Week 2: Characterisations of Sobolev spaces

Sobolev spaces (continued)

We illustrate the potentially pathological nature of Sobolev functions with two more examples.

Example 2.1. Consider the mapping $u:\Omega\rightarrow\mathbb{R}$ such that $x\mapsto |x|^{-\alpha}$, where $\Omega:=B(0,1)$. This function is smooth on $\Omega\backslash\{0\}$ and

A quick computation shows that for $p\geq 1$,

which is finite iff $n-1-p(\alpha+1)>-1\Leftrightarrow p\cdot(\alpha+1)<n$. If this is the case, then by the usual integration by parts formula,

for every $\eps>0$ and $\varphi\in C_{0}^{\infty}(\Omega)$. Now, using the definition of $u$ and crudely estimating, we see that

Since $u,\partial_{i}u\in \lloc^{1}$, we may take limits above to obtain the relation

i.e. whenever $p\cdot (\alpha+1)<n$, we have that $u\in W^{1,p}(\Omega)$.

Example 2.2. Let $\{r_{k}\}_{k=1}^{\infty}\subset \Omega:=B(0,1)$ be dense and define $u:\Omega\rightarrow\mathbb{R}$ such that $x\mapsto \sum_{k=1}^{\infty}2^{-k}\cdot |x-r_{k}|^{-\alpha}$. It may be shown that $u\in W^{1,p}(\Omega)$ iff $p\cdot(\alpha+1)<n\Leftrightarrow \alpha<\frac{n-p}{p}$. If $\alpha>0$ as well, then $u$ is not bounded on any open subset of $\Omega$.

Sobolev spaces via smooth functions

We now turn our attention to alternative characterisations of weak derivatives and Sobolev spaces.

Derivatives in the $L^{p}$ sense ($p<\infty$)

Definition 2.3. Suppose $u\in\lloc^{p}(\Omega)$. We say that $\D^{\alpha}u$ exists in the $L^{p}$ sense if there is a $v\in \lloc^{p}(\Omega)$ such that for every $U\Subset\Omega$, there exists a sequence $\{u_{j}\}_{j=1}^{\infty}\subset C^{|\alpha|}(\Omega)$ such that $u_{j}\xrightarrow{j\rightarrow\infty}u$ and $\D^{\alpha}u_{j}\xrightarrow{j\rightarrow\infty}v$ in $L^{p}(U)$. We then say $v=\D^{\alpha}u$ in the $L^{p}$ sense and call $v$ the $\alpha$th $L^{p}$ derivative of $u$.

Exercise 2.4. If $v=\D^{\alpha}u$ in the $L^{p}$ sense, then $v=\D^{\alpha}u$ in the weak sense.

As it turns out, the converse is also true for $u,\D^{\alpha}u\in\lloc^{p}(\Omega)$.

Lemma 2.5. If $u\in \lloc^{p}(\Omega)$ and $\D^{\alpha}u$ exists in the weak sense and is in $\lloc^{p}(\Omega)$, then $\D^{\alpha}u$ is the $\alpha$th $L^{p}$ derivative of $u$.

Proof. Let $U\Subset\Omega$ be fixed. We know from Theorem 1.1 that $J_{\eps}u\xrightarrow{\eps\searrow 0}u$ in $L^{p}(U)$. We now show that for sufficiently small $\eps>0$, $\D^{\alpha}J_{\eps}u=J_{\eps}\D^{\alpha}u$ so that $u_{j}:=J_{1/j}u$ is the desired sequence.

Now, by definition, for each $x\in U$, by the dominated convergence theorem,

$\D^{\alpha}J_{\eps}u(x)=\eps^{-n}\int_{\Omega}\D^{\alpha}_{x}\left(\rho\left(\frac{x-z}{\eps}\right)\right)\cdot u(z)dz=\eps^{-n}\cdot (-1)^{|\alpha|}\int_{\Omega}\D^{\alpha}_{z}\left(\rho\left(\frac{x-z}{\eps}\right)\right)\cdot u(z)dz.$

In order to shift the derivative $\D_{z}^{\alpha}$ onto $u$, we need to verify that $z\mapsto \rho(\frac{x-z}{\eps})$ is compactly supported in $\Omega$ for sufficiently small $\eps>0$. Now, we know that $\supp (z\mapsto \rho(\frac{x-z}{\eps}))\subset B(x,\eps)$. Taking $\eps<\dist(U,\partial\Omega)$, we see that $B(x,\eps)\Subset \Omega$. Therefore, we may throw the derivative onto $u$:

$\Rightarrow \D^{\alpha}J_{\eps}u(x) = \eps^{-n}\int_{\Omega}\rho\left(\frac{x-z}{\eps}\right)\cdot \D^{\alpha}u(z)dz= J_{\eps}\D^{\alpha}u(x).$

The claim now follows from our remarks above.

