Week 1: Preliminaries and Sobolev spaces

References

[Fr]: Partial Differential Equations by Avner Friedman
[E]: Partial Differential Equations by Lawrence C. Evans
[Z2A]: Nonlinear Functional Analysis and its Applications II/A by Eberhard Zeidler

Outline

We wish to better understand second-order elliptic boundary-value problems:

\begin{aligned} Lu:=\sum_{i,j=1}^{n}a^{ij}\frac{\partial^{2}u}{\partial x^{i}\partial x^{j}} + \sum_{i=1}^{n}b^{i}\frac{\partial u}{\partial x^{i}} + c\cdot u &= f\ \textup{in}\ \Omega\\ \left.u\right|_{\partial\Omega}&\ \textup{given}, \end{aligned} \tag{\#}

where $a^{ij}$ , $b^{i}$ , $c$ and $\left.u\right|_{\partial\Omega}$ are sufficiently 'nice' functions and $(a^{ij})_{i,j=1}^{n}$ is a positive-definite matrix.

...as well as their heat equation counterpart:

\begin{aligned} \partial_{t}u(\cdot,t)&=Lu(\cdot,t)\ \textup{in}\ \Omega\ \textup{for}\ t\in[0,T]\\ \left.u\right|_{\Omega\times\{0\}}&,\left.u\right|_{\partial\Omega\times[0,T]}\ \textup{given}. \end{aligned}

Aspects of such problems that we wish to consider are:

Existence: Under what conditions can we guarantee the existence of solutions to (#)?
Regularity: How smooth can we expect $u$ to be, either in the interior ( $\Omega$ ) or up to the boundary ( $\partial\Omega$ )?
Example: if $-\Delta u = f$ with $f\in C^{k}(\Omega)$ , is $u\in C^{k+2}(\Omega)$ ?
Solution methods: When can we (rigorously) find explicit solutions to (#)?
Approximation methods: Can we 'discretise' the system (#) in some manner to approximate solutions and establish convergence of such a discretisation scheme?

Notation

Analysis

For $x=(x^{1},\dots,x^{n})\in \mathbb{R}^{n}$ and $r>0$ , write $B(x,r):=\{y\in \mathbb{R}^{n}:|y-x|=\left(\sum_{i=1}^{n}(x^{i}-y^{i})^{2}\right)^{1/2}<r\}$ .
Given a measurable set $U\subset\mathbb{R}^{n}$ , write $\mu(U)$ for its Lebesgue measure.
Write $\mathbb{K}$ for $\mathbb{R}$ or $\mathbb{C}$ . We will always deal with functions defined on (subsets of) $\mathbb{R}^{n}$ taking their values in $\mathbb{K}$ .

Ω will always be a nonempty open set in Rⁿ!

Multiïndex notation

A multiïndex is an $n$ -tuple $\alpha=(\alpha_{1},\dots,\alpha_{n})\in\mathbb{Z}^{n}$ , $\alpha_{i}\geq 0$ . Addition is defined in the usual way. We set:

$|\alpha|:=\sum_{i=1}^{n}\alpha_{i}$
$\alpha!:=\alpha_{1}!\cdot\dots\cdot\alpha_{n}!$
$v^{\alpha}:=(v^1)^{\alpha_{1}}\cdot\dots\cdot(v^{n})^{\alpha_{n}}$ for $v=(v^{1},\dots,v^{n})\in\mathbb{R}^{n}$

We also introduce a partial ordering on multiïndices such that $\alpha\leq\beta$ iff for all $i\in\{1,\dots,n\}$ , $\alpha_{i}\leq \beta_{i}$ .

Using multiïndices $\alpha$ with $|\alpha|\leq k$ , we may abbreviate derivatives of $C^{k}$ functions $u$ as

\D^{\alpha}u:=\frac{\partial^{|\alpha|}u}{(\partial x^{1})^{\alpha_{1}}\dots(\partial x^{n})^{\alpha_{n}}}=\partial^{\alpha_{1}}_{1}\dots \partial_{n}^{\alpha_{n}}u.

$\rightsquigarrow$ Integration by parts: If $\varphi$ vanishes close to $\partial\Omega$ , then

\int_{\Omega}\D^{\alpha}f\cdot \varphi = (-1)^{|\alpha|}\int_{\Omega}f\cdot \D^{\alpha}\varphi.

