96758 - PDEs

Course Unit Page

SDGs

This teaching activity contributes to the achievement of the Sustainable Development Goals of the UN 2030 Agenda.

Quality education Partnerships for the goals

Academic Year 2021/2022

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, in person or online. Students should register on alma esami  with at least two days of advance.

Teaching tools

Notes of the teacher in virtuale

Office hours

See the website of Annamaria Montanari