Some notes of the classes
For all students
A - Introduction
B - Classification of (quasi-)linear second order PDEs
C - Some recalls (divergence theorem)
D - Definizione di spazio di Sobolev e qualche esempio di funzione di Sobolev in dimensione 1
D - Definizione di spazio di Sobolev e qualche esempio di funzione di Sobolev in dimensione maggiore di 1
E - A short introduction to distributions
G - ''Variational nature'' of Laplace equations
H - The fundamental solution
I - Properties of harmonic functions
L - Green function on the ball and half space, Harnack's inequality, Liouville's theorem
N - The Dirichlet problem in a general domain
Examples of regular boundary points - Wiener criterion and a very brief mention to the Dirichlet problem in external domains
Alcuni altri esercizi proposti
The gradient and the Laplacian in polar coordinates
Optional material, just for curiosity
D - Continuity equation or equation of conservation of mass
G - ''Variational nature'' of Laplace equations: the example due to Hadamard
L - An example of a function that is not C
2
and whose Laplacian is continuous