Let $U\subset \mathbb{R}^n$ be a bounded domain. For $f\in H^{-1}(U)$ (see \ref{def:pde-hminus1-u-2026c}), consider the weak problem: $\int_U \nabla u\cdot\nabla v\,dx = \langle f, v\rangle \quad \text{for all } v\in H_0^1(U).$ Here $H_0^1(U)$ is as in \ref{def:pde-h01-u-2026c}. Then there exists a unique $u\in H_0^1(U)$ solving the weak Dirichlet Poisson problem. This is an application of \ref{thm:pde-lax-milgram-2026c} with $V=H_0^1(U)$ and $a(u,v)=\int_U \nabla u\cdot\nabla v\,dx$.