$\rightsquigarrow$ Taylor's theorem: Whenever $\Omega$ is open and connected, $x,x_{0}\in \Omega$ and $f\in C^{k}(\Omega)$ ,

f(x)=\sum_{|\alpha|\leq k}\frac{\D^{\alpha}f(x_{0})}{\alpha!}\cdot (x-x_{0})^{\alpha} + O(|x-x_{0}|^{k})\ \textup{as}\ x\rightarrow x_{0}.

Preliminaries

Inequalities

The following inequalities will be used and abused:

Cauchy's inequality: For all $a,b\in\mathbb{R}$ ,

$a\cdot b \leq\frac{1}{2}(a^{2}+b^{2}).$
An immediate consequence is the Peter-Paul inequality:

$a\cdot b \leq \eps a^{2} + \frac{b^{2}}{4\eps}.$
Young's inequality: For all $a,b\in\mathbb{R}^{+}$ and $p,q>1$ such that $\frac{1}{p}+\frac{1}{q}=1$ ,

$a\cdot b\leq \frac{a^{p}}{p} + \frac{b^{q}}{q}.$
An immediate consequence is Young's inequality with $\eps$ or the generalised Peter-Paul inequality:

$a\cdot b \leq \eps a^{p} + C(\eps,p)b^{q}$
Cauchy-Schwarz inequality: For $x,y\in\mathbb{R}^{n}$ , $|x\cdot y|\leq |x|\cdot |y|$ . More generally, if $(a_{ij})$ is a non-negative definite symmetric $n\times n$ matrix, then

$\left(\sum_{i,j=1}^{n}a_{ij}x^{i}y^{j} \right)^{1/2}\leq (\sum_{i,j=1}^{n}a_{ij}x^{i}x^{j})^{1/2}\cdot (\sum_{i,j=1}^{n}a_{ij}y^{i}y^{j})^{1/2}.$
Minkowski's inequality: For $x,y\in\mathbb{R}^{n}$ ,

\left(\sum_{i=1}^{n}|x^{i}+y^{i}|^{p} \right)^{1/p}\leq \left(\sum_{i=1}^{n}|x^{i}|^{p}\right)^{1/p} + \left(\sum_{i=1}^{n}|y^{i}|^{p}\right)^{1/p}.

Integral convergence theorems

We will often interchange limits and integrals. To this end, we recall the following:

Fatou's Lemma: Suppose $\{f_{k}\}_{k=1}^{\infty}$ and $\liminf_{k\rightarrow\infty}f_{k}$ are summable (i.e. $\int|f_{k}|,\int\liminf_{k\rightarrow\infty}|f_{k}|<\infty$ ). Then

$\int \liminf_{k\rightarrow\infty} f_{k} \leq \liminf_{k\rightarrow\infty}\int f_{k}.$
Monotone convergence theorem: Suppose $\{f_{k}\}_{k=1}^{\infty}$ is a sequence of measurable functions on $\mathbb{R}^{n}$ with

$f_{1}\leq f_{2}\leq\dots\leq f_{k} \leq f_{k+1}\leq \dots\ .$
Then $\displaystyle \int\lim_{k\rightarrow\infty} f_{k}=\lim_{k\rightarrow\infty}\int f_{k}$ .
Dominated convergence theorem: Suppose $\{f_{k}\}_{k=1}^{\infty}$ is a sequence of integrable functions (i.e. measurable and $\int f^{-}$ or $\int f^{+}$ finite) such that $f_{k}\rightarrow f$ almost everywhere. If $|f_{k}|\leq g$ a.e. f for some summable $g$ , then

$\lim_{k\rightarrow\infty}\int f_{k} = \int f.$

Topological vector spaces

We will study and make heavy use of spaces of functions equipped with different notions of convergence.

Basic notions.

Let $X$ be a (real or complex) vector space.

A seminorm is given by a nonnegative function $||\cdot||:X\rightarrow\mathbb{R}$ s.t.
- for all $\lambda\in \mathbb{K}$ and $v\in X$ , $||\lambda v||=|\lambda|\cdot||v||$ (positive homogeneïty); and
- for all $v,w\in X$ , $||v+w||\leq ||v|| + ||w||$ (triangle inequality).
A norm is a semi-norm $||\cdot||$ such that $||v||=0\Rightarrow v=0$ .
Given a countable family $\{|\cdot|_{i}\}_{i=0}^{\infty}$ of semi-norms such that $|v|_{i}=0$ for all $i$ $\Rightarrow$ $v=0$ , we naturally obtain a metric on $X$ :

