Week 5: Sobolev embedding theorems (continued)

Morrey's inequality

We now turn our attention to $W^{1,p}(\Omega)$ in the case $p>n$. In this case, we obtain considerably more regularity. We first establish the decisive inequality.

Theorem 5.1 (Morrey's inequality). Suppose $n>1$ and $p\in]n,\infty]$, and let $\gamma=1-\frac{n}{p}$. There exists a constant $C$ depending only on $n$ and $p$ such that for any $u\in C^{1}(\mathbb{R}^{n})$,

Proof. Fix $x\in \mathbb{R}^{n}$ and $r>0$. We proceed in steps:

Step 1. There exists a constant $C>0$ depending only on $n$ such that

$\frac{1}{\mu(B(x,r))}\int_{B(x,r)}|u(y)-u(x)|dy \leq C\cdot \int_{B(x,r)}\frac{|\D u(y)|}{|y-x|^{n-1}}dy.$

Subproof. We compute, using the coarea formula:

$\begin{aligned}\frac{1}{\mu(B(x,r))}\int_{B(x,r)}|u(y)-u(x)|dy &= \frac{1}{\mu(B(x,r))}\cdot \int_{0}^{r}\int_{\partial B(x,t)}|u(y)-u(x)|dy\ dt\\&=\underbrace{\frac{1}{\mu(B(x,r))}\int_{0}^{r}t^{n-1}\int_{\partial B(0,1)}|u(x+tz) - u(x)|dz\ dt}_{\leq\int_{\partial B(0,1)}\int_{0}^{t}|\D u|(x+sz)ds\ dz}\\ &\leq \frac{1}{\omega_{n}r^{n}}\int_{0}^{r}t^{n-1}\int_{0}^{t}\underbrace{\int_{\partial B(0,1)}|\D u|(x+sz)dz}_{=\int_{\partial B(x,s)|\D u|\cdot s^{1-n}}}\ ds\ dt\\ &=\frac{1}{\omega_{n}r^{n}}\int_{0}^{r}t^{n-1}\underbrace{\int_{0}^{t} \int_{\partial B(x,s)}\frac{|\D u|(z)}{|x-z|^{n-1}}dz\ ds}_{\leq \int_{B(x,r)} \frac{|\D u|(y)}{|x-y|^{n-1}}dy}\ dt\\& = \frac{1}{n\omega_{n}}\int_{B(x,r)}\frac{|\D u|(y)}{|x-y|^{n-1}}dy.\end{aligned}$

Step 2. There exists a constant $C>0$ depending only on $n$ and $p$ such that

$\sup_{\mathbb{R}^{n}}|u|\leq C\cdot ||u||_{1,p}.$

Subproof. Fix $x\in\mathbb{R}^{n}$ and note that

$\omega_{n}\cdot |u(x)| = \int_{B(x,1)}|u(x)|dy \leq \int_{B(x,1)}|u(x)-u(y)|dy + \int_{B(x,1)}|u(y)|dy.$

On the one hand, the second term may be controlled using Hölder's inequality:

$\int_{B(x,1)}|u|\leq \omega_{n}^{1-\frac{1}{p}}||u||_{p,\mathbb{R}^{n}}$

On the other hand, by Step 1, we may also handle the first term using Hölder's inequality:

$\begin{aligned}\int_{B(x,1)}|u(x)-u(y)|dy &\leq \int_{B(x,1)}\frac{|\D u|(y)}{|y-x|^{n-1}}dy\\ &\leq ||\D u||_{p,B(x,1)}\cdot \left(\int_{B(x,1)}\frac{1}{|y-x|^{(n-1)p/(p-1)}}dy \right)^{1-1/p}\\&=\left(n\cdot \omega_{n}\cdot \frac{p-1}{p-n} \right)^{1-\frac{1}{p}}\cdot ||\D u||_{p,B(x,1)}\end{aligned}.$

Altogether, we are left with the inequality

$|u(x)|\leq \omega_{n}^{-1/p}\cdot\left(n\cdot \frac{p-1}{p-n} \right)^{1-\frac{1}{p}} \left(||u||_{p} + ||\D u||_{p} \right)$

