# MATS340 Partial differential equations 2, 5cr, spring 2019

# MATS340 Osittaisdifferentiaaliyhtälöt 2, 5 op, kevät 2019

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. The weak theory of non divergence form equations arising for example in financial or control theory applications is covered in Viscosity Theory MATS424 that can be considered as PDE3. Viscosity theory -course starts right after this course.

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

Following the feedback, the Thursday lecture is moved to **Tuesday 14.15-16.00. The exercises are on Tue 10.15-12 at MaA203 (except on 12.2 at MaA204).**

The first set of exercises published, see the **Exercise folder**.

Tomorrow 10.1, the lecture will be moved to 10.15-12.00 MaD380.

First lecture: Wed 9.1.2019

### Schedule

Lectures: Wed 12.15-14.00 and Thursdays 14.15-16.00 at MaD381. The first lecture is on Wed 9.1.

Exercise sessions: Thursdays 10.15-12.00 MaD380. First session 17.1.

### Korppi

https://korppi.jyu.fi/kotka/r.jsp?course=231683

### Lectures

- week: Elliptic second order linear partial differential equation: physical interpretation. Review on Sobolev spaces: lecture exercises.
- 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.

### Lecturers

Mikko Parviainen, mikko.j.parviainen@jyu.fi, MaD306 (first half)

Angel Arroyo, angel.a.arroyo@jyu.fi, MaD344 (second half)

### Completion mode

Course exam and exercises, or alternatively final exam.

In the exercises, the exercises are denoted on the list, and one of the students works out the exercise on the black board. You get points from exercises to the course exam according to the following:

35% exercises -> 1 additional point in exam

45% exercises -> 2 additional points in exam

...

85% exercises -> 6 additional points in exam

### Recommended prerequisites

Measure and integration (part 1), Partial differential equations, Sobolev-spaces. If you do not have all the recommended prerequisites, and would like to attend the course, please contact the lecturer of the first half to get additional material.

### Literature

Outline of lectures will appear on the website. The lectures mostly follow the lecture note.

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.