Remark 2.6. In analogy with $\lloc^{p}$, we say that $u\in\wloc^{k,p}(\Omega)$ if $u\in \wloc^{k,p}(U)$ for every $U\Subset \Omega$, and given a sequence $\{u_{i}\}\subset \wloc^{k,p}(\Omega)$, we say that $u_{i}\xrightarrow{i\rightarrow\infty}u$ in $\wloc^{k,p}(\Omega)$ if $u_{i}\xrightarrow{i\rightarrow\infty}u$ in $W^{k,p}(U)$ for every $U\Subset \Omega$. In particular, the above lemma tells us that for every $u\in \wloc^{k,p}(\Omega)$, we may find a sequence $\{u_{i}:=J_{1/i}u\}_{i=1}^{\infty}\subset C^{\infty}(\Omega)$ converging to $u$ in $\wloc^{k,p}(\Omega)$.

The spaces $H^{k,p}$ and $H_{0}^{k,p}(\Omega)$

In light of the previous section, we know that we may approximate Sobolev functions locally by means of smooth functions. Recall from Theorem 1.1 that for $p<\infty$, $L^{p}(\Omega)=\overline{\cs(\Omega)}$. Motivated by this, we set

and introduce the subspaces

of $W^{k,p}(\Omega)$. Note that $C^{k,p}(\Omega)$ is not closed in $W^{k,p}(\Omega)$.
[[Counterexample: $\Omega=]-1,1[$, $k=1$, $p=1$, $f_{n}(x)=\sqrt{x^{2}+\frac{1}{n}}$]]

It is clear that $H_{0}^{k,p}(\Omega)\subset H^{k,p}(\Omega)\subset W^{k,p}(\Omega)$. We will show that the latter inclusion is actually an equality. We first recall the following technical lemma.

Lemma 2.7. Let $\{U_{\alpha}\}$ be an open cover of a set $A\subset\mathbb{R}^{n}$. There exists a countable family of smooth functions $\{\psi_{i}:\mathbb{R}^{n}\rightarrow[0,\infty[ \}$ such that

• for each $p\in A$, there exists an open set $U_{p}\ni p$ such that $\left.\psi_{i}\right|_{U_{p}}\equiv 0$ for all but finitely many $i$;
• $\sum_{i}\psi_{i}\equiv 1$; and
• for each $i$, there exists an $\alpha$ such that $\supp\psi_{i}\Subset U_{\alpha}$.

If the cover $\{U_{\alpha}\}$ is countable and locally finite, then we may find a collection of functions $\{\psi_{\alpha}\}$ with the same index set as the cover satisfying the above conditions. In either case, we call the collection $\{\psi_{i}\}$ a partition of unity.

Theorem 2.8. If $p<\infty$, then $H^{k,p}(\Omega)=W^{k,p}(\Omega)$.

Proof. We will show that any $u\in W^{k,p}(\Omega)$ may be approximated by a $v\in C^{\infty}(\Omega)\cap C^{k,p}(\Omega)$. This will then establish the claim. For each $k\in\mathbb{N}$, let $\Omega_{k}=B(0,k)\cap \{x\in\Omega:\dist(x,\partial\Omega)>\frac{1}{k}\}$. Clearly $\Omega=\bigcup_{k}\Omega_{k}$ and $\Omega_{k}\Subset \Omega_{k+1}\Subset\Omega$ for each $k$. Now set $\Omega_{0}:=\emptyset$ and

$U_{i}:=\Omega_{i+1}\backslash\overline{\Omega}_{i-1}$

for each $i\in\mathbb{N}$. We now have that $\bigcup_{i=1}^{k}U_{i}=\Omega_{k+1}$ so that $\{U_{i}\}_{i=1}^{\infty}$ forms an open cover of $\Omega$. Moreover, clearly $U_{i}\cap \Omega_{j}=\emptyset$ for all $j\leq i-1$, which implies that $\{U_{i}\}_{i=1}^{\infty}$ is a locally finite cover of $\Omega$. Let $\{\psi_{i}\}_{i=1}^{\infty}$ be a partition of unity subordinate to this cover (with the same index).