Step 3. There exists a constant $C>0$ depending only on $n$ and $p$ such that

$[u]_{\gamma}\leq C\cdot ||\D u||_{p}.$

Subproof. Let $x,y\in\mathbb{R}^{n}$ with $x\neq y$ and set $\Omega=B(x,|x-y|)\cap B(y,|x-y|)$. As before, we first note that

$\mu(\Omega)\cdot |u(x)-u(y)| \leq \int_{\Omega} |u(x)-u(z)|dz + \int_{\Omega} |u(y)-u(z)|.$

By step 1, we may estimate the first term as

$\int_{\Omega}|u(x)-u(z)|dz\leq\int_{B(x,|x-y|)}|u(x)-u(z)|dz \leq \frac{\mu\left(B(x,|x-y|)\right)}{n\omega_{n}}\cdot \int_{B(x,|x-y|)}\frac{|\D u|(z)}{|x-z|^{n-1}}dz.$

Now, Hölder's inequality implies that

$\int_{B(x,|x-y|)}\frac{|\D u|(z)}{|x-z|^{n-1}}dz \leq ||\D u||_{p}\cdot \left(\int_{B(x,|x-y|)} \frac{1}{|x-z|^{(n-1)p/(p-1)}}dz \right)=(n\omega_{n})^{(p-1)/p}|x-y|^{\gamma}\cdot ||\D u||_{p}.$

Arguing similarly for the second term, we are left with the inequality

$\frac{|u(x)-u(y)|}{|x-y|^{\gamma}}\leq (n\omega_{n})^{-\frac{1}{p}}\cdot\frac{\mu(B(x,|x-y|)) }{\mu(\Omega)}\cdot ||\D u||_{p}.$

It now suffices to show that $\frac{\mu(B(x,|x-y|))}{\mu(\Omega)}$ is bounded from above by a constant depending only on $n$. To see that this is the case, note that $z:=x+\frac{1}{2}(y-x)\in \Omega$ and $B(z,\frac{1}{2}|y-x|)\subset\Omega$ so that

$\frac{\mu(B(x,|x-y|))}{\mu(\Omega)}\leq \frac{\mu(B(x,|x-y|))}{\mu(B(z,\frac{1}{2}|y-x|))} = 2^{n}.$

Combining all of the above then establishes the claim.

As with the Gagliardo-Nirenberg-Sobolev inequality, we may use this inequality to prove an embedding theorem.

Corollary 5.2. Suppose $n>1$, $V\subset\mathbb{R}^{n}$ is open, $p\in\left]n,\infty\right[$ and $\gamma=1-\frac{n}{p}$. There exists a continuous embedding

such that for all $u\in H_{0}^{1,p}(V)$, $\iota(u)\equiv u$ almost everywhere. In particular, if $\Omega\subset\mathbb{R}^{n}$ is a bounded domain of class $C^{1}$, then we also obtain the existence of a continuous embedding $\iota:W^{1,p}(\Omega)\hookrightarrow C^{\gamma}(\overline{\Omega})$ such that for all $u\in W^{1,p}(\Omega)$, $\iota(u)\equiv u$ almost everywhere.

Proof. For clarity, we write the elements of $H_{0}^{1,p}(V)$ as equivalence classes $[u]$.

Consider the inclusion $\iota:C_{0}^{1}(V)\subset H_{0}^{1,p}(V)\rightarrow C^{\gamma}(\overline\Omega)$ defined by $[u]\mapsto u$. Morrey's inequality implies that for all $u\in C_{0}^{1}(V)\subset C^{1}(\mathbb{R}^{n})$,

$[\iota([u])]_{\gamma}\leq c(n,p)\cdot ||[u]||_{1,p}.$

By Lemma 4.5, $\iota$ extends uniquely to a continuous linear mapping $\iota:H_{0}^{1,p}(V)\rightarrow C^{\gamma}(\overline\Omega)$. If $\{u_{n}\}$ is an approximating sequence for some $[u]\in H_{0}^{1,p}(V)$, then since both $\{[u_{n}]\}$ and $\{[\iota([u_{n}])]=[u_{n}]\}$ converge in $\lloc^{1}(V)$ by Hölder's inequality, we must have that $\iota([u])\equiv u$ almost everywhere for all $u\in H_{0}^{1,p}(V)$, i.e. $\iota$ is a continuous embedding with the desired property. The latter claim follows from choosing $V$ such that $\Omega\Subset V$ and considering the following chain of continuous embeddings, the last one being a restriction:

