Degree course: 
Corso di First cycle degree in MATHEMATICS
Academic year when starting the degree: 
Academic year in which the course will be held: 
Course type: 
Compulsory subjects, characteristic of the class
First Semester
Standard lectures hours: 
Detail of lecture’s hours: 
Lesson (48 hours), Exercise (12 hours)

Basis of integral and differential calculus in one or several variables. The notions of Mathematical Analysis I and II will be taken for granted. It is recommended that the students will know the topics of Mathematical Analysis III.

Final Examination: 

Oral examination. The exam consists of a discussion on the partial differential equations presented during the course, and on the methods used to solve the associated problems. The purpose of the exam is to verify: the level of knowledge and deepening of the topics addressed; the full understanding of the solving techniques and of the properties of the solutions; the ability to state theorems and expose proofs in a mathematically rigorous way; the ability to discuss the examples presented in class. The student who aspires to excellence must be able to: justify all the assumptions in the statements of the theorems and each individual step of the proofs; discuss several particular cases that are not part of the general theory; use all the solving techniques presented during the course. The mark is expressed in thirtieths

Voto Finale

Partial differential equations describe a wide range of natural phenomena, among which: vibrations of solids; propagation of acoustic and electromagnetic waves, heat diffusion, dynamics of quantum particles.
The goal of the course is to give an introduction to the theory of partial differential equations. Three fundamental equations of mathematical physics will be analyzed: the wave equation, the heat equation, and the Poisson equation. Physical motivations for the study of these equations will be provided. The description of natural phenomena leads to the study of several typical problems associated to the equations: problems with initial and/or boundary data. Using methods from functional analysis, these problems will be solved at a rigorous mathematical level.
At the end of the course the students will be able to:
- obtain representation formulae for the solutions of the problems associated to the three partial differential equations;
- state and prove theorems concerning the existence, uniqueness, and stability of the solutions;
- discuss the fundamental properties of the solutions;
- give examples of physical applications.

The course is essentially divided in four, mutually connected parts:
1. Heat, wave, and Poisson equation in the whole space;
2. Harmonic functions and Poisson equation in bounded domains;
3. Distributions;
4. Heat and wave equation in bounded domains. Eigenvalues and eigenvectors.
The detailed program of the course is the following:
- Transport equation.
- Classification of second order partial differential equations.
- The wave equation on the real line.
-- Derivation from physical considerations. The D’Alembert formula. Causality. Conservation of the energy. Wave equation on the half-line.
- Heat equation on the real line.
-- Derivation from physical considerations. Fundamental solution and solution of the Cauchy problem. Maximum principle. Uniqueness and stability.
- Heat equation in R^n.
- Divergence theorem and Green identities.
- Wave equation in R^n.
-- Spherical means and Euler-Poisson-Darboux equation. Kirchhoff formula. The method of descent. Causality and uniqueness.
- Duhamel principle.
- Poisson equation in R^n.
- Laplace equation and Harmonic functions.
-- Fundamental solution. Mean value theorem. Liouville theorem. Uniqueness.
- Poisson equation on bounded domains.
-- Maximum principle. Green function. Symmetry of the Green function. Green function for the half-space. Green function for the ball. Dirichlet principle. Green function for the Neumann problem.
- Distributions.
-- Test functions and distributions. Functions in Schwartz class. Fourier transform of functions in Schwartz class. Tempered distributions. Operations with distributions: derivation and Fourier transform. Fundamental solutions.
- Fourier series.
-- Heat and wave equation on an interval. Separation of variables. Point convergence of the Fourier series. Solution of the heat equation, uniform convergence. Solution of the heat equation with continuous initial data. Solution of the wave equation.
- Heat equation and wave equation in bounded domains. Eigenvalues and eigenfunctions.

- L.C. Evans, Partial Differential Equations, American Mathematical Society.
- R. S. Strichartz, A Guide to Distribution Theory and Fourier Transforms, World Scientific Publishing Company, 2003.
- W. Strauss, Partial differential equations: An Introduction, Wiley&Sons.

The course consists of 48 hours of frontal theoretical lectures and 24 hours of frontal complementary lectures. During the complementary lectures the teacher will discuss several examples and exercises. Class attendance is not mandatory, but it is strongly recommended.
Due to the Covid-19 emergency, most probably teaching will be on-line. The lectures will be delivered through Microsoft Teams.