Università Roma Tre

Preliminaries
- Weak topologies and weak convergence, weak lower semi-continuity of the norm
- L^P spaces: reflexivity, separability, criteria for strong compactness.

Sobolev spaces and variational formulation of boundary value problems in dimension one
- Motivations
- The Sobolev space W^{1,p} (I)
- The space W^{1,p}_0 (I)
- Some examples of boundary value problems
- Maximum principle

Sobolev spaces and variational formulation of boundary value problems in dimension N
- Definition and basic properties of the Sobolev spaces W^{1,p} (Ω)
- Extension operators
- Sobolev inequalities
- The space W^{1,p}_0 (Ω)
- Variational formulation of some elliptic boundary value problems
- Existence of weak solutions
- Regularity of weak solutions
- Maximum principle

Analisi funzionale, H. Brézis, Liguori Editore