Week 3: Approximation by $C^{k}$ functions up to the boundary

We now take up the question of whether $u\in H^{k,p}(\Omega)$ may be approximated by a function $v\in C^{k}(\overline{\Omega})$, where we assume $\overline{\Omega}$ is bounded. It turns out that this is possible provided $\Omega$ is sufficiently regular.

Definition 3.1. An open bounded set $\Omega\subset\mathbb{R}^{n}$ is said to be a domain of class $C^{k}$ if for every $x\in \partial\Omega$, there exists an open neighbourhood $N_{x}\ni x$ and a $C^{k}$-diffeomorphism $\Phi_{x}:N_{x}\rightarrow B(0,1)\subset\mathbb{R}^{n}$ with the following properties:

• $\Phi_{x}(x)=0$;
• $\Phi_{x}(\partial\Omega\cap N_{x}) = B(0,1)\cap\{x^{n}=0\}$; and
• $\Phi_{x}(\Omega\cap N_{x})=B(0,1)\cap\{x^{n}>0\}$.

In a certain sense, the approximation properties of Sobolev spaces on domains of class $C^{k}$ may be deduced from those of Sobolev spaces on the upper half-ball $B^{+}(0,r):=B(0,r)\cap \{x^{n}>0\}$. We establish this in a sequence of lemmas.

Lemma 3.2. Suppose $\{U_{i}\}_{i=1}^{N}$ is a collection of open sets such that $\overline{\Omega}\subset \bigcup_{i=1}^{N}U_{i}$ and let $\Omega_{i}:=U_{i}\cap \Omega$. If $u\in H^{k,p}(\Omega_{i})$ for all $i$, then $u\in H^{k,p}(\Omega)$.

Proof. Let $\{\psi_{i}\}_{i=1}^{N}$ be a partition of unity subordinate to the cover $\{U_{i}\}$. Now, $u\in H^{k,p}(\Omega_{i})$ iff there is a sequence $\{u^{i}_{j}\}_{j=1}^{\infty}\subset C^{k,p}(\Omega)$ with $u^{i}_{j}\xrightarrow{j\rightarrow\infty}u$ in $W^{k,p}(\Omega_{i})$. Set

$u_{j}:=\sum_{i=1}^{s}u^{i}_{j}\cdot \psi_{i}.$

Clearly $u_{j}\in C^{k,p}(\Omega)$, and since we may write $u=\sum_{i=1}^{s}u\cdot \psi_{i}$, by the triangle inequality

$||u_{j}-u||_{k,p} \leq \sum_{i=1}^{s} ||(u^{i}_{j}-u)\cdot \psi_{i}||_{k,p,\Omega_{i}}\leq C(\psi,k,p)\cdot \sum_{i=1}^{s}||u^{i}_{j}-u||_{k,p,\Omega_{i}}\xrightarrow{j\rightarrow\infty}0,$

where we have used the Leibniz rule as in Theorem 2.11.

Remark 3.3. Lemma 3.2 may be viewed as the converse to the fact that $u\in H^{k,p}(\Omega)\Rightarrow u\in H^{k,p}(V)$ for any $V\subset\Omega$.

Lemma 3.4. Let $U\subset\mathbb{R}^{n}$ be open and bounded such that $\partial U=\Gamma_{1}\cup\Gamma_{2}$, $\Gamma_{1}\cap \Gamma_{2}=\emptyset$ and $\gamma_{1}=V\times\{0\}$, $V\subset\mathbb{R}^{n-1}$ open. If $u\in H^{k,p}(U)$, then for any $A\subset U$ with $\overline{A}\subset U\cup \Gamma_{1}$, there exists a sequence $\{u_{m}\}\subset C^{\infty}(\overline{A})$ such that $u_{m}\xrightarrow{m\rightarrow\infty}u$ in $W^{k,p}(A)$.

Proof. For $u\in L^{p}(U)$ and $\eps>0$, define the mapping $J_{\eps}'u:\overline{A}\rightarrow\mathbb{K}$ by

$J_{\eps}'u(x):=\eps^{-n}\int_{\mathbb{R}^{n}}\rho\left(\frac{x+2\eps e_{n}-z}{\eps} \right)\cdot u(z)dz = \int_{\mathbb{R}^{n}}\rho(w)\cdot u(x+2\eps e_{n}-\eps w)dw,$

where as before we assume $u\equiv 0$ outside $U$. Note that $z\mapsto \rho(\frac{x+2\eps e_{n}-z}{\eps})$ is supported in $B(x+ 2\eps e_{n},\eps)$ so that it does not intersect $\Gamma_{1}$ and is compactly contained in $\Omega$ for $\eps<\frac{1}{3}\dist(A,\Gamma_{2})$.

