**1st semester**
** **
**A. A. 1998/99**

**Dott. S. Baldo**

**OBJECTIVE**

My purpose is to introduce the students to "modern" techniques in the study of elliptic P.D.E.s.

A minimal prerequisite is a one-semester course in Measure Theory, Real Analysis and elementary Banach and Hilbert Spaces.

**PROGRAMME**

(i) A few variational problems in space-dimension 1.

Minimizing geodesics on a riemannian manifold. The Ascoli-Arzelà
Theorem. Sobolev spaces in dimension 1. Existence and regularity of minimizing
geodesics.

(ii) Sobolev Spaces in dimension n. Boundary values problem for elliptic
P.D.E.s in divergence form: weak formulation, variational formulation.

Weak semicontinuity of integral functionals, existence of minimizers.

Existence of weak solutions to elliptic P.D.E.s in divergence form.
Hilbert-space regularity of weak solutions.

The last part of the course will be devoted to some "monographic" argument,
chosen by the students.