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Elliptic Boundary Value Problem: Lapalce and Polar Coordinates

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(lap) means the Laplacian
Vrr means the second derivative of V with respect to r
V(theta theta) means the second derivative of V with respect to theta

(lap)V(r,theta)= Vrr+(1/r)Vr+(1/r^2)V(theta theta)=0
0 < r < 1, -(pi) < theta < pi

V(1,theta) = {1, -(pi/2) < theta < (pi/2)
{0, elsewhere

Please show all work including the derivation of any eigenvalues or eigenvectors.
Thank you

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Solution Summary

This shows how to solve an elliptic boundary value problem with Laplace and polar coordinates. The means for the second derivative with respect to theta are given.

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Please see the attached doc file.

We have:

We consider:

Therefore by substitution we have:

and we can simplify as:

This must be a constant. The constant can be positive, negative or zero. In these kinds of problem, the positive choice works. We call that k^2. Then:

k must be a whole number according ...

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