$J_{\eps}'u$ possesses the basic properties of $J_{\eps}u$ of Theorem 1.1, specifically:

• $J_{\eps}'u\in C^{\infty}(\overline{A})$: Since the support of $z\mapsto \rho(\frac{x+2\eps e_{n}-\eps z}{\eps})$ is compact and this function is smooth, we may apply the dominated convergence theorem to interchange integral and limit to show smoothness.

• $||J_{\eps}'u||_{p,A}\leq ||u||_{p,U}$: Exactly as before, we estimate using Hölder's inequality:

  $\begin{aligned} &|J_{\eps}'u(x)|=\left|\int_{B(0,1)}\rho(w)\cdot u(x+2\eps e_{n}-\eps w)dw\right|\leq \left(\int_{B(0,1)}\rho(z)\cdot |u(x+2\eps e_{n}-\eps z)|^{p}dz \right)^{1/p}\\ &\rightsquigarrow \int_{\mathbb{R}^{n}}|J_{\eps}'u|^{p}\leq \int_{\mathbb{R}^{n}}\int_{B(0,1)}\rho(z)\cdot |u(x+2\eps e_{n}-z)|^{p}dz dx = ||u||_{p}^{p}. \end{aligned}$

• If $u\in C^{0}(\overline{U})$, then $J_{\eps}'u\xrightarrow{\eps\searrow 0}u$ in $C^{0}(\overline{A})$: Analogously to before, using the fact that $B(x+2\eps e_{n},\eps)\subset B(x,3\eps)$, we have that for $x\in \overline{A}$,

  $|J_{\eps}'u(x) - u(x)|=\left | \int_{B(0,1)}\rho(z)\cdot \left(u(x+2\eps e_{n}-\eps z) - u(x)\right)dz \right|\leq \sup_{y\in B(x,3\eps)\cap U}|u(y)-u(x)|$

  so that $\sup_{A}|J_{\eps}'u-u|\leq \sup_{x\in A} \sup_{y\in B(x,3\eps)\cap U}|u(y)-u(x)|\xrightarrow{\eps\searrow 0}0$ by the uniform continuity of $u$.

• $J_{\eps}'u\xrightarrow{\eps\searrow 0}u$ in $L^{p}(A)$: The same argument using the preceding two assertions and the density of $C^{0}_{0}(U)$ in $L^{p}(U)$ applies here.

• $J_{\eps}'u\xrightarrow{\eps \searrow 0}u$ a.e. on $A$: As before, we estimate from above and use $B(x+2\eps e_{n},\eps)\subset B(x,3\eps)$ and appeal to the Lebesgue differentiation theorem:

  $|J_{\eps}'u(x)-u(x)|\leq \sup\rho\cdot \eps^{-n}\int_{B(x,3\eps)}|u(y)-u(x)|dy \xrightarrow{\eps\searrow 0}0.$

• If $\eps<\frac{1}{3}\dist(A,\Gamma_{2})$, $u\in W^{k,p}(U)$ and $|\alpha|\leq k$, we have $\D^{\alpha}J_{\eps}'u=J_{\eps}'D^{\alpha}u$ on $A$: As remarked earlier, for such $\eps$ we have that $\supp (z\mapsto \rho(\frac{x+2\eps e_{n}-z}{\eps}))\Subset U$ so that by `integrating by parts' as in the case of $J_{\eps}u$,

  $D^{\alpha}J_{\eps}'u(x)= (-1)^{|\alpha|}\eps^{-n}\int_{U}D_{z}^{\alpha}\left( \rho\left(\frac{x+2\eps e_{n}-z}{\eps} \right) \right)\cdot u(z)dz=\eps^{-n}\int_{U}\rho\left(\frac{x+2\eps e_{n}-z}{\eps}\right)\cdot D^{\alpha}u(z)dz=J_{\eps}'D^{\alpha}u(x).$

Combining all of the above, we see that if $u\in W^{k,p}(U)$, then $\{u_{m}:=J_{1/m}'u\}$ is a sequence with the desired property.

