UMD PDE Qualifying Exams/Jan2007PDE
a) Show that the function is a solution in the distribution sense of the equation
b) Use part (a) to write a solution of
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.
We guess . Then by part (a),
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.
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 .