$\rho(v,w):=\sum_{i=0}^{\infty}2^{-i}\cdot \frac{|v-w|_{i}}{1+|v-w|_{i}}.$
The topology induced by this metric is independent of the ordering of the semi-norms. If $X$ is equipped with such a family of semi-norms, call the pair $(X,\{|\cdot|_{i}\}_{i=0}^{\infty})$ a countably normed space.
If $X$ is equipped with only a single norm $||\cdot||$ , call the pair $(X,||\cdot||)$ a normed space. Here we use the equivalent metric $\rho(v,w)=||v-w||$ .
If $(X,||\cdot||)$ is a normed space and there exists a Hermitian form $\left<\cdot,\cdot\right>:X\rightarrow\mathbb{K}$ such that for all $v\in X$ , $||v||=\sqrt{\left<v,v\right>}$ , call the pair $(X,\left<\cdot,\cdot\right>)$ an inner product space. The Cauchy-Schwarz inequality holds in such spaces:

$|\left<v,w\right>|\leq ||v||\cdot ||w||$
A sequence $\{v_{n}\}\subset (X,\{|\cdot|_{i}\}_{i=0}^{\infty})$ is said to be
- a Cauchy sequence if $\rho(v_{n},v_{m})\xrightarrow{n,m\rightarrow\infty}0$ ( $\Leftrightarrow\forall i\ |v_{n}-v_{m}|_{i}\xrightarrow{n,m\rightarrow\infty}0$ );
- a (strongly) convergent sequence if there exists a $v\in X$ such that $\rho(v_{n},v)\xrightarrow{n,m\rightarrow\infty}0$ ( $\Leftrightarrow\forall i\ |v_{n}-v|_{i}\rightarrow 0$ ). This $v$ is unique and we write
  
  $'v_{n}\xrightarrow{n\rightarrow\infty}v\textup{ in }X.'$
A convergent sequence is always a Cauchy sequence. If every Cauchy sequence in $X$ converges, we call it a complete space. Complete countably normed, normed and inner product spaces are referred to as Fréchet, Banach and Hilbert spaces respectively.
A Fréchet, Banach or Hilbert space is separable if it contains a countable, dense subset, i.e. if there exists a countable subset $\{v_{n}\}_{n=1}^{\infty}$ such that for any $\eps>0$ and $v\in X$ , there exists an $n$ such that $\rho(v_{n},v)<\eps$ .
Separable Hilbert spaces admit orthonormal bases, i.e. vectors $\{v_{n}\}_{n=1}^{\infty}\subset X$ such that
- $\left<v_{n},v_{m}\right>=\delta_{nm}$ ; and
- for any $v\in V$ , $\displaystyle v=\sum_{n=1}^{\infty}\left<v_{n},v\right>v_{n}$ .
The dual space of a Banach or Hilbert space $X$ , which is itself a Banach space, is given by

$X^{\ast}:=\{f:X\rightarrow\mathbb{\mathbb{K}}\ \textup{linear}: \sup_{||v||=1}|f(v)|<\infty \}.$
The condition $\sup\limits_{||v||=1}|f(v)|<\infty$ is equivalent to the condition

$v_{n}\xrightarrow{n\rightarrow\infty}v\ \textup{in }X\Rightarrow f(v_{n})\xrightarrow{n\rightarrow\infty}f(v).$
A sequence $\{v_{n}\}_{n=1}^{\infty}\subset X$ in a Banach space is said converge weakly to $v\in X$ , written $v_{n}\xrightharpoondown{n\rightarrow\infty} v$ if for all $f\in X^{\ast}$ , $f(v_{n})\xrightarrow{n\rightarrow\infty} f(v)$ . Weak limits are also unique by the Hahn-Banach Theorem.

Examples

Continuous and differentiable functions. For $k\in \mathbb{N}$ , we define the sets

$\begin{aligned} C^{0}(\Omega)&:=\{u:\Omega\rightarrow\mathbb{K}\ \textup{continuous}\}\\ C^{k}(\Omega)&:=\{u:\Omega\rightarrow\mathbb{K}:\ \forall|\alpha|\leq k\ \ \D^{\alpha}u\ \textup{exists and is in}\ C^{0}(\Omega)\}\end{aligned}$
as well as

