Introduction to PDE
PhD School of Mathematics
Universities of Rome "Sapienza", "Tor Vergata", "Tre"

A.A. 2024/2025


Prof. D. Bartolucci
Department of Mathematics
University of Rome "Tor Vergata"
Room 1107, Flat 1 - A1
Tel: 0672594689
E-mail: bartoluc (at) mat.uniroma2.it
E-mail: daniele.bartolucci (at) uniroma2.eu

D.Bartolucci Home Page 

Monday Tuesday Thursday
11:00 - 13:00 11:00 - 13:00 11:00 - 13:00
Aula 1200
Aula 1200 Aula 1200

There will be three lessons of two hours each a week, starting Monday March 03. Lecture notes of the course will be available. The Lectures will be delivered in presence, possibly in mixed (online, Teams platform) form if needed.
PLEASE CONTACT THE TEACHER VIA MAIL FOR ANY FURTHER INFORMATION.



Program of the course


- Laplace and Poisson equations. Harmonic functions. Fundamental solutions.

- Mean value formulas. Maximum principles, uniqueness. Mollifiers, convolutions and smoothing.

- Regularity and local estimates for harmonic functions. The Liouville Theorem, classification of solutions of the Poisson equation in R^N, N >= 2.

- The Harnack inequality for harmonic functions. The Green function. The Green function on a ball. The Poisson Kernel.

- Variational (Energy) methods. The Dirichlet principle.

- The Heat equation. The fundamental solution. The Cauchy problem for the homogeneous and non homogeneous equation. Mean value formula and the heat ball

- Maximum principle for the heat equation. Uniqueness. Regularity of solutions of the heat equation.

- Transport equations. The Wave equation. D’Alambert formula (N=1), Euler-Poisson-Darboux equation, Kirchoff’s formula (N=3). Descent method, Poisson’s formula(N=2). Nonhomogeneous wave equations, retarded potentials. Energy methods, finite speed propagation.



Textbooks and Lecture Notes

  • L.C. Evans, Partial Differential Equations. Second Edition. American Mathematical Society 2010.

  • D. Bartolucci, Lecture Notes of the course.