Programme Partial differential equations, fall 2003
First week. General introduction, see overview.Study Appendix C.4 and Section 5.1. Home work: 5.10: 1,2,3,4. Hand in to Jan Bouwe (or his mailbox) before September 12, 16.30.
Second week. 5.2-5.5. Home work: these exercises and 5.10: 6,8,13. Hand in to JB before September 26, 16.30. For what I really did see here. I inserted a discussion of Lp spaces (see appendix B.2 for the inequalities) and Green's formula (appendix C.2), before giving the definition of weak derivatives in L1 loc. In the example I also needed polar coordinates for integrating over a ball (app. C.3). I finished with the fundamental theorem of calculus in the weak form. This is not treated in Evans. Approximation arguments, traces, extensions postponed to week 4.
Third week. 5.6 5.7. I am in Nottingham this week. Sarah Day will give both the exercise class and the course. Note the new deadline for 2nd home work (one week later). In the course she will focus on the Gagliardo-Nirenberg-Sobolev and Morrey estimates for smooth functions. This is what she did.
Week 4. 5.2-5.7 continued, see here. 3rd homework set 5.10: 14,15,16,17. Hand in to JB before October 3, 16.30. For 14 consider smooth functions which are zero away from the boundary and equal to one on the boundary. For 15 read and imitate the proof of 5.8.1, Thm 1. In 16 you have to use the regularisations of u and the fact that F'(u_eps) converges a.e. so that the dominated convergence theorem can be applied. In 17 I would first do (ii), no regularisation needed, but handle F_eps'(u) as in 16. Part (i) and (iii) follow by applying (ii) to -u and adding and subtracting.
Week 5. 6.1-6.2. Did this. 4th homework set 6.6: 1,2,3. For 2 look at 6.3 (3). Hand in to JB before October 10, 16.30.
Week 6. Intended: 6.4-6.5. Actually only done more of 6.1-6.2. 5th homework set 6.6: 5,6,7,8,9. Hand in to JB before October 31, 16.30.
Week 7. Finished 6.1-6.2. 6.4 until 6.4.3. Discussed spectral theorem for positive selfadjoint compact operators in relation to the inverse of L in 6.5.1 (2) on L^2 and H^1_0 (with inner product B). Lost the pictures...
Week 8. No class but office hours (R3.40).
Week 9. 7.1, see pics . 6th homework set 7.5: 1,2,3,6,7. Hand in to JB before November 6.
Week 10. 7.1 continued, 7.4, see pics.
Week 11,12 (lost count??). 7.4 concluded, 8.1, see pics and pics. 7th homework set 8.6: 1,2,4,5. Hand in to JB (before) November 28. Exercise 5 relies on 4 and Sard's theorem (almost every value of a smooth map is a regular value). You may thus assume first that each point in the preimage of x_0 has a nonzero Jacobian determinant and hence there are only finitely many such points).
Week 13 (25-10). 8.2,....
Week 14 (2-12, last course)