Elliptic Boundary Value Problem

Uxx means second derivative with respect to x
Uyy means second derivative with respect to y

Uxx + Uyy = 0, 0 < x < pi,  0 < y < pi

U(x,0) = 0,  U(x,pi) = 1,  0 < x < pi

U(0,y) = 0,  U(pi,y) = 1  0 < y < pi

I know the problem has to be broken into 2 separate problems using U = V + W with zero conditions on 3 sides for each problem W and V. So please show the problem in this manner. Please show all work including the derivation of any eigenvalues or eigenvectors.
THank you