$W^{1,p}(\Omega)\hookrightarrow H_{0}^{k,p}(V)\hookrightarrow C^{\gamma}(\overline V)\hookrightarrow C^{\gamma}(\overline \Omega)$

Remark 5.3. In the case where $n=1$, we have the inclusion

which depends only on an application of Hölder's inequality and the characterisation of $W^{1,1}(]a,b[)$ in terms of absolutely continuity. In fact, for $u\in W^{1,p}(]a,b[)$, there exists a $v\in C^{1-\frac{1}{p}}([a,b])$ such that $u\equiv v$ a.e. and

for all $x,y\in]a,b[$.

Remark 5.4. Note that if $\Omega$ is a bounded domain of class $C^{1}$ and $u\in W^{1,\infty}(\Omega)$, then $u\in W^{1,p}(\Omega)$ for all $p<\infty$ so that $u\in C^{1-\frac{n}{p}}(\overline\Omega)$ for every $p>n$ after possibly redefining $u$ on a set of measure zero. By the proof of Morrey's inequality, we actually have the estimate

for $x,y\in\Omega$ with $x\neq y$. We may take the limit $p\rightarrow\infty$ on both sides to obtain

This implies that $u$ is actually Lipschitz continuous. The same argument applies for $u\in H_{0}^{1,p}(V)$ provided $V$ is bounded.

Higher-order Sobolev inequalities

By appropriately successively applying the Gagliardo-Nirenberg-Sobolev inequality and Morrey's inequality, we obtain the following higher-order Sobolev embeddings:

Theorem 5.5. Suppose $\Omega\subset\mathbb{R}^{n}$ is a bounded domain of class $C^{1}$.

• If $k<\frac{n}{p}$ and $q\in\mathbb{R}$ satisfies $\frac{1}{q}=\frac{1}{p}-\frac{k}{n}$, then there is a continuous embedding $\iota:W^{k,p}(\Omega)\hookrightarrow L^{q}(\Omega)$ with $\iota(u)\equiv u$ almost everywhere.
• If $k>\frac{n}{p}$, then there is a continuous embedding $\iota:W^{k,p}(\Omega)\hookrightarrow C^{K,\gamma}(\overline \Omega)$ with $\iota(u)\equiv u$ almost everywhere, where $K=k-\lfloor\frac{n}{p}\rfloor -1$ and

Remark 5.6. The idea here is that depending on how weakly differentiable $u$ is relative to $n$ and $p$, it is either in a higher $L^{p}$ space or a (higher) Hölder space. For example, fixing $m\in\mathbb{N}$ and $p=2$, we may ask the following question: How large should $k$ be so that $W^{k,2}(\Omega)\hookrightarrow C^{m}(\Omega)$? According to the latter case, we ought to have $K\geq m$, i.e. $k\geq m+\lfloor\frac{n}{2}\rfloor +1$. Therefore, if $u\in W^{k,2}(\Omega)$ for all $k\in\mathbb{N}$, then $u\in C^{\infty}(\Omega)$. These higher embedding theorems will be crucial in proving classical differentiability of weak solutions to elliptic PDE later on in the course.

Compact embeddings

Recall that the Gagliardo-Nirenberg-Sobolev inequality implies the continuous embedding

whenever $\Omega$ is a bounded domain of class $C^{1}$ and $p\in\left[1,n\right[$, where $p^{\ast}=\frac{np}{n-p}$ and $q\in[1,p^{\ast}[$. It turns out we can say more about the latter embedding.

