Existence and uniqueness of weak solution for Laplace equation with Dirichlet BC Considering equation
$$-\Delta u=f\ \mathrm{in}\ \Omega$$
$$u=g\ \mathrm{on}\ \partial\Omega$$
where $f\in L^2(\Omega)$ and there exists $G\in H^1(\Omega)$ such that $G=g$ on $\Omega$.
I now want to prove the ellipticity of $\Delta$, which I suppose I need to do using the Friedrichs inequality. But that only holds for functions from $H^1_0(\Omega)$. So I suppose that the solution $u$ has the form $u=G+w$, where $w\in H^1_0$. The equation than takes the form
$$-\Delta w=f+\Delta G=F$$
But I don't know if $F\in L^2(\Omega)$. So how do I prove the existence and uniqueness of solution?
 A: It is actually clear when I write down the weak form:
\begin{align}
-\int_\Omega\Delta w\ v&=\int_\Omega fv+\int_\Omega\Delta G\ v\\
\int_\Omega\nabla w\nabla v&=\int_\Omega fv-\int_\Omega\nabla G\nabla v\\
\int_\Omega\nabla w\nabla v&=\int_\Omega(fv-\nabla G\nabla v)\ \ \ \forall v\in H^1_0
\end{align}
So I have equation in the form
\begin{equation}
a(w,v)=F(v)\ \ \ \forall v\in H^1_0,
\end{equation}
where
\begin{align}
a(w,v)&=\int_\Omega\nabla w\nabla v\\
F(v)&=\int_\Omega(fv-\nabla G\nabla v)
\end{align}
The ellipticity (coercivity) of $a(\cdot,\cdot)$ follows from the Friedrichs inequality, which I can now use, because $w\in H^1_0(\Omega)$. Boundedness is trivial in this case.
As for the functional $F$, it is also bounded, because $f,\ \nabla G\in L^2(\Omega)$. My mistake was assuming, that I need $G\in L^2(\Omega)$, but as it turns out, I only need $\nabla G\in L^2(\Omega)$, which is satisfied, because $G\in H^1(\Omega)=W^{1,2}(\Omega)$.
Now I can use the Lax-Milgram theorem to show, that there is a unique $w\in H^1_0$ such that
\begin{equation}
a(w,v)=F(v)\ \ \ \forall v\in H^1_0,
\end{equation}
and the solution to the equation is then $u=w+G$.
A: You have $F \in H^{-1}(\Omega)$, and you can use the Lax-Miligram Theorem to get existence and uniqueness (actually it's just the Riesz-Representation Theorem in this case). 
