Geometric Measure Theory II

Period of duration of course
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Course info
Number of course hours
40
Number of hours of lecturers of reference
40
Number of hours of supplementary teaching
0
CFU 6
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Type of exam

Oral exam

Lecturer

View lecturer details

Prerequisites

Last year of the laurea magistrale/bachelor, PhD

Programme

The space BV
BV functions of one variable
Sets of finite perimeter
Embedding theorems and isoperimetric inequalities
Structure of sets of finite perimeter
Approximate continuity and differentiability
Fine properties of BV functions
Boundary trace theorems
Decomposition of derivative and rank one properties
The chain rule in BV 191
One dimensional restrictions of BV functions
Theory of currents
Basic operations
Normal, Flat, Rectifiable currents
Slicing
Isoperimetric inequality and deformation theorem
Closure and boundary rectifiability theorems
Basic theory of varifolds
Compactness and partial regularity for mass-minimizing varifolds

Educational aims

The goal of the course is to provide an introduction to modern Geometric Measure Theory. Starting from the codimension 1 theory, namely the theory of sets of finite perimeter and BV functions, we move to the study of (generalized) oriented surfaces of arbitrary dimension and codimension. We illustrate basic structure and compactness results, leading to solutions to Plateau's problem, and in the final part of the course we deal with non-oriented surfaces (varifolds) and partial regularity results. 

Bibliographical references

L.Ambrosio, N.Fusco, D.Pallara: Functions of bounded variation and free discontinuity problems

P.Mattila: Geomety of sets and measures in Euclidean spaces

F.Morgan: Geometric Measure Theory: a beginner's guide

H.Federer: Geometric Measure Theory

S.G.Krantz, H.Parks: Geometric Integration Theory

L.Simon: Lectures on Geometric Measure Theory