Master Course Partial Differential Equations

Lecturers: Arjen Doelman, Stephan van Gils, Joost Hulshof

This is
the webpage maintained by Arjen

In week 42 I will start with
harmonic functions and related topics.

I basically did the mean value inequality subharmonic fucntions
and the Poisson integral formula
for harmonic functions on balls.
Homework you can do is 1.13, 1.19, 1.20, 1.23.

In week 43 I will do
Perron's method.

Last time I did Perron's method for the cosntruction of
an harmonic function satisfying the boundary conditions
as the pointwise sup of all supersolutions.

Exercise 2.12 is
recommended. A 4th equivalent characterisation is that
the mean value inequalities hold with averages over the full balls.

Next time (week 44) I will do potential theory and its application to
the solution of Delta u =f on a domain with boundary condition u=phi.

Unfortunately I was ill on November 3. November 10 I did Section 3
of the notes, but there is no time left for Section 4.
The proof of Theorem 4.6 relies on Schauder's
fixed point theorem. You can find the original paper on
Guido Sweers home page, number 4 in his publication list. The proof
is completely elementary. It uses the weak solution concept, where both
derivatives are put on the test function.

HAND IN HOMEWORK (deadline December 1):
Exercise 2.12, including the 4th characterisation mentioned above.
For the construction of approximating smooth functions use the procedure in
Robinson, Section 1.3.1.

I ended with an introduction on what we are going to do from the
book of Robinson, see his
webpage
for course notes.
See the beginning of Chapter 6.
November 17,24 and December 1 I will go through Chapters 5 and 6 of the book.
Chapters 1-4 are considered as familiar, unless you tell me otherwise.

November 17 I briefly discussed Section 4 of the notes (the Clement-Sweers
result on existence of solutions between sub-and supersolutions for semilinear
problems. I then went Chapter 5 in the book. Explained how you find and prove
the basic Sobolev Embedding: If L^p-norm (p

Here
is the last homework set for this semester.