Ellipctic Boundary Value problem using laplace and polar cordinates

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

Solve:
(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 inluding the derivation of any eigenvalues or eigenvectors.
Thank you