Master Course Partial Differential Equations

Semester: Fall 2004, first session 9-9-2004, afternoon

Place: Utrecht

Lecturers: Arjen Doelman and Joost Hulshof

This is the webpage maintained by Arjen

Grades of exercises for Chapter 6 and 7 so far.

Next semester: JF's course on
perturbations methods

Next semester: Rob's course on topological methods
for nonlinear differential equations,

more information coming soon on
Rob's home page

This is what I did on October 21, the applications of Green part of Chapter 6. See the last board for homework. Next time I hope to finish Chapter 6, in particular maximum principles for general elliptic operators.

This is what I did on November 4, I inserted a discussion
about applying the Mean Value Property to the gradient of a harmonic function,
Perron's method to construct harmonic functions on a more general class
of domains, and Harnack's inequality. Then I continued with Section 6.3,
maximum
principles for general elliptic operators.

Homework to be handed in December 9:
Excercises 1,2,3 + an this exercise
here shortly. Here are my grades.

This is what I did on November 18, It more or less corresponds to section 7.2, of which I still have to do general elliptic operators and quadratic forms, and the dual space of a Sobolev space. Not in the course notes is a discussion of the Sobolev embedding and the critical exponent which I inserted. Have a look at this nice exercise.

This is what I did on November 25, spectral theory for the Laplacian with zero Dirichlet boundary data.

This is what I did on December 2, discussion of compactness of the Sobolevembedding, dual spaces uses Fourierseries representations, and subsequently solutions/ apriori estimates of the inhomogeneous linear heat equation on a domain with zero Dirchlet boundary data.

This is what I did on December 9, weak formulation of the inhomogeneous linear heat equation, existence, uniqueness, and semilinear equations with contraction argument.

This is what I did on December 16, Navier-Stokes in the same "fashion" as the inhomogeneous linear heat equation, wellposedness in dimension 2 (sketchy...). Final homework set: from Section 7.7 in the course notes of Hans en Arjen, 2,5,11,18. You have to read a little bit in Chapter 7 to do them.