Let $U\subset\mathbb{R}^n$ be bounded and let $f\in H^{-1}(U)$ as in \ref{def:pde-hminus1-u-1772661803}. Then there exists a unique $u\in H_0^1(U)$ such that $\int_U \nabla u\cdot \nabla v\,dx = \langle f,v\rangle \quad\text{for all } v\in H_0^1(U).$ In particular this is an application of \ref{thm:pde-lax-milgram-real-hilbert-1772661803}.