### Course Overview

**Teacher: **Kari Astala

**Scope:** 10 cr

**Registration:** via WebOodi

**NOTE**: *The tutorials on Friday 30.11. are changed to Wednesday 5.12. at 10-12, Room *Y313.

**NOTE**: Small clarification added to Exercise 9, Problem 4.

**NOTE**: *Lecture rooms have been changed* !!! as follows:

From Oct. 29 on, Monday lectures will be in the room M2 (M233) and Tuesday lectures the room M3 (M234).

**NOTE**: Below you find links to following notes

- A note giving an example of a set of finite perimeter, with boundary of positive volume; in addition, an example of reduced boundary points.
- Notes on isoperimetric inequality, with Sobolev and Poincare inequalities for BV. (Notes now contains also local isoperimetric inequality, p.8)
- Notes on Compactness in BV(U), when U is a Lipschitz domain
- Notes on weak*-convergence in BV(U).
- Notes on Gauss-Green theorem for Lipschitz domains. (
**REMARK**: correction made to p.4) - Notes on BV-functions on the real line.

**Here**also short list of properties of vector measures and their variation measures.

**Teaching: **The course extends over periods I and II. ** **Lectures are on Mondays and Tuesdays at 12 - 14, room M2 (M233) on Mondays and room M3 (M234) on Tuesdays.

First **homework here** and in the materials folder (to be returned by Fri. 21.9). [correction to problem 3: Need to assume \mu(F) is positive and finite]

Second **homework here** and in the materials folder (to be returned by Fri. 5..10).

Third **homework here** and in the materials folder (to be returned by Monday 15.10)

Fourth **homework here** and in the materials folder (to be returned by Monday 22.10)

Fifth **homework here **and in the materials folder (to be returned by Monday 5.11) [**Note**: Misprints corrected in Problem 2 and 4 (Thu. Oct. 18)]

Sixth **homework here **and in the materials folder (to be returned by Monday 12.11)

Seventh **homework here** and in the materials folder (to be returned by Monday 19.11)

Eight **homework here** and in the materials folder (to be returned by Monday 26.11) (small misprint corrected in problem 1)

Ninth **homework here** and in the materials folder (to be returned by Monday 10.12). (small clarification added to Problem 4; small misprint corrected)

Homeworks should be returned to Vito Buffa (office M229).

**Grading**: There will be no exam for this course; grading is based on homework (50 % = 1/5; 90 % = 5/5) and course activity. Homework is given weekly and returned in writing.

**Homework Tutorials: **Fridays 10 - 12, room Y313

The tutorials are given by Vito Buffa.

**Prerequisites**: MS-E1280 Measure and integral; also MS-E1281 Real Analysis is highly recommended, but not obligatory. However, required facts will be recalled when needed.

**COURSE MATERIAL: **The course will mainly follow the book "L. Evans and R. Gariepy: Measure Theory and Fine Properties of Functions (CRC Press, Revised edition 2015)", Chapters 2, 3 and 5 from there. *Having access to the book will be necessary !! * In particular, the Evans-Gariepy book will serve as the course notes, only some notes on some additional/occasional material will be given on these pages.

**NOTE**: The Evans-Gariepy book is now available electronically via Aalto library !

**LECTURE DIARY**: *In the first week* we covered Section 2.1 from the book of Evans-Gariepy, and began discussion on self-similarity.

*In the second week*, we explained the self-similar sets and proved that their Hausdorff dimension can be explicitly calculated. For details, see *Lecture notes 1 *which is now completed.* *We also* *briefly discussed the Steiner symmetrisation from Evans-Gariepy, pp. 87-89.

**Note**: Since there were no lectures the week of Sept. 24 - 25, the few remaining points from Chapter 2 in Evans-Gariepy were left for self-study. More detailed instructions are **here**.

*In the third week,* we started Chapter 3, and covered Section 3.1, 3.2 and Lemma 3.1 from Section 3.3. Details of linear algebra from Section 3.2 were left for self-study. For Rademacher's theorem we gave a proof different from that in Evans-Gariepy; you can find it **here**. That uses Ascoli-Arzela theorem, whose proof (in Finnish) can be found **here**.

