Functions of Bounded Variation & Sets of Finite Perimeter (WS24)

About the course.

  We will study functions of bounded variation, which are functions whose weak first partial derivatives are Radon measures. This is essentially the weakest definition of a function to be differentiable in the measure-theoretic sense. After discussing the basic properties of them, we move on to the study of sets of finite perimeter, which are Lebesgue measurable sets in the Euclidean space whose indicator functions are BV functions. Sets of finite perimeter are fundamental in the modern Calculus of Variations as they generalize in a natural measure-theoretic way the notion of sets with regular boundaries and possess nice compactness, thus appearing in many Geometric Variational problems. If time permits, we will discuss the (capillary) sessile drop problem as one important application.

Basic knowledge in measure theory and analysis is required.

Dates: Monday, 14:00-16:00 in SR 127.   Tutorial: Monday, 16:15-18:15 in SR 127.

Office hours: Wednesday, 11:00-12:00.

Lecture notes [last updated: 18/11/2024]

Exercise sheets:

• Week 1: Exercises 1.4, 1.9, 2.4(*)

• Week 2: Exercise 4.6

• Week 3: Exercises 5.10, 5.11, 5.12, 5.13, 5.16

• Week 4: Exercises 5.22, 5.25, 5.26

• Week 5: Exercises 6.4, 6.9(*) 

• Week 6: Exercise 8.9

References.

• Maggi, Francesco. Sets of finite perimeter and geometric variational problems: an introduction to Geometric Measure Theory. No. 135. Cambridge University Press, 2012.

• Evans, Lawrence C. and Gariepy, Ronald F. Measure theory and fine properties of functions. CRC Press, Boca Raton, FL, 2015.

• Ambrosio, Luigi, Fusco, Nicola, and Pallara, Diego. Functions of bounded variation and free discontinuity problems. Oxford Math. Monogr. The Clarendon Press, Oxford University Press, New York, 2000.