Let $U\subsetneq\mathbb R^n$ be bounded and open. Let $B:H_0^1(U)\times H_0^1(U)\to\mathbb R$ be bounded and coercive. For every $F\in H^{-1}(U)$, there exists a unique $u\in H_0^1(U)$ such that
\begin{align*}
B[u,v]&=F(v)
\end{align*}
for every $v\in H_0^1(U)$.