Università Roma Tre

1. Laplace's equation

Mean-value inequalities, maximum and minimum principle, the Harnack inequality, the Green representation, the Poisson integral, convergence theorems, interior estimates of derivatives, the Dirichlet problem and the method of sub-harmonic functions

2. The classical maximum principle

Weak maximum principle, strong maximum principle, a-priori bounds, symmetry properties and the method of moving planes

3. Poisson's equation and the Newtonian potential

Hölder continuity, the Dirichlet problem for Poisson's equation, Hölder estimates for second derivatives, estimates at the boundary, Hölder estimates for the first derivatives

4. Classical solutions: the Schauder approach

Schauder interior estimates, boundary and global estimates, the Dirichlet problem, interior and boundary regularity

"Elliptic Partial Differential Equations of Second Order", by David Gilbarg and Neil S. Trudinger, Classics in Mathematics,volume 224, Springer-Verlag Berlin Heidelberg, 2nd Edition, 2001

"Elliptic Partial Differential Equations: Second Edition", by Qing Han and Fanghua Lin, Courant Lecture Notes, volume 1, AMS American Mathematical Society, 2nd Edition, 2011

"Partial Differential Equations: Second Edition", by Lawrence C. Evans, Graduate Studies in Mathematics, volume 19, AMS American Mathematical Society, 2010