Mathematical Analysis 4

The course intends to provide the students the fundaments of the modern theory of partial differential equations (PDE).
In the first part the notion of function space will be studied: the L^p spaces of Lebesgue, Banach spaces, Hilbert spaces. In the second part it is shown that these spaces are the natural environment in which one obtains existence and uniqueness theorems for a large class of PDE's.

At the end of the course the student will have acquired the following capabilities: he/she will
1) know the structure and properties of the Lebesgue spaces L^p
2) have a good knowledge of the properties of Banach and Hilbert spaces
3) know the properties of Sobolev spaces
4) have seen the fundamental theorems of compactness: the theorems of Ascoli-Arzela and of Rellich-Kondrachov
5) know the weak formulation of elliptic equations of second order
6) will know to apply the Dirichlet principle to linear and nonlinear elliptic equations
7) will know the importance of the regularity theorems for weak solutions, and will have seen the corresponding theorems
8) will have studied the heat eqqation and the representation formulas of the solutions
9) will know the modern theory of second order parabolic equations, weak solutions and energy estimates
10) will have studied the basics of hyperbolic equations