The cousre aim at exposing the students to some of the fundamental concepts of Geometric Measure Theory that are at the basis of classical results as well as central tools in some important topics of the current mathematical research.
Basic Measure Theory and Functional Analysis
Course contents summary
The course will focus on some basic topics of Geometric Measure Theory.
An indicative list of the topics that will be treated is the following:
- Review and complements of Measure Theory
-Covering theorems and their application to the proof of the Lebesgue and Besicovitch Differentiation Theorems.
- Theory of sets of finite perimeter.
- The Isoperimetric problem
-The Plateau problem
-The problem of capillarity and the supported drop
The topics covered in the second part
will be chosen from those listed below, depending on the amount of time left and on students' interests:
-Partial Regularity Theory for quasi-minimiser of the perimeter
- Theory of Currents and (outline of) their application to the Plateau problem
- Free discontinuity functionals.
There will be no real reference text. Students will however find it useful to consult the following textbooks:
1) L.C Evans and R.F. Gariepy: "Measure Theory and Fine Properties of Functions"
2) F. Maggi: "Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory"