I am trying to show that a smooth function u:\Omega \rightarrow \mathbb{R} satisfying the poisson equation and a "mixed" boundary condition \alpha u + \frac{\partial u}{\partial n} = 0 on the boundary \partial \Omega where \alpha > 0 is a constant, also satisfies the variational formulation:

find u \in H^1(\Omega) such that,

\displaystyle\int_\Omega \nabla u . \nabla v + \alpha \displaystyle\int_ {\partial \Omega} uv = \displaystyle\int_\Omega vf for all v \in H^1(\Omega)

and show that a weak solution is unique....

I am having real difficulties with both parts of the problem as I havnt dealt with mixed boundary conditions before. I would really appreciate any help..