Partial differerential equations 2, 9cr, spring 2013

Exercise points given so far, see Korppi. You can get your solutions returned by visiting MaD306.

New deadline for Set 2, part 1: 2.4.2013. No lectures on 26. and 28.3, Easter break.

The first round of exercises has been graded. Exercises are returned during the lectures; otherwise visit MaD306.

Lectures on Tuesdays 10.15-12.00 and Thursdays 10.15-12.00 at MaD381. The first lecture is on Tu 15.1.

- Weeks 1-4: Weak derivatives, Sobolev spaces, local and global approximations, Sobolev spaces with zero boundary values, difference quotients, Sobolev type inequalities
- Weeks 5-6. Review, Poincare-type inequlities, Morrey's inequality, Rellich-Kontrachov compactness theorem, weak solution to linear elliptic PDEs.
- Week 7: Existence to linear elliptic PDEs by using Hilbert space methods, variational formulation, Dirichlet's principle, existence of a minimizer, examples/counterexamples of the weak solutions.
- Week 8: Extension of class of test functions, uniqueness, regularity, comparison principle.
- Week 9: Weak and strong max principles, Hopf lemma, parabolic equations, density of smooth functions in a parabolic space, time-mollification.
- Week 10: Local and global weak formulation for parabolic problem, existence for parabolic PDE: Galerkin method, mollified weak formulation by standard mollifiers and Steklov averages.
- Week 11: Uniqueness of parabolic weak solutions, sub- and supersolutions, energy estimates, Harnack's inequality: Moser's iteration.
- Week 12: Moser's iteration continues, estimates for supersolutions, weak Harnack's ie, Harnack's ie, counterexample for "elliptic Harnack", Hölder continuity of a solution. Remarks on higher regularity.
- Week 13: Local Schauder estimates in the elliptic case.

Lecturer Mikko Parviainen, mikko.j.parviainen@jyu.fi, MaD306

Lectures on Tuesdays 10.15-12.00 and Thursdays 10.15-12.00 at MaD381.

Course is passed by solving a sufficient number of exercises, and returning solutions to the lecturer. There will be three exercise sets (about 20-25 problems each). You can return exercises to the lecturer at the lectures, at lecturer's office MaD306 (mailbox outside the office), or you can scan exercises and send them via email.

The course will be graded as follows

50% problems solved -> grade 1

....

90% problems solved -> grade 5

Prerequisites: Measure and integration 1, PDE1.

- Evans: Partial differential equations
- Wu, Yin, Wang: Elliptic and parabolic equations
- Gilbarg, Trudinger: Elliptic partial differential equations of second order