UMD PDE Qualifying Exams/Jan2007PDE

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Problem 1[edit]

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

.

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

Solution[edit]

(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 6[edit]

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.

Solution[edit]

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 .