Corollary 3.5. If $\Omega$ is a domain of class $C^{k}$, then $H^{k,p}(\Omega)=\overline{C^{k}(\overline\Omega)}$.

Proof. For each $x\in \partial\Omega$, let $\Phi_{x}$ be as in the definition of a domain of class $C^{k}$ and set $\Omega_{\delta}:=\{x\in \Omega:\dist(x,\partial\Omega)>\delta \}$. Clearly

$\overline\Omega\subset \bigcup_{x\in \partial\Omega}\Phi_{x}^{-1}(B(0,\frac{1}{2})) \cup \bigcup_{\delta>0}\Omega_{\delta}.$

By compactness, we may find $\{x_{i}\}_{i=1}^{N}\subset \partial\Omega$ and a single $\delta>0$ such that

$\overline\Omega\subset \Omega_{\delta}\cup \bigcup_{i=1}^{N}\Phi_{x_{i}}^{-1}(B(0,\frac{1}{2})).$

Let $\{\psi_{0},\psi_{1},\dots,\psi_{N}\}$ be a partition of unity subordinate to this cover and $v\in C^{k,p}(\Omega)$. Define

$v_{m}:= \psi_{0}\cdot v + \sum_{i=1}^{N}\psi_{i}\cdot J_{1/m}'(v\circ \Phi_{x_{i}}^{-1})\circ \Phi_{x_{i}},$

where we take $U:=B^{+}(0,1)$ and $A:=B^{+}(0,\frac{1}{2})$ in the definition of $J_{1/m}'(v\circ \Phi_{x_{i}}^{-1})$. Clearly $v_{m}\in C^{k}(\overline{\Omega})$, since $J_{1/m}'(v\circ \Phi_{x_{i}}^{-1})\in C^{\infty}(\overline{B^{+}(0,\frac{1}{2})})$. Using the same tricks as before, we compute that

$||v_{m}-v||_{k,p}\leq C(k,p,\psi)\cdot\sum_{i=1}^{N}||J_{1/m}'(v\circ \Phi_{x_{i}}^{-1})\circ \Phi_{x_{i}} - v||_{k,p,\Phi_{x_{i}}^{-1}(B(0,\frac{1}{2}))}.$

To see that this tends to $0$, we compute for each $i$, setting $\Phi:=\Phi_{x_{i}}$ and $A:=B^{+}(0,\frac{1}{2})$ for simplicity of notation:

$\int_{\Phi^{-1}A}|\D^{\alpha}(J_{1/m}'(v\circ \Phi^{-1})\circ \Phi - v)|^{p}= \int_{A}|\D^{\alpha}(J_{1/m}'(v\circ \Phi^{-1})\circ \Phi - v)|^{p}\circ \Phi^{-1}\cdot |\det\D \Phi|^{-1}\circ \Phi.$

Now, noting that $|\det\D \Phi|$ is bounded from below by a positive constant, using the chain rule and liberally bounding from above, we may estimate this expression from above:

$\leq C(F,\alpha,p)\sum_{\beta\leq\alpha}\int_{A}|\D^{\beta}(J_{1/m}'(v\circ F^{-1}) - (v\circ F^{-1})) |^{p}\xrightarrow{m\rightarrow\infty} 0$

since we also have that $v\circ F^{-1}\in C^{k,p}(A)$. Therefore, for any $\eps>0$, we may choose $m$ large enough so that $||v_{m}-v||<\eps$.

Another useful characterisation of $H^{k,p}(\Omega)$ functions is in terms of compactly supported functions. We first establish the following lemmas.

Lemma 3.6. There exists a constant $C>0$ depending on $k$ and $p$ such that for any $u\in C^{k}(\overline{B^{+}(0,r)})$, we may find a $u'\in C^{k}(\overline{B(0,r)})$ such that $\left.u'\right|_{\overline{B^{+}(0,r)}}\equiv u$ and $||u'||_{k,p}\leq C\cdot ||u||_{k,p}$.

