# PDE2

PDE2 — Matematiikan ja tilastotieteen laitos

## MATS340 Osittaisdifferentiaaliyhtälöt 2, 9 op

Partial differential equations 2, 9cr, spring 2017

The theory of partial differential equations (PDE) 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.

In this course we will deal with roughly speaking: Sobolev spaces and inequalities, weak derivatives, 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

Lecture note updated below (on 13.3.2017): typo fixes and contains global regularity and De Giorgi's method.

The exercise points can be found at Korppi. The answers are returned at the lectures after the grading.

### Schedule

Lectures: Wed 12.15-14.00 and Thursdays 14.15-16.00 at MaD380. The first lecture is on Wed 18.1.

Exercise sessions: Thursdays 12.15-14.00 MaD355. First session 26.1. At the exercise sessions the idea is to work with the exercises and later return them to the lecturer. Lecturers are there to help if needed.

### Korppi

https://korppi.jyu.fi/kotka/r.jsp?course=202979

### Lectures

- week: Introduction, Lp-spaces and useful inequalities, weak derivatives, Sobolev spaces, examples of Sobolev functions.
- week: Standard mollification and its properties, global approximation in Sobolev spaces, W_0^{1,p} and examples, min and max of Sobolev functions, difference quotients and Sobolev spaces,
- week: End of difference quotients, Sobolev type inequalities, Poincare type inequalities, Morrey type inequality, Rellich-Kontrachov type compactness theorem, linear elliptic PDEs in divergence form.
- week: Examples of weak solutions and their regularity, existence through Riesz representation theorem, examples of nonexistence, variational method, Dirichlet principle, existence of a minimiser
- week: W^(1,2)_0 test functions, uniqueness, L^2-regularity theory for the elliptic equations, global regularity (This material is added to lecture note below)
- week: counterexamples for the global regularity, L^p-regularity, Hölder-regularity using De Giorgi method. (This material is added to lecture note below).
- week: end of Hölder-regularity using De Giorgi method, comparison and maximum principles. (This material is added to lecture note below).
- week: Parabolic spaces, weak formulation for parabolic equations, existence of weak solutions via Galerkin's method, mollification in time, ->p98,
- week: Steklov average, uniqueness of weak solutions, strategy of Moser to obtain Harnack estimates, Energy estimate and parabolic Sobolev inequality and how to use them to derive a reverse Hölder estimate, Moser iteration.->p107.
- week: estimates for super-solutions, weak Harnack Harnack's inequality , Harnack's inequality, Hölder continuity of weak solutions, weak and strong maximum principles. Remarks on higher regularity.->p115. (notes for maximum principles).
- week: Introduction to Companato spaces, relation with Hölder spaces, Schauder estimates for Poisson equation and remarks for more general elliptic equations using the ''freezing- coefficients'' technique->p125
- week: introduction to BMO spaces, L^p estimates for the Poisson equation for p>2 using the Stampacchia interpolation theorem, counterexamples for p=1 and p=\infty, proof of a Marcinkievicz interpolation theorem and a part of the proof of Stampacchia's theorem. (notes for L^p regularity).
- week: Ending lecture. Continuation of the proof of Stampacchia's theorem, Calderón-Zygmund decomposition and proof of Fefferman Stein theorem, Continuity method and application to existence of classical solutions. (Notes for continuity method).

Lecturers:

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

Amal Attouchi, amal.a.attouchi@jyu.fi, MaD305, (second half)

### Course requirements

Course is passed by solving a sufficient number of exercises that are handed out each weak (about 5-7 exercise a week), and returning solutions to the lecturers. You can return exercises to the lecturer at the lectures, exercise sessions or at lecturers' office (mailbox outside the office).

The course will be graded as follows

50% problems solved -> grade 1

....

90% problems solved -> grade 5

In addition, at least 1 problem of each set must be solved.

Prerequisites: Measure and integration (part 1) and Partial differential equations are recommended. Also the course on the Sobolev-spaces is useful but not mandatory.

### Exercise sets

Please return the exercises at latest at the beginning of the exercise session on the deadline day. The paper return is preferable, but also email returns in one file are accepted. Please, staple the papers together.

- exercise
- exercise
- exercise
- exercise
- exercise
- exercise
- exercise
- exercise
- exercise
- exercise
- exercise
- exercise
- exercise

### 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.