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 some basic results on Fredholm alternative and spectral theory, some useful notes on functional analysis, a brief presentation of Sobolev spaces and their properties, saddle point problems, parabolic equations and hyperbolic equations. Many 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.
Author(s): Alberto Valli
Series: UNITEXT 126
Edition: 1
Publisher: Springer Nature Switzerland
Year: 2021
Language: English
Pages: 235
Tags: Partial Differential Equations, PDE
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 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 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
9.3 Maximum Principle for Parabolic Problems
9.4 Exercises
10 Hyperbolic PDEs
10.1 Abstract Problem
10.1.1 Application to Hyperbolic PDEs
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
