# Variational Formulation - inhomogeneous

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 $$u_o$$.

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.
• I was thinking more along the lines of maybe splitting the solution by calling a $u_g=u-u_o$ then have the new problem be, $-\Delta u = f + u_g$ with the new boundary being $u_g=0$. This would then let me try Green's formula by splitting the domain @daw – lnbmoco Oct 24 '18 at 17:31