 Sobolev space with a partial Dirichlet condition

Let $U$ be a bounded Lipschitz domain and ${\Gamma }_D$ a measurable portion of its boundary. The spaceis a closed vector subspace of the H1 space. The Sobolev trace operator maps $H^1(U)$ continuously into $L^2(\partial U)$, and restriction to ${\Gamma }_D$ is bounded, so $V$ is the kernel of a continuous linear map. Boundary regularity is what makes this ordinary trace formulation available.

No positive-measure assumption is needed for closedness. If $U$ is also connected and ${\Gamma }_D$ has positive surface measure, the Poincare inequality with a partial Dirichlet boundary controls the full Sobolev norm by the gradient norm on $V$.