Geometric Measure Theory

Prerequisites:

Measure theory, Real analysis, and preferably Functional analysis

Course description:

In this course, students will be presented with a modern theory of measure in Polish spaces; Lipschitz mappings; Hausdorff measures; rectifiable sets and measures in Euclidean spaces, and functions of bounded variation. These tools are basic to many fields of study in analysis and partial differential equations.

Learning objectives:

After completing the course, students will be able to:

a. Be aware of basics of descriptive set theory as it applies to analysis, for instance, recognize the importance and ubiquity of Cantor spaces;

b. Understand and apply weak* convergence of Radon measures in Polish spaces;

c. Understand various covering theorems and know when to apply them, for instance, to compute Radon-Nikodym derivatives;

d. Be able to apply extension and differentiability properties of Lipschitz mappings;

e. Be familiar with topics relevant to Hausdorff measures, such as the area formula and apply these to basic variational problems, for instance, involving minimization of length in different contexts;

f. Be familiar with some applications of the Brunn-Minkowski inequality, such as the isodiametric inequality, the isoperimetric inequality, and Poincaré-Sobolev inequalities.

g. Understand and apply the notion of rectifiability whether it be that of sets or of measures in Euclidean spaces and its interpretations: tangent measures; density 1; projections;

h. Be able to analyse examples of purely unrectifiable sets (fractals);

i. Understand and be able to apply basic facts of the theory of functions of bounded variation, such as compactness, isoperimetric inequalities, coarea formula, and te reduced boundary and Gauss-Green formula.

Detailed topics covered:

a. Polish and Cantor Spaces

i. Polish spaces

ii. Cantor Space, Brouwer's theorem, Cantor-Bendixson theorem (comments about closed sets and Lusin's theorem)

b. Finite Borel Measure in Polish Spaces

i. Outer regularity and tightness

ii. Riesz-Markoff representation theorem

iii. Weak* covergence and compactness in locally compact Polish spaces

iv. Differentiation bases, Radon-Nikodym derivatives

v. Vitali metric covering theorem (applications to doubling measures)

vi. Besicovitch covering theorem in Euclidean spaces

c. Lipschitz Mappings (extensions, differentiability, locally flat Lipschitz functions)

i. Extension theorems

ii. Differentiability theorems: Lebesgue and Rademacher

iii. Optimality of Lebesgue's differentiability theorem in one variable

iv. Whitney extension theorem and Lusin-type approximation by $C^1$ functions

v. Locally flat Lipschitz functions

d. Hausdorff Measures (basics, dimension, Vitali metric covering theorem, densities, Brunn-Minkowski)

i. Definition, Lipschitz mappings, dimension

ii. Brunn-Minkowski and isodiametric inequalities, link with Lebesgue's measure

iii. Jacobians

iv. Decomposition of Lipschitz mappings

v. Kirchheim's area formula

vi. Example: minimization problems involving curves

e. Rectifiable Measures in Euclidean Spaces:

i. Rectifiable and purely-unrectifiable sets

ii. Characterization via tangent measures

iii. Characterization via density 1

iv. Characterization in 1 dimension via the little Lipschitz uniform separation property

v. Example: Four-corner Cantor set

f. Functions of bounded variation

i. Definition and lower semincontinuity

ii. Poincaré and isoperimetric inequalities, compactness

iii. Coarea formula

iv. Sets of finite perimeter, reduced boundary, rectifiability

v. Nöbeling's example

References:

a. P. Mattila, Geometry of Sets and Measures in Euclidean Spaces

b. L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions

c. L. Simon, Lectures on Geometric Measure Theory