Theorem 5.7 (Rellich-Kondrachov). Suppose $\Omega\subset\mathbb{R}^{n}$ is a bounded domain of class $C^{1}$. If $p\in[1,n[$ and $q\in[1,p^{\ast}[$, then the embedding $W^{1,p}(\Omega)\hookrightarrow L^{q}(\Omega)$ is compact.

Proof. We shall show that the embedding $H_{0}^{1,p}(V)\hookrightarrow L^{q}(V)$ is compact for $V\subset\mathbb{R}^{n}$ open and bounded. Choosing $V$ so that $V\Supset\Omega$ then establishes the claim (why?). Let $\{u_{m}\}_{m=1}^{\infty}\subset H_{0}^{1,p}(V)$ be a bounded sequence. We proceed in steps.

Step 1. $J_{\eps}u_{m}\xrightarrow{\eps\searrow 0}u_{m}$ in $L^{q}(V)$ uniformly in $m$.

Subproof. Fix $v\in C_{0}^{\infty}(V)$ and compute for $x\in V$ that

$(J_{\eps}v-v)(x)=\int_{B(0,1)}\rho(y)\cdot (v(x-\eps y)-v(x))dy=-\eps\int_{B(0,1)}\rho(y)\int_{0}^{1}\D v(x-\eps t y)\cdot y\ dt\ dy,$

whence

$\int_{V}|J_{\eps}v - v|\leq \eps\int_{V}\int_{B(0,1)}\int_{0}^{1}\rho(y)\cdot |\D v(x-\eps t y)|dt\ dy\ dx.$

By Tonelli's theorem, we may interchange the integrals at will. In particular, since

$\int_{V} |\D v(x-\eps t y)|dx= \int_{V-\eps t y}|\D v|\leq \int_{V}|\D v|$

due to the fact that $\supp v\Subset V$, we are left with the inequality

$\int_{V}|J_{\eps}v - v|\leq \eps\cdot \int_{V}|\D v|.$

By approximation, we deduce that this inequality holds for all $v\in H_{0}^{1,p}(V)$ so that

$\int_{V}|J_{\eps}u_{m} - u_{m}|\leq \eps||\D u_{m}||_{1} \leq \eps\cdot ||\D u_{m}||_{p}\cdot (\mu(V))^{1-\frac{1}{p}}.$

Therefore, $J_{\eps}u_{m}\xrightarrow{\eps\searrow 0}u_{m}$ in $L^{1}$ uniformly in $m$. To establish uniform convergence in $L^{q}(V)$, we make use of the interpolation inequality

$||J_{\eps}u_{m}-u_{m} ||_{q}\leq ||J_{\eps}u_{m}-u_{m}||_{1}^{\theta} \cdot ||J_{\eps}u_{m}-u_{m}||_{p^{\ast}}^{1-\theta},$

where $\theta\in\left]0,1\right[$ is such that $\frac{1}{q} = \theta + \frac{1-\theta}{p^{\ast}}$. Since $||J_{\eps}u_{m}||_{p^{\ast}}\leq ||u_{m}||_{p^{\ast}}$ and by Gagliardo-Nirenberg-Sobolev $||u_{m}||_{p^{\ast}}\leq \textup{const}\cdot ||\D u_{m}||_{p}\leq\textup{const}$, we immediately see that $J_{\eps}u_{m}\xrightarrow{\eps\searrow 0}u_{m}$ in $L^{q}$ uniformly in $m$.

Step 2. For fixed $\eps>0$, the sequence $\{J_{\eps}u_{m}\}_{m=1}^{\infty}$ is uniformly bounded and equicontinuous.

Subproof. Note that for $x\in\mathbb{R}^{n}$,

$|J_{\eps}u_{m}(x)|=\eps^{-n}\left|\int_{\mathbb{R}^{n}}\rho(\frac{x-z}{\eps})u(z)dz \right|\leq\eps^{-n}\cdot \sup\rho\cdot ||u_{m}||_{1}\leq C(\eps,\sup_{m}||u_{m}||_{p}).$

This establishes boundedness. As for equicontinuity, we use the same argument and appeal to Exercise 1 on Tutorial sheet 1:

$|\D J_{\eps}u_{m}(x)|= \eps^{-(n+1)}\left| \int_{\mathbb{R}^{n}}\D \rho(\frac{x-z}{\eps})u(z)dz \right|\leq C(\eps,\sup_{m}||u_{m}||_{p}).$

Step 3. For fixed $\delta>0$ there exists a subsequence $\{u_{m_{j}}\}$ of $\{u_{m}\}$ such that $\limsup_{j,k\rightarrow\infty}||u_{m_{j}}-u_{m_{k}}||_{q}\leq \delta$.

