96758 - PDEs

Learning outcomes

At the end of the course, students will be able to study linear PDEs of first and second order, mainly by classical methods. This knowledge is fundamental to all the theoretical and modelling. applications.

Course contents

Generalities on PDEs

1. What is a PDE?
2. Examples of PDEs
3. Classification of second order PDEs
4. Well-posed problems

First-order PDEs

1. Linear equations with constant coefficients
2. The Method of the Characteristics
3. Existence and Uniqueness for the Cauchy Problem

Second order PDEs: Harmonic functions

1. The Laplace operator. Harmonic functions
2. Harmonic functions in open subsets of R2
3. Some integral identities
4. Radial harmonic functions in RN, N ≥ 2
5. The fundamental solution of the Laplacian and a representation formula
6. Mean Value Theorems for harmonic functions
7. Surface and solid average operators for continuous functions
8. Mean Value properties imply harmonicity
9. Some convergence theorems
10. The weak Laplacian
11. The Harnack inequality
12. Monotone sequences of harmonic functions
13. Strong maximum principle and boundary estimates for harmonic functions
14. Analyticity of the harmonic functions
15. Liouville Theorems
16. Maximum Principles for linear second order PDOs with nonnegative characteristic form

The Dirichlet problem for the Laplace operator: the Perron method

1. Introduction
2. Preliminaries: the Green function
3. The Green functions for the Euclidean ball
4. The Poisson kernel for the Euclidean ball
5. The solution of the Dirichlet problem on the Euclidean balls
6. Superharmonic functions
7. The Perron-Wiener solution of the Dirichlet problem
   
8. Boundary behavior of the Perron-Wiener solution

The Heat operator

1. The Heat operator. Caloric functions
2. Fundamental solution of H. Solvability of the Cauchy problem
3. Green identity for H
4. Some representation formulas in terms of H
5. Smoothness of caloric functions and some convergence theorems
6. Weak caloric functions
7. Mean value Theorem for caloric functions
8. Reverse of the Mean Value Property
9. The caloric strong Maximum Principle
10. The weak caloric Harnack inequality
11. Monotone sequences of caloric functions
12. The caloric Harnack inequality
13. The parabolic weak maximum principle for the heat operator
14. Uniqueness for the Cauchy problem
15. Representation theorems on strips
16. Liouville theorems for caloric functions

The wave operator

1. The wave operator
2. The Cauchy problem for the wave equation in R×]0, ∞[. D’Alembert formula.
3. Some properties of the surface average. Darboux formula.
4. The Cauchy problem for the wave equation n R3 ×]0, ∞[. Kirchhoff formula.
5. Energy estimate and uniqueness for compactly supported data.

Readings/Bibliography

Lawrence C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Volume 19 American Mathematical Society.

Teaching methods

The course consists of lessons describing the fundamental concepts of the program. Lessons are completed with examples illuminating the theoretical content.

Assessment methods

Oral exam

Teaching tools

Notes of the teacher on virtuale [http://virtuale.unibo.it/]

Office hours

See the website of Eleonora Cinti