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) mat.uniroma2.it E-mail: 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), 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
|