$\begin{aligned} C^{0}(\overline{\Omega})&:=\{u:\overline{\Omega}\rightarrow\mathbb{K}\ \textup{continuous}\}\\ C^{k}(\overline\Omega)&:=\{u:\overline\Omega\rightarrow\mathbb{K}:\forall|\alpha|\leq k\ \D^{\alpha}u\ \textup{exists, is in }C^{0}(\Omega)\ \textup{and is continuously extensible to}\ \overline\Omega\}.\end{aligned}$
For $i\in\{0,\dots,k\}$ and a family of $\{K_{j}\Subset\Omega\}_{j=0}^{\infty}$ with $\bigcup_{j=0}^{\infty}K_{j}=\Omega$ , introduce the seminorm

$|u|_{i,j}:=\sup_{|\alpha|=i}\sup_{K_{j}}|\D^{\alpha}u|.$
Properties:
- $(C^{k}(\Omega),\{|\cdot|_{i,j}:i\in\{0,\dots,k\},j\in\mathbb{N}\cup\{0\}\})$ is a Fréchet space. The underlying topology is independent of the choice of compact sets. Convergence in this space is referred to as local uniform convergence.
- If $\Omega$ is bounded, the norm $||u||_{k,\Omega}:=\sum_{i=0}^{k}\sup_{|\alpha|=i}\sup_{\Omega}|\D^{\alpha}u|$ gives $C^{k}(\overline\Omega)$ the structure of a Banach space.
- A very useful compactness criterion in $C^{0}(\overline\Omega)$ is the Arzéla-Ascôli theorem: If $\{u_{n}\}_{n=1}^{\infty}$ is bounded, i.e. $||u||_{0,\Omega}\leq C$ , and equicontinuous, i.e. for all $\eps>0$ there is a $\delta > 0$ such that for all $n\in \mathbb{N}$ ,
  
  $|x-y|<\delta\Rightarrow |u_{n}(x)-u_{n}(y)|<\eps,$
  then there is a subsequence $\{u_{n_{k}}\}$ of $\{u_{n}\}$ and a $u\in C^{0}(\overline\Omega)$ such that $u_{n_{k}}\xrightarrow{k\rightarrow\infty}u$ in $C^{0}(\overline{\Omega})$ (i.e. $u_{n_{k}}$ converges to $u$ uniformly).
Smooth functions: We similarly set $C^{\infty}(\Omega):=\bigcap_{k=0}^{\infty}C^{k}(\Omega)$ and $C^{\infty}(\overline{\Omega}):=\bigcap_{k=0}^{\infty}C^{k}(\overline\Omega)$ . Both of these sets are Fréchet spaces when equipped with the family of semi-norms $\{|\cdot|_{i,j}\}_{i,j\in\mathbb{N}\cup\{0\}}$ and $\{|\cdot|_{i,\Omega}\}_{i\in\mathbb{N}\cup\{0\}}$ respectively.

A distinguished subset of both of these spaces is the space of compactly supported test functions $C_{0}^{\infty}(\Omega)$ :

$C_{0}^{\infty}(\Omega):=\{u\in C^{\infty}(\Omega):\supp u\Subset \Omega\}$
⟦Recall that the support of a function $u:\Omega\rightarrow\mathbb{K}$ is given by $\supp u=\overline{\{x\in\Omega:u(x)\neq 0\}}$ .⟧

This set may be given a natural topology with the following notion of convergence:
- A sequence $\{u_{n}\}_{n=1}^{\infty}\subset C_{0}^{\infty}(\Omega)$ is said to converge in $C_{0}^{\infty}(\Omega)$ to $u\in C_{0}^{\infty}(\Omega)$ if there is some compact set $K\subset\Omega$ such $(\forall k)\ \supp u_{k}\subset K$ and $\left.u_{k}\right|_{K}\xrightarrow{k\rightarrow\infty} \left.u\right|_{K}$ in $C^{\infty}(K)$ .
Lebesgue spaces: For each $p\in[1,\infty]$ and measurable function $u:\Omega\rightarrow \mathbb{K}$ , set

$||u||_{p}:=\begin{cases}\displaystyle\left(\int_{\Omega}|u|^{p}\right)^{1/p},&p<\infty\\ \esssup_{\Omega} |u|,&p=\infty \end{cases}$
where $\esssup_{\Omega} f:=\inf\{K\in\mathbb{R}:f\leq K\ \textup{a.e.}\}$ for a real-valued function $f$ .

We now define the Lebesgue spaces $L^{p}(\Omega)$ (or $L^{p}(\Omega,\mathbb{K})$ ) as

$L^{p}(\Omega):=\{f\ \textup{measurable}: ||f||_{p}<\infty\}/\sim,$
where $f\sim g\Leftrightarrow f=g$ almost everywhere (a.e.). We will generally omit mention of equivalence classes and refer to the functions in question directly.

