonometric functions Invertible matrix Irreducibility (mathematics) Lebesgue measure Lie group Linearity Lp space Martin Bridson Mathematical physics MATHEMATICS MATHEMATICS / Applied Measure (mathematics) Metric space Mixed boundary condition Neumann boundary condition Open set Partial differential equation Peter Sarnak Physics Pointwise Probability Resolvent set Ricci curvature Riemannian manifold Riesz representation theorem Riesz transform SCIENCE / Physics / Mathematical & Computational Self-adjoint Self-adjoint operator Semigroup Sesquilinear form Sign (mathematics) Singular integral Smoothness Sobolev inequality Special case Spectral theory Square root Stochastic calculus Theorem Theory Time derivative Upper and lower bounds Variable (mathematics) Weight function Open set Convex set Adjoint Boundedness Bounded operator Partial differential equation Self-adjoint operator Hermitian adjoint Invertible matrix Characterization (mathematics) Hilbert space Sesquilinear form Resolvent set Mathematical physics Time derivative Approximation Estimation Heat kernel Analytic semigroup Neumann boundary condition Semigroup Riesz representation theorem Mixed boundary condition Riemannian manifold Existential quantification Cauchy problem Boundary value problem Weight function Linearity Lie group Special case Lp space Martin Bridson Sign (mathematics) Ellipse Metric space Heat equation Theory Riesz transform Peter Sarnak Mathematics Dirichlet boundary condition C0 Upper and lower bounds Lebesgue measure Singular integral Pointwise Theorem Euclidean space Sobolev inequality Inverse trigonometric functions Coefficient Square root Probability Equation Functional analysis Smoothness Complex number Variable (mathematics) Spectral theory Self-adjoint Stochastic calculus Ricci curvature Elliptic operator Differential operator Measure (mathematics) Irreducibility (mathematics)
ISBN:	9781400826483, 9780691120164, 0691120161, 1400826489
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Analysis of heat equations on domains (LMS-31) -- CONTENTS -- PREFACE -- NOTATION -- CHAPTER ONE: SESQUILINEAR FORMS, ASSOCIATED OPERATORS, AND SEMIGROUPS -- CHAPTER TWO: CONTRACTIVITY PROPERTIES -- CHAPTER THREE: INEQUALITIES FOR SUB-MARKOVIAN SEMIGROUPS -- CHAPTER FOUR: UNIFORMLY ELLIPTIC OPERATORS ON DOMAINS -- CHAPTER FIVE: DEGENERATE-ELLIPTIC OPERATORS -- CHAPTER SIX: GAUSSIAN UPPER BOUNDS FOR HEAT KERNELS -- CHAPTER SEVEN: GAUSSIAN UPPER BOUNDS AND Lp-SPECTRAL THEORY -- CHAPTER EIGHT: A REVIEW OF THE KATO SQUARE ROOT PROBLEM -- BIBLIOGRAPHY -- INDEX.
Front Matter Table of Contents Preface Notation Chapter One: SESQUILINEAR FORMS, ASSOCIATED OPERATORS, AND SEMIGROUPS Chapter Two: CONTRACTIVITY PROPERTIES Chapter Three: INEQUALITIES FOR SUB-MARKOVIAN SEMIGROUPS Chapter Four: UNIFORMLY ELLIPTIC OPERATORS ON DOMAINS Chapter Five: DEGENERATE-ELLIPTIC OPERATORS Chapter Six: GAUSSIAN UPPER BOUNDS FOR HEAT KERNELS Chapter Seven: GAUSSIAN UPPER BOUNDS AND ${L^p}$-SPECTRAL THEORY Chapter Eight: A REVIEW OF THE KATO SQUARE ROOT PROBLEM Bibliography Index
Cover -- Contents -- Preface -- Notation -- Chapter 1. SESQUILINEAR FORMS, ASSOCIATED OPERATORS, AND SEMIGROUPS -- 1.1 Bounded sesquilinear forms -- 1.2 Unbounded sesquilinear forms and their associated operators -- 1.3 Semigroups and unbounded operators -- 1.4 Semigroups associated with sesquilinear forms -- 1.5 Correspondence between forms, operators, and semigroups -- Chapter 2. CONTRACTIVITY PROPERTIES -- 2.1 Invariance of closed convex sets -- 2.2 Positive and L[sup(p)]-contractive semigroups -- 2.3 Domination of semigroups -- 2.4 Operations on the form-domain -- 2.5 Semigroups acting on vector-valued functions -- 2.6 Sesquilinear forms with nondense domains -- Chapter 3. INEQUALITIES FOR SUB-MARKOVIAN SEMIGROUPS -- 3.1 Sub-Markovian semigroups and Kato type inequalities -- 3.2 Further inequalities and the corresponding domain in L[sup(p)] -- 3.3 L[sup(p)]-holomorphy of sub-Markovian semigroups -- Chapter 4. UNIFORMLY ELLIPTIC OPERATORS ON DOMAINS -- 4.1 Examples of boundary conditions -- 4.2 Positivity and irreducibility -- 4.3 L&amp -- #8734 -- -contractivity -- 4.4 The conservation property -- 4.5 Domination -- 4.6 L[sup(p)]-contractivity for 1 &lt -- p &lt -- &amp -- #8734 -- -- 4.7 Operators with unbounded coefficients -- Chapter 5. DEGENERATE-ELLIPTIC OPERATORS -- 5.1 Symmetric degenerate-elliptic operators -- 5.2 Operators with terms of order 1 -- Chapter 6. GAUSSIAN UPPER BOUNDS FOR HEAT KERNELS -- 6.1 Heat kernel bounds, Sobolev, Nash, and Gagliardo-Nirenberg inequalities -- 6.2 H&amp -- #246 -- lder-continuity estimates of the heat kernel -- 6.3 Gaussian upper bounds -- 6.4 Sharper Gaussian upper bounds -- 6.5 Gaussian bounds for complex time and L[sup(p)]-analyticity -- 6.6 Weighted gradient estimates -- Chapter 7. GAUSSIAN UPPER BOUNDS AND L[sup(p)]-SPECTRAL THEORY -- 7.1 L[sup(p)]-bounds and holomorphy
7.2 L[sup(p)]-spectral independence -- 7.3 Riesz means and regularization of the Schr&amp -- 246 -- dinger group -- 7.4 L[sup(p)]-estimates for wave equations -- 7.5 Singular integral operators on irregular domains -- 7.6 Spectral multipliers -- 7.7 Riesz transforms associated with uniformly elliptic operators -- 7.8 Gaussian lower bounds -- Chapter 8. A REVIEW OF THE KATO SQUARE ROOT PROBLEM -- 8.1 The problem in the abstract setting -- 8.2 The Kato square root problem for elliptic operators -- 8.3 Some consequences -- Bibliography -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- N -- O -- P -- Q -- R -- S -- U -- W
Chapter Six. Gaussian Upper Bounds for Heat Kernels
Chapter Seven. Gaussian Upper Bounds and Lp-Spectral Theory
Index
Notation
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Chapter Five. Degenerate-Elliptic Operators
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Contents
Chapter Three. Inequalities for Sub-Markovian Semigroups
Chapter Eight. A Review of the Kato Square Root Problem
Frontmatter --
Chapter Two. Contractivity Properties
Chapter Four. Uniformly Elliptic Operators on Domains
Preface
Chapter One. Sesquilinear Forms, Associated Operators, and Semigroups
Bibliography