Solution
= Solution
The <zero-boundary Sobolev space> is
$$
\boxed{H_0^1(\Omega)=\overline{C_c^\infty(\Omega)}^{\,H^1(\Omega)}}.
$$
Here $H^1(\Omega)$ consists of the $L^2$ functions whose first <weak derivatives> also belong to $L^2$, with norm
$$
\lVert u\rVert_{H^1}^2=\int_\Omega\bigl(|u|^2+|\nabla u|_2^2\bigr).
$$
On a sufficiently regular domain, the closure condition is equivalent to having zero <Sobolev trace theorem>[boundary trace].