In Strauss's Partial Differential Equations, the eigenvalue problem $-\Delta v=\lambda v,\qquad v\big|_{\partial \Omega}=0$ is solved by separating the $x,y,z$ variables: $v=X(x)Y(y)Z(z)$, $ \frac{X''}{X}+\frac{Y''}{Y}+\frac{Z''}{Z}=-\lambda$
The separated BCs are $X(0)=X(\pi)=Y(0)=Y(\pi)=Z(0)=Z(\pi)=0$
Here are my questions:
How do we deduce that the solutions are $v(x,y,z)=\sin lx\sin my \sin nz$ where $l^2+m^2+n^2=\lambda\quad (1\leq m,l,n<\infty)$
Why the shape of $\Omega=[a,b]\times[c,d]\times[e,f]$ is needed for this kind of method?