Zero-boundary Sobolev space as a norm closure

Assume Countable Choice. Let $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ be open with $n\ge 1$, let $k\in {\mathbb{N}}_0$, let $1\le p\le \mathrm{\infty }$, and let $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$.

Every $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{K})$ belongs to $W^{k,p}(\mathrm{\Omega };\mathbb{K})$. Indeed $\phi$ is bounded with compact support, hence lies in $L^p(\mathrm{\Omega };\mathbb{K})$ for every exponent in range, and each of its classical partial derivatives is again smooth and compactly supported, hence lies in $L^p(\mathrm{\Omega };\mathbb{K})$ and is the corresponding weak derivative by Classical derivatives agree with weak derivatives.

Define the zero-boundary Sobolev space $W_0^{k,p}(\mathrm{\Omega };\mathbb{K}):=\overline{C_c^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{K})},$ the closure taken in the normed space $(W^{k,p}(\mathrm{\Omega };\mathbb{K}),\mathrm{\parallel }\cdot {\mathrm{\parallel }}_{W^{k,p}(\mathrm{\Omega })})$ of Integer-order Sobolev spaces and their norms. The norm is a genuine norm on classes by The Sobolev norm descends to equivalence classes, so this is the usual metric closure of a subset of a normed space: $u\in W_0^{k,p}(\mathrm{\Omega };\mathbb{K})$ if and only if for every $\delta >0$ there is $\phi \in C_c^{\mathrm{\infty }}(\mathrm{\Omega };\mathbb{K})$ with $\mathrm{\parallel }u-\phi {\mathrm{\parallel }}_{W^{k,p}(\mathrm{\Omega })}<\delta$. Write $H_0^k(\mathrm{\Omega }):=W_0^{k,2}(\mathrm{\Omega };\mathbb{K})$ when the scalar field is fixed by context.

Three warnings are part of the definition. First, this is a closure of almost-everywhere classes, and all its equalities are equalities of Sobolev classes. Second, no pointwise boundary values and no trace characterization are asserted: a description of $W_0^{k,p}$ by vanishing boundary data belongs to the later trace theory and is not used here. Third, the closure is defined for every $1\le p\le \mathrm{\infty }$, but for $p=\mathrm{\infty }$ it is not claimed that every compactly supported $W^{k,\mathrm{\infty }}(\mathrm{\Omega })$ function lies in $W_0^{k,\mathrm{\infty }}(\mathrm{\Omega })$; the definition merely names the closure of the test functions.

Kinnunen, Definition 1.23 and Remarks 1.24, printed pp. 21–22, defines $W_0^{k,p}$ as the closure of the test functions and warns that this is not yet a boundary-value statement. Laugesen, Definition 3.11, printed p. 59, uses the same closure convention for $W_0^{k,p}$.