Boundary regularity required by the constructed extension

The four approximation and extension statements on this page carry deliberately different regularity hypotheses on $\mathrm{\partial }\mathrm{\Omega }$, and none of them may be strengthened by accident. This remark records what each construction actually uses. The comparisons retain the cited Choice hypotheses: Countable Choice for the density and zero-extension interfaces, and the Axiom of Choice for the constructed bounded-domain extension and boundary-density interfaces.

Approximation needs no boundary regularity. Meyers–Serrin density Meyers–Serrin density on an arbitrary open set, under Countable Choice, assumes only that $\mathrm{\Omega }$ is open and that $1\le p<\mathrm{\infty }$: no boundary chart, no extension operator and no unboundedness of the derivatives of a cutoff near $\mathrm{\partial }\mathrm{\Omega }$ enters, because the argument exhausts $\mathrm{\Omega }$ by compactly contained pieces. Likewise the zero extension of Zero extension of W_0^{1,p} has no boundary derivative is defined by extending a class in $W_0^{1,p}(\mathrm{\Omega })$ by zero, and the limit definition of the closure Zero-boundary Sobolev space as a norm closure requires no boundary regularity; the price is the membership hypothesis in $W_0^{1,p}$, not a hypothesis on $\mathrm{\partial }\mathrm{\Omega }$. The interior mollification Interior mollification commutes with weak derivatives is available only on the shrinking sets ${\mathrm{\Omega }}_{\epsilon }=\{x\in \mathrm{\Omega }:\mathrm{dist}(x,{\mathbb{R}}^n\setminus \mathrm{\Omega })>\epsilon \}$. For a general class on $\mathrm{\Omega }$, this interior construction alone supplies convergence on compactly contained subsets, not convergence in the norm on all of $\mathrm{\Omega }$. A licensed extension to the whole space changes that conclusion: for $u\in W_0^{1,p}(\mathrm{\Omega })$ and $1\le p<\mathrm{\infty }$, zero extension lies in $W^{1,p}({\mathbb{R}}^n)$ by Zero extension of W_0^{1,p} has no boundary derivative. Whole-space mollification then converges in $W^{1,p}({\mathbb{R}}^n)$, and restriction gives convergence on $\mathrm{\Omega }$, including regions arbitrarily close to its boundary. Indeed, the whole-space density corollary Compactly supported smooth functions are dense in W^{k,p}(R^n) supplies ambient compactly supported smooth restrictions converging to $u$ without any boundary regularity. These restrictions need not themselves lie in $C_c^{\mathrm{\infty }}(\mathrm{\Omega })$ or have zero boundary values.

Extension needs exactly the chart regularity of its order. The half-space operator and the $C^k$ flattening lemma C^k boundary flattening preserves local W^{k,p} are combined by Bounded C^k domains admit integer-order Sobolev extension under the hypothesis that $\mathrm{\Omega }$ is a bounded $C^k$ domain in the graph sense of Bounded C^k domains and boundary charts: at each boundary point the flattening chart and the graph function $h$ must have bounded derivatives through order $k$, because the pullback of a $W^{k,p}$ class needs bounded derivatives of the chart through that same order, and the reflected moment formula cancels interface terms involving derivatives up to order $k-1$. For fixed $k$ the hypothesis is "boundary of class $C^k$": a $C^1$ boundary suffices for the first-order theorem and gives no control of second or higher derivatives, and a $C^{\mathrm{\infty }}$ boundary is needed only when one wants the construction at every order simultaneously. Consequently the smooth-up-to-the-boundary density statement Ambient smooth restrictions are dense on bounded C^k domains, which is obtained by extending first and mollifying afterwards, inherits the same bounded-$C^k$ hypothesis and the same fixed $k$.

Excluded endpoints and unproved strengthenings. The density statements are for $1\le p<\mathrm{\infty }$; at $p=\mathrm{\infty }$ the excluded endpoint is recorded in Meyers–Serrin excludes the W^{k,∞} norm endpoint, where the one-dimensional corner $\mathrm{\mid }x\mathrm{\mid }$ shows that no smooth sequence converges in the $W^{1,\mathrm{\infty }}$ norm. Nothing here asserts extension theorems for Lipschitz domains, for domains with less regular boundary, or a single operator simultaneously bounded on all orders $k$: such results are separate theorems, not consequences of the $C^k$ chart-and-reflection argument displayed on this page, and no step of that argument supplies the uniform higher-order estimates they would require.