Local smooth approximation in integer-order Sobolev spaces

Given: Countable Choice; an open set $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ with $n\ge 1$; $k\in {\mathbb{N}}_0$; $1\le p<\mathrm{\infty }$; $\mathbb{K}\in \{\mathbb{R},\mathbb{C}\}$; a class $u\in W^{k,p}(\mathrm{\Omega };\mathbb{K})$; a nonnegative unit-mass $\rho \in C_c^{\mathrm{\infty }}({\mathbb{R}}^n)$ with support in ${\bar{B}}_1(0)$; and an open set $U\subset \subset \mathrm{\Omega }$, so that $\bar{U}$ is a compact subset of $\mathrm{\Omega }$.

Interior mollification commutes with weak derivatives: with ${\mathrm{\Omega }}_{\epsilon }=\{x\in \mathrm{\Omega }:\mathrm{dist}(x,{\mathbb{R}}^n\setminus \mathrm{\Omega })>\epsilon \}$ and $D^{\alpha }u$ extended by zero, one has $D^{\alpha }{u}_{\epsilon }={\rho }_{\epsilon }\ast (D^{\alpha }u)$ on ${\mathrm{\Omega }}_{\epsilon }$ for every $\mathrm{\mid }\alpha \mathrm{\mid }\le k$, and $u_{\epsilon }$ is smooth there (Interior mollification commutes with weak derivatives).

The family ${\rho }_{\epsilon }$ is the mollifier family generated by the unit-mass bump $\rho$ (The mollifier family generated by a unit-mass smooth bump).

Approximate identity convergence: the family $({\rho }_{\epsilon })$ is an $L^1$ approximate identity (A unit-mass smooth bump generates an $L^1$ approximate identity); if $f\in L^p({\mathbb{R}}^n)$ and $1\le p<\mathrm{\infty }$, then $\mathrm{\parallel }f\ast {\rho }_{\epsilon }-f{\mathrm{\parallel }}_{L^p({\mathbb{R}}^n)}\to 0$ as $\epsilon \to 0^{+}$ (Every $L^1$ approximate identity converges to the identity in $L^p$ for $1\le p<\mathrm{\infty }$), and the same holds for complex-valued $f$ with the complex convolution conventions, including the case of a complex scalar field (Complex translation, convolution, approximate identities, and mollification).

For $w\in L^p(\mathrm{\Omega };\mathbb{K})$ the extension by zero $E_{0}w$ lies in $L^p({\mathbb{R}}^n;\mathbb{K})$ with $\mathrm{\parallel }{E}_{0}w{\mathrm{\parallel }}_{L^p({\mathbb{R}}^n)}=\mathrm{\parallel }w{\mathrm{\parallel }}_{L^p(\mathrm{\Omega })}$, because the integral over a measurable set is the integral of the indicator product (Integral over a measurable subset, Complex Lp classes and Euclidean test-function conventions).

The $W^{k,p}$ norm is the ${\mathrm{\ell }}^p$ sum of the $L^p$ norms of all derivative classes $D^{\alpha }u$ with $\mathrm{\mid }\alpha \mathrm{\mid }\le k$; for $p=\mathrm{\infty }$ it is the maximum of the essential bounds (Integer-order Sobolev spaces and their norms).

Compact containment: $\bar{U}\subseteq \mathrm{\Omega }$ compact with $\mathrm{\Omega }$ open implies $d:=\mathrm{dist}(\bar{U},{\mathbb{R}}^n\setminus \mathrm{\Omega })>0$, and for $0<\epsilon <d$ every $x\in U$ satisfies $\mathrm{dist}(x,{\mathbb{R}}^n\setminus \mathrm{\Omega })\ge d>\epsilon$, i.e. $U\subseteq {\mathrm{\Omega }}_{\epsilon }$; Here distance to the empty set is $+\mathrm{\infty }$, including when $U=\mathrm{\varnothing }$ or $\mathrm{\Omega }={\mathbb{R}}^n$; the containment still holds. This is elementary metric topology and uses no choice.

Choice use. Countable Choice is used through the approximate-identity and mollification interfaces of [F1] and [F3]; the compact-containment constant of [F6] is explicit and choice-free.