UMD PDE Qualifying Exams/Jan2007PDE

Problem 1Edit

a) Show that the function   is a solution in the distribution sense of the equation

 .

b) Use part (a) to write a solution of

 

SolutionEdit

(a)Edit

We want to show   for every test function  .

One can compute   and  . Therefore, away from 0, we have  , that is,   a.e. and  .

We now compute by an integration by parts:

 

A similar calculation gives

 

So we have shown that for all  

  which gives the desired result.


(b)Edit

We guess  . Then by part (a),

 .

Problem 6Edit

Let   be the unit ball in  . Consider the eigenvalue problem,

 

where   denotes the normal derivative on the boundary  . Show that all eigenvalues are positive and the eigenfunctions corresponding to different eigenvalues are orthogonal to each other.

SolutionEdit

Multiply the PDE by   and integrate:

 .

Of course we know that   is an eigenvalue of   corresponding to a constant eigenfunction. But a constant function has   which implies   by the boundary condition. Hence   is no longer an eigenvalue. This forces  .

To see orthogonality of the eigenfunctions, let   be two eigenfunctions corresponding to distinct eigenvalues  , respectively. Then by an integration of parts,

 

So by the PDE,

 .

Since   this implies that   are pairwise orthogonal in  .