941 Partial Differential Equations

Fall 2007

Fall 2007

Office Hours: Wed 1:30- 2:30, Friday 2:30-3:45 or by appointment

Syllabus

Comparison table for the four main

Homework 2 (pdf) - due October 25

Homework 3 (pdf) - due November 15 (extension until November 19, if necessary)

Homework 4: (due December 6): Problems 5, 6, 13, 16 pg. 290-291

Comments:

09/11 - Regarding Kyle's question on Harnack's Inequality. One can rephrase the question as:

If the infimum of a harmonic nonnegative function is zero on a set U, then by Harnack's

inequality we would get that the supremum has to be zero, hence u is constant zero. This seems

to come in contradiction with the fact that there are plenty of nonnegative harmonic functions.

First of all, the assumption that inf u is strictly positive is not needed (as Kyle pointed out this

assumption could be eliminated anyway by shifting u so that the new infimum is above zero). The

key to solving this apparent contradiction is given by the fact that Harnack's inequality holds for

every compactly embedded subset V in U. For a set to be compactly embedded we have that its

closure (which is compact) is contained in the open set, hence V can never get too close to the boundary

of U (in fact, the distance from V to the boundary of U is strictly positive). Thus, we can never touch the

point on the boundary of U where the minimum is attained (by the minimum principle).

10/16 - Regarding the Chain Rule for distributions/ weakly differentiable functions. In order to have

d (F(u))/dx=F'(u)du/dx it is sufficient to have F Lipschitz (W^{1,\infty} and u differentiable in the weak

sense (W^{1,p}) (Mizel). Much stronger results are available (Leoni & Morini).

Announcements:

- The midsemester exam will be in class on Tuesday, October
30; open notes and books will be permitted.