By Remark 2.4, $J_{\delta}\psi_{i}\cdot u\xrightarrow{\delta\searrow 0}\psi_{i}\cdot u$ in $W^{k,p}(\Omega)$ so that for each $\eps>0$ and $i\in \mathbb{N}$, we may choose a $\delta_{i}<\frac{1}{(i+1)(i+2)}<\dist(\Omega_{i+1},\partial\Omega)$ such that

$||J_{\delta_{i}}(\psi_{i}\cdot u) - \psi_{i}\cdot u||_{k,p}<\frac{\eps}{2^{i+1}}.$

This choice of $\delta_{i}$ ensures that $\supp J_{\delta}(\psi_{i}\cdot u)\subset \Omega_{i+2}\backslash\Omega_{i-2}$ so that for each $j\in\mathbb{N}$ we have the following equalities when restricting to $\Omega_{j}$:

$\begin{aligned} u&=\sum_{i=1}^{\infty}\psi_{i}\cdot u=\sum_{i=1}^{j+2}\psi_{i}\cdot u\\ v&:=\sum_{i=1}^{\infty}J_{\delta_{i}}(\psi_{i}\cdot u)=\sum_{i=1}^{j+2}J_{\delta_{i}}(\psi_{i}\cdot u) \end{aligned}$

Hence, for each $j\in\mathbb{N}$, the triangle inequality implies that

$||u- v||_{k,p,\Omega_{j}} \leq \sum_{i=1}^{j+2}||J_{\delta_{i}}(\psi_{i}\cdot u)-\psi_{i}\cdot u||_{k,p}< \frac{\eps}{2}.$

By the monotone convergence theorem, we may take the limit of both sides as $j\rightarrow\infty$ to obtain $||u-v||_{k,p}\leq \frac{\eps}{2}<\eps$.

Remark 2.9. The above claim is false for $p=\infty$ for much the same reason $L^{\infty}$ functions cannot be approximated by continuous functions.

Exercise 2.10. $u\in W^{1,1}(]a,b[)$ iff $u'$ exists almost everywhere, $u'\in L^{1}(]a,b[)$ and the fundamental theorem of calculus holds, i.e. for any $x,y\in\left]a,b\right[$ with $x<y$,

The case $\Omega=\mathbb{R}^{n}$ is particularly special.

Theorem 2.11. If $p<\infty$, then $W^{k,p}(\mathbb{R}^{n})= H_{0}^{k,p}(\mathbb{R}^{n})$.

Proof. Let $v\in C^{\infty}(\mathbb{R}^{n})\cap W^{k,p}(\mathbb{R}^{n})$ be such that $||u-v||_{k,p}< \frac{\eps}{2}$. Let $\varphi:\mathbb{R}^{n}\rightarrow [0,1]$ be a smooth function such that $\varphi\equiv 1$ on $B(0,1)$ and $\varphi \equiv 0$ outside $B(0,2)$. For each $\delta>0$, define the function

$v_{\delta}(x):=v(x)\cdot \varphi(\delta\cdot x).$

Clearly $v_{\delta}\in C_{0}^{\infty}(\mathbb{R}^{n})$. Set $V_{\delta}:=\{x\in\mathbb{R}^{n}:|x|>\frac{1}{\delta}\}$. The triangle inequality implies that

$||v-v_{\delta}||_{k,p}=||v-v_{\delta}||_{k,p,V_{\delta}}\leq ||v||_{k,p,V_{\delta}} + ||v_{\delta}||_{k,p,V_{\delta}}.$

Now, by the product rule,

$|\D^{\alpha}v_{\delta}|=\left|\sum_{\beta\leq \alpha}\left(\begin{matrix}\alpha\\ \beta \end{matrix} \right)\D^{\beta}v\cdot \delta^{|\alpha-\beta|}(\D^{\alpha-\beta}\varphi)(\delta x) \right| \leq C\cdot \sum_{\beta\leq\alpha}|\D^{\beta}v|,$

where $C$ is a constant depending only on $\alpha$, where we assume that $\delta\leq 1$. From the above and another application of the triangle inequality, we are left with the inequality $||v-v_{\delta}||_{k,p}\leq C_{0}\cdot ||v||_{k,p,V_{\delta}}$. However, if $w\in L^{p}(\mathbb{R}^{n})$, then by the monotone convergence theorem,

$\int_{\mathbb{R}^{n}}|w|^{p}=\lim_{\delta\searrow 0}\int_{\mathbb{R}^{n}\backslash V_{\delta}}|w|^{p}\Leftrightarrow \lim_{\delta\searrow 0}\int_{V_{\delta}}|w|^{p}.$

Therefore, we may choose $\delta$ sufficiently small so that $||v||_{k,p,V_{\delta}}<\frac{\eps}{2C_{0}}$. This then implies that

$||u-v_{\delta}||_{k,p}\leq ||u-v||_{k,p} + ||v-v_{\delta}||_{k,p} < \eps.$