Problems on Partial Differential Equations (Problem Books in Mathematics)

Description

This book covers a diverse range of topics in Mathematical Physics, linear and nonlinear PDEs. Though the text reflects the classical theory, the main emphasis is on introducing readers to the latest developments based on the notions of weak solutions and Sobolev spaces. In numerous problems, the student is asked to prove a given statement, e.g. to show the existence of a solution to a certain PDE. Usually there is no closed-formula answer available, which is why there is no answer section, although helpful hints are often provided.   This textbook offers a valuable asset for students and educators alike. As it adopts a perspective on PDEs that is neither too theoretical nor too practical, it represents the perfect companion to a broad spectrum of courses. Preface Introduction The Content of the Chapters About the Authors Notation Contents 1 Preliminaries 1.1 Integration by Parts 1.1.1 Theoretical Background Basic Formulas Surface Integrals A Quick Review of Differential Forms 1.1.2 Worked-Out Problems 1.1.3 Problems 1.2 Convolutions 1.2.1 Theoretical Background 1.2.2 Worked-Out Problems 1.2.3 Problems 1.3 Bibliographical Remarks 2 Distributions, Sobolev Spaces and the Fourier Transform 2.1 The Fourier Transform 2.1.1 Theoretical Background The Schwartz Space The Fourier Transform Properties of the Fourier Transform Parseval's Identity and Plancherel's Theorem 2.1.2 Worked-Out Problems 2.1.3 Problems 2.2 The Theory of Distributions 2.2.1 Theoretical Background Distributional Derivatives Tempered Distributions Convolutions Fundamental Solutions 2.2.2 Worked-Out Problems 2.2.3 Problems 2.3 Sobolev Spaces 2.3.1 Theoretical Background Sobolev Spaces Wm,p Spaces Hs(Rn) Spaces Hs(Ω) Duality and the Spaces W-m,p'(Ω) Trace of a Sobolev Function Embedding Theorems 2.3.2 Worked-Out Problems 2.3.3 Problems 2.4 Bibliographical Remarks 3 Common Methods 3.1 Weak Convergence 3.1.1 Theoretical Background Weak Convergence in Hilbert Spaces Dual Spaces Weak and Weak* Convergence in Banach Spaces Banach Space-Valued Functions 3.1.2 Worked-Out Problems 3.1.3 Problems 3.2 The Separation of Variables Method 3.2.1 Theoretical Background Hyperbolic Problems Comments on the Regularity of u Given by Formula (3.21) Hyperbolic Problems: A Non-homogeneous Case Parabolic Problems 3.2.2 Worked-Out Problems 3.2.3 Problems Hyperbolic Equations Parabolic Equations 3.3 Galerkin Method 3.3.1 Theoretical Background 3.3.2 Worked-Out Problems 3.3.3 Problems 3.4 Bibliographical Remarks 4 Elliptic Equations 4.1 Classical Theory of Harmonic Functions 4.1.1 Theoretical Background Definitions Green's Functions Holomorphic Functions and Harmonic Functions Subharmonic Functions General Elliptic Operators 4.1.2 Worked-Out Problems The Laplace Operator in Different Coordinate Systems Green's Functions Harmonic and Holomorphic Functions 4.1.3 Problems Harmonic Functions: Basic Properties The Laplace Operator in Different Coordinate Systems Green's Functions Harmonic and Holomorphic Functions Subharmonic Functions 4.2 Weak Solutions 4.2.1 Theoretical Background Physical Motivation Weak Formulation Variational Approach on Hilbert Spaces Lax–Milgram's Lemma Regularity of Weak Solutions 4.2.2 Worked-Out Problems 4.2.3 Problems 4.3 Bibliographical Remarks 5 Evolution Equations 5.1 First-Order Equations and the Method of Characteristics 5.1.1 Theoretical Background The Characteristic System Admissible Boundary Conditions Noncharacteristic Initial Conditions Local Solutions 5.1.2 Worked-Out Problems 5.1.3 Problems 5.2 Hyperbolic Problems 5.2.1 Theoretical Background Duhamel's Formula 5.2.2 Worked-Out Problems The Wave Equation Duhamel's Formula Hyperbolic Systems of the First Order The Use of the Fourier Transform Energy Estimates and Uniqueness 5.2.3 Problems The Wave Equation Hyperbolic Systems of the First Order Hyperbolic Equations of the Second Order, Considered More Generally The Use of the Fourier Transform Energy Estimates and Uniqueness 5.3 Parabolic Equations 5.3.1 Theoretical Background Heat Equation Linear Second-Order Parabolic Equations Weak Solutions 5.3.2 Worked-Out Problems 5.3.3 Problems 5.4 Bibliographical Remarks Bibliography