Properties:
- $L^{p}(\Omega)$ is a Banach space for any $p$ , separable for $p<\infty$ , and $L^{2}(\Omega)$ is a Hilbert space with inner product
  
  $\left<f,g\right>:=\int_{\Omega}\bar{f}\cdot g$
- $C^{0}(\Omega)$ is dense in $L^{p}(\Omega)$ , i.e. for all $u\in L^{p}(\Omega)$ and $\eps>0$ , there exists a $v\in C^{0}(\Omega)$ such that $||u-v||_{p}<\eps.$ In fact, $v$ may be taken to be compactly supported in $\Omega$ .
- If $u_{n}\xrightarrow{n\rightarrow\infty}u$ in $L^{p}(\Omega)$ , then there exists a subsequence $\{u_{n_{k}}\}$ of $\{u_{n}\}$ such that $u_{n_{k}}\xrightarrow{n\rightarrow\infty}u$ a.e.
- For $p\in\left[1,\infty\right[$ , $(L^{p}(\Omega))^{\ast}=L^{q}(\Omega)$ , where $q$ is s.t. $\frac{1}{p}+\frac{1}{q}=1$ , i.e. if $F\in (L^{p}(\Omega))^{\ast}$ , then there exists an $f\in L^{q}(\Omega)$ s.t. for all $g\in L^{p}(\Omega)$ ,
  
  $F(g)=\int_{\Omega}f\cdot g.$
- If $\{u_{n}\}_{n=1}^{\infty}\subset L^{p}(\Omega)$ is a bounded sequence, i.e. $||u_{n}||_{p}\leq C$ , then there exists a subsequence $\{u_{n_{k}}\}_{k=1}^{\infty}$ of $\{u_{n}\}$ and a $u\in L^{p}(\Omega)$ such that $u_{n_{k}}\xrightharpoondown{k\rightarrow\infty}u$ .
- The Hölder inequality holds: For $p,q>1$ with $\frac{1}{p}+\frac{1}{q}=1$ , $f\in L^{p}(\Omega)$ and $g\in L^{q}(\Omega)$ ,
  
  $\int_{\Omega}|f\cdot g|:\leq ||f||_{p}\cdot ||g||_{q}$
For later purposes, we define the auxiliary spaces

$\lloc^{p}(\Omega):=\{u:\Omega\rightarrow \mathbb{K}\ \textup{measurable}:\forall K\Subset\Omega\ \ u\in L^{p}(K)\}.$
By Hölder's inequality, we have that $\lloc^{p}(\Omega)\subset \lloc^{q}(\Omega)$ whenever $q<p$ . We say that a sequence converges in $\lloc^{p}(\Omega)$ if its restriction to $K$ converges in $\lloc^{p}(K)$ for every $K\Subset\Omega$ .
Hölder spaces: We call a function $f:\Omega\rightarrow\mathbb{K}$ Hölder continuous with exponent $\gamma\in\left]0,1\right]$ if there is a constant $C\geq 0$ s.t. for all $x,y\in\Omega$ ,

$|f(x)-f(y)|\leq C|x-y|^{\gamma}.$
⟦Note that the case $\gamma=1$ corresponds to Lipschitz functions.⟧

We introduce the $\gamma$ th Hölder seminorm

$[f]_{\gamma}:=\sup\left\{\frac{|f(x)-f(y)|}{|x-y|^{\gamma}}: x,y\in\Omega,\ x\neq y \right\}$
as well as the $k+\gamma$ th Hölder norm

$||f||_{k+\gamma}:=\sum_{|\alpha|\leq k}||\D^{\alpha}f||_{k,\Omega}+\sum_{|\alpha|=k}[\D^{\alpha}u]_{\gamma}.$
The Hölder spaces are then given by

$C^{k+\gamma}(\overline\Omega):=\{f\in C^{k}(\overline\Omega):||f||_{k+\gamma}<\infty\}.$
Exercise. $(C^{k+\gamma}(\overline{\Omega}),||\cdot||_{k+\gamma})$ is a Banach space.

Approximation by smooth functions

Let $\rho\in \cs(\mathbb{R}^{n})$ be such that $\supp\rho\subset \overline{B(0,1)}$ , $\rho\geq 0$ and $\int_{\mathbb{R}^{n}}\rho=1$ ; for example,

\rho(x):=\begin{cases}c\cdot e^{-1/(|x|^{2}-1)},&\ |x|<1\\ 0,&\ \textup{otherwise}\end{cases}

for a suitable constant $c>0$ .

