Let $U\subset\mathbb{R}^n$ be bounded and open. Let $L$ be a divergence form [elliptic operator](/page/Elliptic%20Operator) whose associated [bilinear form](/page/Bilinear%20Form) $B$ is bounded on $H_0^1(U)\times H_0^1(U)$ and coercive on $H_0^1(U)$. Then 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 $v\in H_0^1(U)$. Moreover, there exists $C>0$ such that

\begin{align*} \|u\|_{H_0^1(U)}\le C\|f\|_{H^{-1}(U)}. \end{align*}