Subproof. By Step 1, there exists an $\eps>0$ such that $||J_{\eps}u_{m}-u_{m}||_{q}\leq \frac{\delta}{2}$ for all $m\in\mathbb{N}$, and by Step 2, we may appeal to the Arzelà-Ascoli theorem to pass to a subsequence $\{J_{\eps}u_{m_{j}}\}$ converging in $C(V)$, i.e.

$\lim_{j,k\rightarrow\infty}\sup_{V}|J_{\eps}u_{m_{j}}-J_{\eps}u_{m_{k}}|=0.$

This implies that $\lim_{j,k\rightarrow\infty}||J_{\eps}u_{m_{j}}-J_{\eps}u_{m_{k}}||_{q}=0$. Altogether, we obtain that

$\limsup_{j,k\rightarrow\infty}||u_{m_{j}}-u_{m_{k}}||_{q}\leq \limsup_{j,k\rightarrow\infty}\left(||u_{m_{j}}-J_{\eps}u_{m_{j}}||_{q} + ||J_{\eps}u_{m_{j}} - J_{\eps}u_{m_{k}}||_{q} + ||J_{\eps}u_{m_{k}} - u_{m_{k}}||_{q} \right)\leq \delta.$

Now, applying Step 3 with $\delta=1$, we extract a subsequence $\{u_{m_{j}}^{1} \}$ of $\{u_{m}\}$ with $\limsup_{j,k\rightarrow\infty}||u_{m_{j}}^{1}-u_{m_{k}}^{1}||_{q}\leq 1$. More generally, for each $l\in \{2,3,\dots\}$, we inductively define the subsequence $\{u^{l}_{m_{j}}\}_{j=1}^{\infty}$ of $\{u^{l-1}_{m_{j}}\}_{j=1}^{\infty}$ obtained from Step 3 with $\delta=\frac{1}{l}$, i.e. such that

$\limsup_{j,k\rightarrow\infty}||u^{l}_{m_{j}}-u^{l}_{m_{k}}||_{q}\leq \frac{1}{l}$

for each $l\in\mathbb{N}$. Finally, we consider the sequence $\{v_{j}\}_{j=1}^{\infty}$ such that $v_{j}=u^{j}_{m_{j}}$, which is a subsequence of $\{u_{m}\}$. Moreover, for each $N\in\mathbb{N}$, $\{v_{l}\}_{l=k}^{\infty}$ is a subsequence of $\{u^{N}_{m_{j}}\}_{j=1}^{\infty}$ whenever $k\geq N$ so that

$\limsup_{j,k\rightarrow\infty}||v_{j}-v_{k}||_{q}=\limsup_{j,k\rightarrow\infty}||u^{N}_{m_{j}}-u^{N}_{m_{k}}||_{q}\leq \frac{1}{N}.$

Since $N$ is arbitrary, we conclude that $||v_{j}-v_{k}||\xrightarrow{j,k\rightarrow\infty}0$ so that $\{v_{j}\}$ is the desired subsequence of $\{u_{m}\}$.

Remark 5.8. Since $p^{\ast}=\frac{np}{n-p}>p$ and $p^{\ast}\rightarrow\infty$ as $p\nearrow n$, we have that the embedding $W^{1,p}(\Omega)\hookrightarrow L^{p}(\Omega)$ is compact for $p\in[1,n]$. Moreover, for $p>n$, we may use Morrey's inequality and the Arzelà-Ascoli theorem to conclude that this embedding is compact in the case where $p>n$. Using similar techniques, we also deduce that the embedding $H_{0}^{1,p}(\Omega)\hookrightarrow L^{p}(\Omega)$ is compact for all $p$.