Dirichlet Poisson regularity theorem ($-\Delta :H^2(U)\cap H_0^1(U)\to L^2(U)$)

If $U\subset {\mathbb{R}}^n$ is bounded with $C^2$ boundary, then every $F\in L^2(U)$ determines a unique $v\in H^2(U)\cap H_0^1(U)$ satisfying