This textbook is devoted to second order linear partial differential equations. The focus is on variational formulations in Hilbert spaces. It contains elliptic equations, including the biharmonic problem, some useful notes on functional analysis, a brief presentation of Sobolev spaces and their properties, some basic results on Fredholm alternative and spectral theory, saddle point problems, parabolic and linear Navier-Stokes equations, and hyperbolic and Maxwell equations. Almost 80 exercises are added, and the complete solution of all of them is included. The work is mainly addressed to students in Mathematics, but also students in Engineering with a good mathematical background should be able to follow the theory presented here. This second edition has been enriched by some new sections and new exercises; in particular, three important equations are now included: the biharmonic equation, the linear Navier-Stokes equations and the Maxwell equations.
Author(s): Alberto Valli
Series: UNITEXT 154
Edition: 2
Publisher: Springer Nature Switzerland
Year: 2023
Language: English
Pages: 262
City: Cham
Tags: Linear PDEs, Sobolev Spaces, Weak Solutions, Saddle Point Problems
Preface
Contents
1 Introduction
1.1 Examples of Linear Equations
1.2 Examples of Non-linear Equations
1.3 Examples of Systems
1.4 Exercises
2 Second Order Linear Elliptic Equations
2.1 Elliptic Equations
2.2 Weak Solutions
2.2.1 Two Classical Approaches
2.2.2 An Infinite Dimensional Linear System?
2.2.3 The Weak Approach
2.3 Lax–Milgram Theorem
2.4 Exercises
3 A Bit of Functional Analysis
3.1 Why Is Life in an Infinite Dimensional Normed Vector Space V Harder than in a Finite Dimensional One?
3.2 Why Is Life in a Hilbert Space Better than in a Pre-Hilbertian Space?
3.3 Exercises
4 Weak Derivatives and Sobolev Spaces
4.1 Weak Derivatives
4.2 Sobolev Spaces
4.3 Exercises
5 Weak Formulation of Elliptic PDEs
5.1 Weak Formulation of Boundary Value Problems
5.2 Boundedness of the Bilinear Form B(·,·) and the linear functional F(·)
5.3 Weak Coerciveness of the Bilinear Form B(·,·)
5.4 Coerciveness of the Bilinear Form B(·,·)
5.5 Interpretation of the Weak Problems
5.6 A Higher Order Example: The Biharmonic Operator
5.6.1 The Analysis of the Neumann Boundary ValueProblem
5.7 Exercises
6 Technical Results
6.1 Approximation Results
6.2 Poincaré Inequality in H10(D)
6.3 Trace Inequality
6.4 Compactness and Rellich Theorem
6.5 Other Poincaré Inequalities
6.6 du Bois-Reymond Lemma
6.7 f = 0 implies f = const
6.8 Exercises
7 Additional Results
7.1 Fredholm Alternative
7.2 Spectral Theory
7.3 Maximum Principle
7.4 Regularity Issues and Sobolev Embedding Theorems
7.4.1 Regularity Issues
7.4.2 Sobolev Embedding Theorems
7.5 Galerkin Numerical Approximation
7.6 Exercises
8 Saddle Points Problems
8.1 Constrained Minimization
8.1.1 The Finite Dimensional Case
8.1.2 The Infinite Dimensional Case
8.2 Galerkin Numerical Approximation
8.2.1 Error Estimates
8.2.2 Finite Element Approximation
8.3 Exercises
9 Parabolic PDEs
9.1 Variational Theory
9.2 Abstract Problem
9.2.1 Application to Parabolic PDEs
9.2.2 Application to Linear Navier–Stokes Equations for Incompressible Fluids
9.3 Maximum Principle for Parabolic Problems
9.4 Exercises
10 Hyperbolic PDEs
10.1 Abstract Problem
10.1.1 Application to Hyperbolic PDEs
10.1.2 Application to Maxwell Equations
10.2 Finite Propagation Speed
10.3 Exercises
A Partition of Unity
B Lipschitz Continuous Domains and Smooth Domains
C Integration by Parts for Smooth Functions and Vector Fields
D Reynolds Transport Theorem
E Gronwall Lemma
F Necessary and Sufficient Conditions for the Well-Posedness of the Variational Problem
References
Index
