Meyers–Serrin excludes the W^{k,∞} norm endpoint

The local mollification theorem Local smooth approximation in integer-order Sobolev spaces and the Meyers–Serrin density theorem Meyers–Serrin density on an arbitrary open set both require $1\le p<\mathrm{\infty }$; neither asserts that smooth functions are dense in $W^{k,\mathrm{\infty }}(\mathrm{\Omega })$ in the Sobolev norm. The exclusion is not a defect of the proofs but a genuine endpoint failure, and the one-dimensional example $u(x)=\mathrm{\mid }x\mathrm{\mid }$ on $(-1,1)$ exhibits it.

First, $u\in W^{1,\mathrm{\infty }}(-1,1)$ with weak derivative the sign function $\mathrm{sgn}(x)=1_{(0,1)}-1_{(-1,0)}$. Indeed $x\mapsto x$ is smooth with classical derivative $1$, which is its weak derivative by Classical derivatives agree with weak derivatives, so the truncation calculus Positive, negative, and truncated Sobolev functions applied with $p=\mathrm{\infty }$ gives $\mathrm{\mid }x\mathrm{\mid }\in W^{1,\mathrm{\infty }}(-1,1)$ and $D\mathrm{\mid }x\mathrm{\mid }=\mathrm{sgn}(x)\cdot 1$ almost everywhere; the norm of Integer-order Sobolev spaces and their norms then computes $\mathrm{\parallel }\mathrm{\mid }x\mathrm{\mid }{\mathrm{\parallel }}_{W^{1,\mathrm{\infty }}(-1,1)}=\max \{1,1\}=1$.

Second, no sequence of smooth functions converges to $\mathrm{\mid }x\mathrm{\mid }$ in $W^{1,\mathrm{\infty }}(-1,1)$-norm, hence the finite-$p$ conclusion cannot be extended to $p=\mathrm{\infty }$. Suppose $g_m\in C^{\mathrm{\infty }}(-1,1)$ satisfied $\mathrm{\parallel }{g}_m-\mathrm{\mid }x\mathrm{\mid }{\mathrm{\parallel }}_{W^{1,\mathrm{\infty }}}<1\mathrm{/}(2m)$ or merely $\mathrm{\parallel }{g}_m^{\mathrm{\prime }}-\mathrm{sgn}{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(-1,1)}<1\mathrm{/}2$ for some $m$. On $(0,1)$ the sign equals $1$, so $g_m^{\mathrm{\prime }}>1\mathrm{/}2$ almost everywhere there; since $g_m^{\mathrm{\prime }}$ is continuous, $g_m^{\mathrm{\prime }}\ge 1\mathrm{/}2$ at every point of $(0,1)$, for otherwise continuity would give a whole interval on which $g_m^{\mathrm{\prime }}<1\mathrm{/}2$, a set of positive measure contradicting the essential bound. Applying the same reasoning on $(-1,0)$, where the sign equals $-1$, gives $g_m^{\mathrm{\prime }}\le -1\mathrm{/}2$ on $(-1,0)$. Continuity of $g_m^{\mathrm{\prime }}$ at $0$ then forces the two incompatible limits $g_m^{\mathrm{\prime }}(0)={\lim }_{x\to 0^{+}}{g}_m^{\mathrm{\prime }}(x)\ge 1\mathrm{/}2$ and $g_m^{\mathrm{\prime }}(0)={\lim }_{x\to 0^{-}}{g}_m^{\mathrm{\prime }}(x)\le -1\mathrm{/}2$, a contradiction.

The same phenomenon separates the exponents for local approximation: the mollifications of $\mathrm{\mid }x\mathrm{\mid }$ converge to $\mathrm{\mid }x\mathrm{\mid }$ in $W_{\mathrm{l}\mathrm{o}\mathrm{c}}^{1,q}(-1,1)$ for every finite $q$ by Local smooth approximation in integer-order Sobolev spaces, while the derivative error at the corner stays of size at least one half in $L^{\mathrm{\infty }}$ for every mollification scale. Local convergence in every finite $W^{1,q}$ therefore does not imply convergence at the $W^{1,\mathrm{\infty }}$ endpoint norm; the companion examples page computes the mollifications of $\mathrm{\mid }x\mathrm{\mid }$ explicitly.