Master Course Partial Differential Equations
Lecturers: Arjen Doelman, Stephan van Gils, Joost Hulshof
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.