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 Lax-Milgram 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.