title: Some applications of geometric measure theory

short course delivered at the Spring school 
"Geometric Measure Theory: Old and New" 
Les Diablerets (Switzerland), April 3-8, 2005.


Abstract: The purpose of my lectures is to illustrate the 
application of Geometric Measure Theory to problems coming 
from different areas of analysis. Rather than giving a general 
overview, I will focus on few chosen examples. 

1. 
Survey of basic properties of BV functions.
The compactness theorem for SBV functions. 
Existence of minimizers for image segmentations 
problems.

2. 
Variational models of phase separation.
Heuristic justifications. 
Gamma-convergence and the Modica-Mortola theorem.

3. 
Coarea formula (standard and oriented versions)
Distributional Jacobian of Sobolev maps. 
The case of maps valued into spheres: relation to 
singularity.

4. 
Jerrard-Soner rectifiability theorem for the Jacobian.
Approximation and lifting problems for Sobolev maps valued 
in $S^1$.

Prerequisites: basic notions about BV functions and finite 
perimeter sets (I will just quickly recall them in the first 
lecture); the notions from the theory of currents needed in 
the last two lectures should be provided by one of the parallel 
courses. 

References: the essentials about BV functions and finite 
perimeters sets can be recovered from Chapter 5 of the book 
by Evans & Gariepy (Measure Theory and Fine Properties of 
Functions); another reference could be Chapters 1-4 of the 
book by Ambrosio, Fusco & Pallara (Functions of Bounded 
Variation and Free Discontinuity Problems).