Given $u\in \lloc^{1}(\Omega)$ , extend $u$ to all of $\mathbb{R}^{n}$ s.t. $\left.u\right|_{\mathbb{R}^{n}\backslash\Omega}\equiv 0$ and for each $\eps>0$ define the mollifier of $u$ to be the function

\begin{aligned} J_{\eps}u&:\mathbb{R}^{n}\rightarrow \mathbb{R}\\ J_{\eps}u(x)&:=\eps^{-n}\int_{\mathbb{R}^{n}}\rho(\frac{x-z}{\eps})\cdot u(z)dz=\int_{B(0,1)}\rho(z)u(x-\eps z)dz. \end{aligned}

Theorem 1.1. $J_{\eps}u\in C^{\infty}(\mathbb{R}^{n})$ and the following hold:

If $\supp u \Subset\Omega$ , then for $\eps<\dist(\supp u,\partial\Omega)$ , $J_{\eps}u\in \cs(\Omega)$ .
For any $u\in L^{p}(\Omega)$ , $p\in[1,\infty]$ , $||J_{\eps}u||_{p}\leq ||u||_{p}$ .
If $u\in C^{0}(\Omega)$ , then $J_{\eps}u\xrightarrow{\eps\searrow 0}u$ locally uniformly, i.e. for each $K\Subset\Omega$ ,

$\lim_{\eps\searrow 0}\sup_{K}|J_{\eps}u - u|=0.$
If $p\in[1,\infty[$ , then $||J_{\eps}u-u||_{p}\xrightarrow{\eps\searrow 0}0$ .
If $p\in[1,\infty]$ , then $\displaystyle\lim_{\eps\searrow 0}J_{\eps}u=u$ almost everywhere.

Proof. Note that $x\mapsto \rho(\frac{x-z}{\eps})$ and all of its derivatives are bounded functions of $z$ supported in $\overline{B(x,\eps)}$ , whence smoothness follows from the dominated convergence theorem. We tackle the other claims:

Let $x\in\mathbb{R}^{n}$ be such that $\dist(x,\supp u)>\eps$ and $z\in B(0,1)$ . The (reverse) triangle inequality implies that $\dist(x-\eps z,\supp u)>\dist(x,\supp u)-\eps>0$ so that

$J_{\eps}u(x)=\int_{B(0,1)}\rho(z)u(x-\eps z)dz=0,$
whence $\supp J_{\eps}u\subset \overline{B_{\eps}(\supp u)}$ . This establishes the claim for small $\eps$ .

We estimate, using Hölder's inequality using the conjugate pair $(\frac{p}{p-1},p)$ :

$\begin{aligned}|J_{\eps}u(x)|&= \left|\int_{B(0,1)}\rho(z)^{\frac{p-1}{p}}\cdot (\rho(z)u(x-\eps z)^{p})^{1/p} dz\right|\\ &\leq\underbrace{(\int_{B(0,1)}\rho)^{\frac{p-1}{p}}}_{=1}\cdot \left(\int_{B(0,1)}\rho(z)\cdot |u(x-\eps z)|^{p}d z\right)^{1/p} \end{aligned}$
Therefore, we see that

$\int_{\mathbb{R}^{n}}|J_{\eps}u|^{p} \leq \int_{\mathbb{R}^{n}}\int_{B(0,1)}\rho(z)\cdot |u(x-\eps z)|^{p}dz\ dx=\int_{B(0,1)}\rho(z)\cdot \int_{\mathbb{R}^{n}}|u(x-\eps z)|^{p}dx\ dz=||u||_{p}^{p}.$
The case $p=\infty$ follows immediately from the definition of $J_{\eps}u$ .

We compute that for $x\in K$ , $|J_{\eps}u(x) - u(x)|=|\int_{B(0,1)}\rho(z)\cdot [u(x-\eps z) - u(x)]dz|\leq \sup_{y\in B(x,\eps)}|u(y)-u(x)|$ , i.e.

