941 Partial Differential Equations
Fall 2007

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

  Comparison table for the four main PDEs 
  Homework 1:  (due September 25) :  Problems 1, 3, 4, 5, 9     pg. 85-87
  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).

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