# variational formulation of poisson equation

• March 8th 2009, 05:18 AM
johnbarkwith
variational formulation of poisson equation
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..