*In the fourth week*, we covered the rest of Section 3.3, pp.115 - 125, i.e. from Lemma 3.2 up to discussion on the applications. The main point was the area formula, Theorem 3.8., but also the necessary auxiliary results and corollaries will be useful later.

*In the** fifth week*, we covered the proof of the coarea formula for Lipschitz mappings, and completed Chapter 3 from Evans-Gariepy. Lemma 3.5 from [EG] was left for self-study.

*In the sixth week*, the first after the exam week, we started the study of BV-functions. We first discussed the BV-functions of one variable, from many different points of view, see the enclosed **notes here**. We then began the study of BV-functions in higher dimensions, and covered the first three pages from Chapter 5 in Evans-Gariepy.

*In the seventh week*, we started a systematic study of BV-functions in higher dimensions, and sets of finite perimeter. **Here are** notes proving the Gauss-Green formula for Lipschitz domains, which explains and motivates the definition of a set of finite perimeter. Last topic covered was approximation (p.199 - 203). Additionally, we discussed the weak*-convergence in BV(U); **Here** you have notes on that topic.

*In the eight week*, we studied compactness properties of BV(U). We gave a self-contained proof showing that for any bounded sequence of BV-functions in a Lipschitz domain, there exists a subsequence (norm)converging in L^{1}. Here we did not follow presentation of Evans-Gariepy, but rather our own notes **here**. We also briefly described how this can be used in different minimisation problems. In addition, we showed that BV-spaces are invariant under compositions with bilipschitz-mappings, and that BV-functions in Lipschitz domains allow extensions to the whole space. We skipped the section "Traces" from Evans-Gariepy, and start next week with coarea formula for BV.

*In the ninth week*, we first proved the coarea formula for BV-functions [EG, Section 5.5]. Then we gave a proof for the Sobolev inequality for p=1, with constant 1; here we did not follow [EG]. For notes on the inequality see **here**. The notes also proved Poincare inequalities for BV; these will be applied next week.

*In the tenth week,* we first used the Poincare inequality to prove the relative isoperimetric inequality, i.e. Thm 5.11. (ii) in [EG]. After that we began the study of reduced boundary, Section 5.7 in [EG]. Last topic discussed was Thm 5.13, whose proof we started and proved there first three steps ([EG] pp. 226-227). In addition, **HERE** you find notes on an example of a set of finite perimeter, with boundary of positive volume; the note also discusses (via an example) of the smoothness of boundary at reduced boundary points.

*In the eleventh* (i.e. last) *week*, we completed Theorem 5.13, on blow-ups of reduced boundary, and then proved De Giorgi's structure theorem, i.e. Thm. 5.15 in [EG], together with the other remaining material from Section 5.7. Whitney's extension theorem (Thm. 6.10 in [EG]) was explained and used, but its proof was left for self-study. Last, we briefly mentioned Section 5.8 and the form Gauss-Green theorem obtains in domains with finite perimeter (Thm. 5.16 in [EG].)

**Topic: **Geometric Measure Theory is the study of geometric properties of (in general non-smooth) sets, via the methods of measure theory.

The origins of the theory come from minimizations problems such as the Plateau problem, which asks if every smooth curve in R^{3 } spans a "soap film", a surface of minimum area with the given curve as the boundary. Another similar cornerstone is the isoperimetric inequality, that the smallest possible circumference for a surface with given area is that of a round circle. Corresponding questions naturally exist in higher dimensions as well.

The course aims for understanding the basic ideas of geometric measure theory, in particular the sets of finite perimeter and functions of bounded variation. These methods and issues have plenty of applications besides to geometry also e.g. to the study of partial differential equations (PDE's).

*Themes of the course include *:

Hausdorff measures and Hausdorff dimension

Self-similar fractals

Rademacher's theorem

Area and Coarea formula

Functions of Bounded variation

Isoperimetric inequalities

Reduced boundary

Structure of sets of finite perimeter

**Literature**:

Evans, L., Gariepy, R: Measure theory and fine properties of functions. CRC Press, Revised edition 2015

Maggi, F: Sets of finite perimeter and geometric variational problems. Cambridge University Press, 2012

Mattila, P.: Geometry of sets and measures in Euclidean spaces. Fractals and rectifiability. Cambridge University Press, 1995