Università Roma Tre

Basics of functional analysis (normed spaces, Hilbert spaces, Banach spaces, linear and limited hearts).
Lp spaces (completeness, duality. Hilbert's space L2).
Regularization and approximation through smooth functions: convolution, approximate delta.
Weak derivatives (test functions, distributions, weak derivatives in Lp).
The spaces of Sobolev Wk, p:
The space of Sobolev W1, p.
The space W01, p.
Some examples of limit problems.
Maximum principle.
Density theorems.
Immersion theorems.
Potential estimates.
Compactness.
Extensions and interpolations.
Sobolev spaces and variational formulation of problems at the limits in dimension N:
Definition and elementary properties of Sobolev spaces W^1, p (D)
Extension operators.
Sobolev inequalities.
The space W^01, p
Variational formulation of some elliptic boundary problems.
Existence of weak solutions.
Regularity of weak solutions.
Maximum principle.

[GT] D. Gilbarg, N.S. Trudinger Elliptic partial differential equations of second order