UMD PDE Qualifying Exams/Jan2006PDE
Problem 1Edit

SolutionEdit
Problem 2Edit

Problem 3Edit

Problem 4Edit
A weak solution of the biharmonic equation,
is a function such that
for all . Assume that is a bounded subset of with smooth boundary and use the weak formulation of the problem to prove the existence of a unique weak solution. 
SolutionEdit
Consider the functional . is bilinear by linearity of the Laplacian. Now, we claim that is also continuous and coercive.
where the first inequality is due to Holder and the second is by the definition of the Sobolev norm. And so is a continuous functional.
To show coercivity, we use the fact that by two uses of integration by parts, which gives
which establishes coercivity.
Thus, by the LaxMilgram Theorem, the weak solution exists and is unique.
Problem 5Edit

Problem 6Edit
This problem has a typo and can't be solved by characteristics as it is written. 