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

A.A. 2023/2024

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

D.Bartolucci Home Page 

Monday Tuesday Friday
14:00 - 16:00 09:00 - 11:00 09:00 - 11:00
Aula 29
Aula D'Antoni Aula 22

There will be three lessons of two hours each a week, starting Monday March 04. Lecture notes of the course will be available. The Lectures will be delivered in presence, possibly in mixed (online) form if needed.

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.