Let $U\subset \mathbb{R}^n$ be bounded with Lipschitz boundary and $f\in H^{-1}(U)$ (see \ref{def:pde-hminus1-u-2026a}). Then there exists a unique $u\in H_0^1(U)$ (see \ref{def:pde-h01-u-2026a}) such that $\int_U \nabla u\cdot\nabla v\,dx = \langle f,v\rangle \quad \text{for all } v\in H_0^1(U).$ Equivalently, with $V=H_0^1(U)$, this is an application of \ref{thm:analysis-lax-milgram-2026a} to $a(u,v)=\int_U \nabla u\cdot\nabla v\,dx$.