# MATS235 Sobolev Spaces (9 cr)

**Study level:**

Advanced studies

**Grading scale:**

0-5

**Language:**

English, Finnish

**Responsible organisation:**

Department of Mathematics and Statistics

**Curriculum periods:**

2020-2021, 2021-2022, 2022-2023

## Description

Sobolev spaces are an important tool in modern analysis and in applied mathematics. The course contains the essential parts of the theory of Sobolev spaces like

- the convolution approximation
- weak (distributional) derivatives
- partition of unity and approximation of Sobolev functions by smooth functions
- Sobolev inequalities
- the ACL-charterization of Sobolev functions
- weak and strong convergence in L^p- and Sobolev spaces
- p-capacity

## Learning outcomes

In the course, the basic properties of Sobolev spaces are studied. After the course, the student can use the definition of the weak derivative and its properties, Sobolev inequalities, approximation of Sobolev functions by smooth functions and different characterizations of Sobolev spaces.

## Description of prerequisites

Measure and integration theory 1&2

## Literature

- L.C. Evans & R.F. Gariepy, Measure Theory and Fine Properties of Functions; ISBN: 9781482242386
- W.P. Ziemer, Weakly Differentiable Functions; ISBN: 978-0-387-97017-2
- G. Leoni, A first course in Sobolev spaces; ISBN: 978-0821847688

## Completion methods

### Method 1

**Evaluation criteria:**

course exam points and exercise points

Select all marked parts

### Method 2

**Evaluation criteria:**

final exam points

Select all marked parts

**Parts of the completion methods**

x

### Teaching (9 cr)

**Type:**

Participation in teaching

**Grading scale:**

0-5

**Evaluation criteria:**

course exam points and exercise points

**Language:**

English, Finnish

**Study methods:**

lectures and exercises

#### Teaching

##### 9/8–12/18/2020 Lectures

##### 12/15–12/15/2020 Exam

x

### Exam (9 cr)

**Type:**

Exam

**Grading scale:**

0-5

**Evaluation criteria:**

final exam points

**Language:**

English, Finnish