finite difference method

I am implementing finite difference scheme for the simple 2D Laplace
equation. I am facing problem in imposing boundary conditions in 9-point
stencil. What boundary conditions should I give at the vertex i.e. at
the corner where two boundaries intersect.

e.g.- For square plate what should be boundary condition at four vertices?

What should be the boundary condition at vertices in Dirichlet problem?either homogeneous or non-homogeneous?

Please help in this regard. 

Hello everybody,

The coupled differential equations need to solve as following:

Could you help me to discretize this systems and find the eigenvalue λ  numerically using finite difference method?

Thanks in advance,

lqkhai