Università Roma Tre

1. Preliminaries
Definition of hyper-surface. Integration on hyper-surfaces. The divergence theorem.

2. The Laplace equation
Mean value inequalities. Minimum and maximum principle. The Harnack inequality. The Green representation. The Poisson integral. Convergence theorems. Interior estimates on the derivatives. The Dirichlet problem:
the method of sub-harmonic functions.

3. The classical maximum principle
The weak maximum principle. The strong maximum principle. The Hopf lemma.

4. The Poisson equation and the Newtonian potential
Hölder-continuity. The Dirichlet problem for the Poisson equation. Hölder estimates for second derivatives. Boundary estimates. Hölder estimates for first derivatives.

5. Banach and Hilbert spaces
The contraction principle. The continuity method.

6. Classical solutions: the Schauder approach
Schauder interior estimates. Boundary and global estimates. The Dirichlet problem. Boundary and interior regularity.

“Elliptic partial differential equations of second order. Reprint of the 1998 edition”, D. Gilbarg e N.S. Trudinger. Classics in Mathematics. Springer-Verlag, Berlin, 2001.