Proof. Define $u':\overline{B(0,r)}\rightarrow\mathbb{K}$ such that

$u'(x)=\begin{cases}u(x),&x^{n}\geq 0\\ \displaystyle\sum_{j=1}^{k+1}c_{j}u(x^{1},\dots,x^{n-1},\frac{-x^{n}}{j}),&x^{n}<0 ,\end{cases}$

where the $\{c_{j}\}_{j=1}^{k+1}$ satisfy $\sum_{j=1}^{k+1}c_{j}\cdot \left(-\frac{1}{j}\right)^{m-1}=1$ for all $m\in\{1,\dots,k+1\}$. It follows that this relation uniquely determines the $\{c_{j}\}$ and ensure that $u'$ is $C^{k}$ across $\{x^{n}=0\}$. Moreover, it is clear that for each $\alpha$ with $|\alpha|\leq k$, $||\D^{\alpha}u'||_{p}\leq C(k,p)||\D^{\alpha}u||_{p}$, which establishes the claim.

Lemma 3.7. Suppose $\Omega$ is a domain of class $C^{k}$. There exists a constant $C>0$ depending on $k$, $p$ and $\Omega$ such that for each $u\in C^{k}(\overline\Omega)$, we may find a $u'\in C_{0}^{k}(\mathbb{R}^{n})$ with $\left.u'\right|_{\Omega}\equiv \left.u\right|_{\Omega}$ and

Proof. As before, we cover $\overline\Omega\subset \Omega_{\delta}\cup \bigcup_{i=1}^{N}\Phi_{x_{i}}^{-1}(B(0,\frac{1}{2}))$ and let $\{\psi_{0},\dots,\psi_{N}\}$ be a subordinate partition of unity. Note that for each $i$, $u\circ \Phi_{x_{i}}^{-1}\in C^{k}(\overline{B^{+}(0,\frac{1}{2})})$. By the preceding lemmas, we may extend this function to a $v_{i}\in C^{k}(\overline{B(0,\frac{1}{2})})$ such that $||v_{i}||_{k,p}\leq C(k,p)\cdot ||u||_{k,p}$. Now let

$u'=\psi_{0}\cdot u + \sum_{i=1}^{N}\psi_{i}\cdot v_{i}\circ \Phi_{x_{i}}.$

It immediately follows that $u'$ possesses the desired properties.

Remark 3.8. Let $\Omega$ be a domain of class $C^{k}$ and $V\Subset\mathbb{R}^{n}$ an open set such that $\Omega\Subset V$ and let $\psi:\mathbb{R}^{n}\rightarrow [0,1]$ be a smooth function such that $\psi\equiv 1$ on $\Omega$ and $\supp\psi\Subset V$. Using the same techniques as before, we may establish the estimate

for $u'$ as in Lemma 3.7. In particular, replacing $u'$ with $\psi\cdot u'$, we deduce that for any $u\in C^{k}(\overline\Omega)$ and $p>1$, there exists a $u'\in C_{0}^{k}(\Omega)$ such that $\left.u'\right|_{\Omega}\equiv \left.u\right|_{\Omega}$, $\supp u' \Subset V$ and $||u'||_{k,p}\leq C(k,p,V)\cdot ||u||_{k,p}$. A cursory inspection of the proofs of Lemmas 3.6 and 3.7 show that the mapping

is linear, and the condition on the norm of $u'$ amounts to its continuity. We call this an extension operator.

Corollary 3.9. For any $u\in H^{k,p}(\Omega)$ and $\eps>0$, there exist $u_{1}\in C_{0}^{k}(\Omega)$ and $u_{2}\in C_{0}^{\infty}(\Omega)$ such that $||u-u_{1}||_{k,p}<\eps$ and $||u-u_{2}||_{k,p}<\eps$.

Proof. By Corollary 3.5, for any $u\in H^{k,p}(\Omega)$ and $\eps>0$, we may find a $v\in C^{k}(\overline{\Omega})$ such that $||u-v||_{k,p,\Omega}<\frac{\eps}{2}$, and by Lemma 3.7 this function is the restriction to $\overline{\Omega}$ by a function $v'\in C_{0}^{k}(\mathbb{R}^{n})$, establishing the equality. To establish the second, consider the sequence $\{w_{m}:=J_{1/m}v'\}\subset C_{0}^{\infty}(\mathbb{R}^{n})$, the inclusion following from Theorem 1.1. By Remark 2.6, we may choose $N>0$ large enough so that $||w_{N}-v'||_{k,p,\mathbb{R}^{n}}<\frac{\eps}{2}$, which immediately implies that $||w_{N}-v'||_{k,p,\Omega}<\frac{\eps}{2}$. By the triangle inequality, we have that $||u-w_{N}||_{k,p,\Omega}<\eps$, establishing the second equality.