Variational Formulation - inhomogeneous

Question

I'm not sure how to get started with the following. Consider,

$- \Delta u=f$ in $\Omega$

$u=u_o$ on $\Gamma$

I need to find a $u \in V(u_o)$ such that

$a(u,v)=(f,v)$ $\forall v\in H^1_o$ where

$V(u_o)$={$v\in H^1 \Omega$: $v = u_o$ on $\Gamma$}

Now I think that I need to use Green's formula here but the only problem I'm running into now is that I can't say that the boundary integral goes to $0$ because in this case it's $u_o$.

Any help on how to go about this would be insightful!

Answer 1

You need that $V(u_0)$ is non-empty, that is, there is $v_0\in V(u_0)$. Then split the unknown $u=v_0 + u_0$, where $u_0\in H^1_0(\Omega)$ solves $$ a(u_0,v) = (f,v) - a(v_0,v) \quad \forall v\in H^1_0(\Omega). $$ Just superposition principle for linear equations.