$\sup_{K}|J_{\eps}u-u|\leq\sup_{x\in K}\sup_{y\in B(x,\eps)}|u(y)-u(x)|\xrightarrow{\eps\searrow 0}0.$

Let $u\in L^{p}(\Omega)$ and fix $\delta>0$ . By the density of $C^{0}(\Omega)$ in $L^{p}(\Omega)$ , we may find a $v\in C^{0}(\Omega)$ with compact support in $\Omega$ such that $||u-v||<\frac{\delta}{3}$ . Therefore, by the triangle inequality,

$||J_{\eps}u - u||_{p} \leq \underbrace{||J_{\eps}u - J_{\eps}v||_{p}}_{\leq ||u-v||_{p}} + ||J_{\eps}v - v||_{p} + ||v-u||_{p}<\frac{2\delta}{3} + ||J_{\eps}v-v||_{p}.$
By the preceding part, we may choose $\eps>0$ sufficiently small so that $||J_{\eps}v-v||_{p}<\frac{\delta}{3}$ .

Note that $|J_{\eps}u(x) - u(x)| = |\eps^{-n}\int_{\Omega}\rho(\frac{x-z}{\eps})\cdot (u(z)-u(x) )dz|\leq \sup\rho\cdot \eps^{-n}\int_{B(x,\eps)}|u(z)-u(x)|dz\xrightarrow{\eps\searrow 0}0$ by the Lebesgue differentiation theorem.

Using mollifiers, we may prove the following density result.

Lemma 1.2. $\cs(\Omega)$ is dense in $L^{p}(\Omega)$ , i.e. for any $u\in L^{p}(\Omega)$ and $\eps>0$ , there exists a $v\in \cs(\Omega)$ such that $||u-v||_{p}<\eps$ .

Proof. First assume $\Omega$ is bounded. For any $\delta>0$ , let $U=\{x\in \Omega: \dist(x,\partial\Omega)>\delta\}$ and set $v:=u\cdot \chi_{U}$ , where $\chi_{U}$ is the characteristic function of $U$ . Since

$||u-v||_{p}=\left(\int_{\Omega\backslash U}|u|^{p} \right)^{1/p}$
and $\mu(\Omega\backslash U)\searrow 0$ as $\delta\searrow 0$ , we may choose $\delta$ small enough so that $||u-v||_{p}<\frac{\eps}{3}$ , and $\supp v\subset\overline{U}\Subset\Omega$ . On the other hand, by the preceding theorem, $||J_{z}v-v||_{p}\xrightarrow{z\searrow 0}0$ and $J_{z}v\in\cs(\Omega)$ $z<\delta$ . Therefore, for sufficiently small $z$ , $||J_{z}v-v||_{p}<\frac{\eps}{3}$ . The triangle inequality now implies $||u-J_{z}v||<\frac{2\eps}{3}$ .

Now suppose $\Omega$ is unbounded. Since $\int_{\Omega}|u|^{p} = \lim_{r\rightarrow\infty}\int_{\Omega\cap B(0,r)}|u|^{p}$ , we may choose $r>0$ large enough so that $||u- u\cdot \chi_{\Omega\cap B(0,r)}||_{p}<\frac{\eps}{3}$ . We may now treat $u\cdot\chi_{\Omega\cap B(0,r)}$ according to the preceding case.

Sobolev spaces

We now turn our attention to $L^{p}$ functions that admit derivatives in a weaker sense.

Weakly differentiable functions

Definition. Let $u\in \lloc^{1}(\Omega)$ . We say that $\D^{\alpha}u$ exists in the weak sense if there exists a $v\in\lloc^{1}(\Omega)$ such that for all $\varphi\in C_{0}^{\infty}(\Omega)$ ,

\int_{\Omega}u\cdot \D^{\alpha}\varphi = (-1)^{|\alpha|}\int_{\Omega}v\cdot \varphi\tag{*}

and write $\D^{\alpha}u:=v$ . If for each $|\alpha|\leq k$ , $\D^{\alpha}u$ exists in the weak sense, we say that $u$ is $k$ -times weakly differentiable.

Remark. When $\D^{\alpha}u$ exists, it is uniquely determined by (*) almost everywhere.

Notation. Whenever $u$ is $k$ -times weakly differentiable, we adopt the usual partial derivative notation, i.e. $\partial_{i}u:=\D^{e_{i}}u$ and $\D^{\alpha}u=\partial_{1}^{\alpha_{1}}\dots\partial_{n}^{\alpha_{n}}u$ . Note that there is no ambiguity in the order of differentiation!

