variational formulation of poisson equation

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

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

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

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..