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 Email: bartoluc (at) mat.uniroma2.it Email: daniele.bartolucci (at) uniroma2.eu 
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), EulerPoissonDarboux 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
