# MATS340 Partial Differential Equations 2

The theory of partial differential equations (PDEs) is an interesting part of analysis and also plays an important role in many applications both mathematical as well as practical. It is a field of active research containing interesting topics for thesis as well.

The pointwise classical definition of a solution is too restrictive in many occasions (see Hilbert's 20th problem). Therefore, the definition has to be relaxed: this course deals with a weak distributional theory of partial differential equations in divergence form. This kind of equations arise for example when modeling physical phenomena involving diffusion.

In this course we will deal with roughly speaking: Sobolev theory (review), Elliptic partial differential equations in divergence form, and their weak solutions, existence of solutions, maximum and comparison principles, uniqueness of solutions, regularity of solutions, parabolic partial differential equations and their weak solutions.

### Announcements

17.1 - The first exercise set has been added.

The 2nd exercise set has been added.

2.2 - The 3rd exercise set has been added.

10.2 - The 4th exercise set has been added.

17.2 - The 5th exercise set has been added.

23.2 - The last exercise set has been added.

The first exercise session is at 23.1 and not 16.1 as stated before.

### Schedule

Lectures in MaD355 on Mondays 14:15-16:00 and Tuesdays 12:15-14:00. The first lecture is at 9.1.

Excercise sessions in MaD380 on Mondays 16:15-18:00. The first session is at 23.1.

### Content of lectures

The lectures mostly follow the lecture notes from 2019 by Mikko Parviainen (some notations may be different).

- week: Elliptic second order linear partial differential equation: physical interpretation. Review on Sobolev spaces.
- week: Weak solutions. Examples of weak solutions. Existence: Hilbert space approach.
- week: Existence: variational method, uniqueness, comparison, start of regularity theory.
- week: L^p regularity, start of the De Giorgi's method for Hölder regularity.
- week: De Giorgi's method: ess-sup estimate, measure decay lemma.
- week: Hölder regularity. Weak and strong maximum principles.
- week: Linear Parabolic equations. Weak solutions. Existence: the Garlekin's Method.

### Completion mode

The course is completed by returning the exercises to the lecturer. The exercises appear on this website. It is possible to get help for the exercises during the exercise sessions on Mondays.

50% exercises => grade 1

60% exercises => grade 2

70% exercises => grade 3

80% exercises => grade 4

90% exercises => grade 5

### Recommended prerequisites

Measure and integration (part 1), Partial differential equations, Sobolev-spaces.

### Literature

Outline of lectures will appear on the website.

Evans: Partial differential equations

Wu, Yin, Wang: Elliptic and parabolic equations

Gilbarg, Trudinger: Elliptic partial differential equations of second order

Giaquinta, Martinazzi: An introduction to the regularity theory for elliptic systems, harmonic maps and minimal graphs.