Semigroups of Bounded Operators and Second-Order Elliptic and Parabolic Partial Differential Equations aims to propose a unified approach to elliptic and parabolic equations with bounded and smooth coefficients. The book will highlight the connections between these equations and the theory of semigroups of operators, while demonstrating how the theory of semigroups represents a powerful tool to analyze general parabolic equations.
Features:
Useful for students and researchers as an introduction to the field of partial differential equations of elliptic and parabolic types
Introduces the reader to the theory of operator semigroups as a tool for the analysis of partial differential equations
Author(s): Luca Lorenzi, Abdelaziz Rhandi
Series: Monographs and Research Notes in Mathematics
Edition: 1
Publisher: CRC Press
Year: 2020
Language: English
Pages: 502
Tags: Operator Semigroups, Partial Differential Equations
Cover
Half Title
Series Page
Title Page
Copyright Page
Dedication
Contents
Symbol Description
Preface
Introduction
1.
Function Spaces
1.1. Spaces of (Hölder) Continuous Functions
1.1.1. Functions defined on the boundary of a smooth open set
1.2. Anisotropic and Parabolic Spaces of Hölder Continuous Functions
1.2.1. Anisotropic spaces of functions defined on the boundary of a set
1.3. Lp- and Sobolev Spaces
1.4.
Besov Spaces
1.5.
Exercises
I:
Semigroups of Bounded Operators
2.
Strongly Continuous Semigroups
2.1. Definitions and Basic Properties
2.2. The Infinitesimal Generator
2.3.
The Hille-Yosida, Lumer-Phillips and Trotter-Kato Theorems
2.4.
Nonhomogeneous Cauchy Problems
2.5.
Notes and Remarks
2.6.
Exercises
3.
Analytic Semigroups
3.1.
Prelude
3.2.
Sectorial Operators and Analytic Semigroups
3.3.
Interpolation Spaces
3.4.
Nonhomogeneous Cauchy Problems
3.5.
Notes
3.6.
Exercises
II:
Parabolic Equations
4.
Elliptic and Parabolic Maximum Principles
4.1.
The Parabolic Maximum Principles
4.1.1.
Parabolic weak maximum principle
4.1.2.
The strong maximum principle
4.2.
Elliptic Maximum Principles
4.3.
Notes
4.4.
Exercises
5. Prelude to Parabolic Equations: The Heat Equation and the Gauss-Weierstrass Semigroup in Cb(ℝd)
5.1.
The Homogeneous Heat Equation in ℝd. Classical Solutions: Existence and Uniqueness
5.2.
The Gauss-Weierstrass Semigroup
5.2.1. Estimates of the spatial derivatives of T(t)f
5.3. Two Equivalent Characterizations of Hölder Spaces
5.4.
Optimal Schauder Estimates
5.5. Notes
5.6.
Exercises
6. Parabolic Equations in ℝd
6.1.
The Continuity Method
6.2.
A priori Estimates
6.2.1.
Solving problem (6:0:1)
6.2.2.
Interior Schauder estimates for solutions to parabolic equations in domains: Part I
6.3.
More on the Cauchy Problem (6:0:1)
6.4.
The Semigroup Associated with the Operator A
6.4.1.
Interior Schauder estimates for solutions to parabolic equations in domains: Part II
6.5.
Higher-Order Regularity Results
6.6.
Notes
6.7.
Exercises
7. Parabolic Equations in ℝd+ with Dirichlet Boundary Conditions
7.1.
Technical Results
7.2.
An Auxiliary Boundary Value Problem
7.3.
Proof of Theorem 7.0.2 and a Corollary
7.4.
More on the Cauchy Problem (7:0:1)
7.5.
The Associated Semigroup
7.6.
Notes
7.7.
Exercises
8. Parabolic Equations in ℝd+ with More General Boundary Conditions
8.1.
A Priori Estimates
8.2.
