1st semester
A. A. 1998/99

Dott. S. Baldo


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.


(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.