Proof of Theorem 8.0.2
8.3. Interior Schauder Estimates for Solutions to Parabolic Equations in Domains: Part III
8.4.
More on the Cauchy Problem (8:0:1)
8.5.
The Associated Semigroup
8.6.
Exercises
9. Parabolic Equations in Bounded Smooth Domains Ω
9.1. Optimal Schauder Estimates for Solutions to Problems (9:0:1) and (9.0.2)
9.2. Interior Schauder Estimates for Solutions to Parabolic equations in Domains: Part IV
9.3. More on the Cauchy Problems (9:0:1) and (9:0:2)
9.4.
The Associated Semigroup
9.5.
Exercises
III:
Elliptic Equations
10. Elliptic Equations in ℝd
10.1. Solutions in Hölder Spaces
10.1.1.
The Laplace equation
10.1.2.
More general elliptic operators
10.1.3.
Further regularity results and interior estimates
10.2. Solutions in Lp(ℝd; ℂ) (p ε (1, ∞))
10.2.1. The Calderón-Zygmund inequality
10.2.2.
The Laplace equation
10.2.3.
More general elliptic operators
10.2.4.
Further regularity results and interior Lp-estimates
10.3. Solutions in L∞(ℝd; ℂ) and in Cb(ℝd; ℂ)
10.4.
Exercises
11. Elliptic Equations in ℝd+ with Homogeneous Dirichlet Boundary Conditions
11.1. Solutions in Hölder Spaces
11.1.1.
Further regularity results
11.2.
Solutions in Sobolev Spaces
11.2.1.
Further regularity results
11.3. Solutions in L∞(ℝd+; ℂ) and in Cb(ℝd+; ℂ)
11.4.
Exercises
12. Elliptic Equations in ℝd+ with General Boundary Conditions
12.1. The Cɑ-Theory
12.1.1.
Further regularity
12.2. Elliptic Equations in Lp(ℝd+; ℂ)
12.2.1.
Further regularity results
12.3. Solutions in L∞(ℝd+; ℂ) and in Cb(ℝd+; ℂ)
12.4.
Exercises
13. Elliptic Equations on Smooth Domains Ω
13.1. Elliptic Equations in Cɑ(Ω; ℂ)
13.1.1.
Further regularity results
13.2. Elliptic Equations in Lp(Ω; ℂ)
13.2.1.
Further regularity results
13.3. Solutions in L∞(Ω; ℂ), in C(Ω; ℂ) and in Cb(Ω; ℂ)
13.4.
Exercises
14.
Elliptic Operators and Analytic Semigroups
14.1. The Semigroup Cb(ℝd; ℂ)
14.2. The Semigroups in Cb(ℝd+; ℂ)
14.2.1.
Proof of Theorems 7.4.1 and 7.4.3
14.3. The Semigroups in Cb(Ω; ℂ)
14.4.
Exercises
15.
Kernel Estimates
15.1.
Dunford-Pettis Criterion and Ultracontractivity
15.2. Gaussian Estimates for Second-Order Elliptic Operators with Dirichlet
Boundary Conditions
15.3. Integral Representation for the Semigroups in Chapters 6, 7 and 9
15.4.
Notes
15.5.
Exercises
IV:
Appendices
A: Basic Notions of Functional Analysis in Banach Spaces
A.1.
Bounded and Closed Linear Operators
A.2.
Vector Valued Riemann Integral
A.3.
Holomorphic Functions
A.4.
Spectrum and Resolvent
A.5.
A Few Basic Notions from Interpolation Theory
A.5.1.
Marcinkiewicz's Interpolation Theorem
A.6.
Exercises
B:
Smooth Domains and Extension Operators
B.1.
Partition of Unity
B.2.
Smooth Domains
B.3.
Traces of Functions in Sobolev Spaces
B.4.
Extension Operators
B.4.1.
Extending functions de ned on open sets
B.4.2.
Extending functions de ned on the boundary of a set
Bibliography
Index