Exercise. Show that weak differentiation is $\mathbb{K}$ -linear, i.e. given $\lambda\in \mathbb{K}$ and $u,v\in \lloc^{1}(\Omega)$ such that both $\D^{\alpha}u$ and $\D^{\alpha}v$ exist in the weak sense, we have that $\D^{\alpha}(\lambda u)= \lambda\cdot\D^{\alpha}u$ and $\D^{\alpha}(u+v)=\D^{\alpha}u + \D^{\alpha}v$ .

Exercise. If $u\in \lloc^{1}(\Omega)$ is such that $\D^{\alpha}u\in \lloc^{1}(\Omega)$ exists weakly and if $\D^{\beta}(\D^{\alpha}u)\in\lloc^{1}(\Omega)$ exists weakly, then $\D^{\alpha+\beta}u$ exists and $\D^{\alpha+\beta}u=\D^{\beta}(\D^{\alpha}u)$ .

Exercise. If $u\in \lloc^{1}(\Omega)$ is $k$ -times weakly differentiable and $v\in C^{k}(\Omega)$ , then for all $\beta$ with $|\beta|\in[0,m]$ ,

\D^{\beta}(uv) = \sum_{\alpha\leq \beta}\left(\begin{matrix}\beta\\\alpha \end{matrix} \right)\D^{\alpha}u\cdot \D^{\beta-\alpha}u.

Examples

If $u\in C^{k}(\Omega)$ , then (*) is just integration by parts formula $\Rightarrow$ $u$ is $k$ -times weakly differentiable.
Consider the function $f:\left]0,2\right[\rightarrow\mathbb{R}$ defined by

$f(x)=\begin{cases}x,& 0<x\leq 1\\ 1,&1\leq x < 2 \end{cases}.$
We claim that $f$ is weakly differentiable with weak derivative $g:\left]0,2\right[\rightarrow\mathbb{R}$ given by

$g(x)=\begin{cases}1,& 0<x\leq 1\\ 0,&1\leq x < 2 \end{cases}.$

We need to show that for any $\varphi\in C_{0}^{\infty}(\left]0,2\right[)$ ,

$\int_{0}^{2}f\cdot \varphi' = -\int_{0}^{2}g\cdot \varphi.\tag{!}$
We compute:

$\int_{0}^{2}f\cdot \varphi' = \underbrace{\int_{0}^{1}x\cdot \varphi'(x)dx}_{\textup{IBP}} + \int_{1}^{2}\varphi'=-\int_{0}^{1}\varphi=-\int_{0}^{2}g\cdot\varphi.$
We now consider a function with a jump, viz. $f:\left]0,2\right[\rightarrow\mathbb{R}$ such that

$f(x)=\begin{cases}x,& 0<x\leq 1\\ 2,&1\leq x < 2 \end{cases}.$
We ask whether $f$ is weakly differentiable.

Suppose there exists a $g\in \lloc^{1}(\left]0,2\right[)$ such that (!) holds for all $\varphi\in\cs(\left]0,2\right[)$ . A quick computation shows that the left-hand side of this equation is $\varphi(1)-\int_{0}^{1}\varphi$ , whereas the right-hand side evaluates to $-\int_{0}^{1}\varphi\cdot g$ . Now, choosing $\varphi\in C_{0}^{\infty}(]0,1[)$ , we find that $g\equiv 1$ a.e. on $]0,1[$ , and choosing $\varphi\in C_{0}^{\infty}(]1,2[)$ , we see that we must have $g\equiv 0$ . However, for general $\varphi$ , we are left with the nonsensical statement

$\forall\varphi\in \cs(\left]0,2\right[\qquad \varphi(1)=0!$

Definition. For each $k\in\mathbb{N}\cup\{0\}$ and $p\in[1,\infty]$ , the Sobolev space $W^{k,p}(\Omega)$ is defined by

W^{k,p}(\Omega):=\{u\in L^{p}(\Omega):\forall|\alpha|\leq k,\ D^{\alpha}u\ \textup{exists in the weak sense and is in }L^{p}(\Omega)\}.

For $u\in W^{k,p}(\Omega)$ , we introduce the Sobolev norm:

||u||_{k,p}:=\begin{cases} \left(\sum_{|\alpha|\leq k}||\D^{\alpha}u||_{p}^{p} \right)^{1/p},&p<\infty\\ \sum_{|\alpha|\leq k}||\D^{\alpha}u||_{\infty},&p=\infty\end{cases}

For $p=2$ , this norm arises from the inner product

\left<u,v\right>_{k}:=\sum_{|\alpha|\leq k}\left<\D^{\alpha}u,\D^{\alpha}v\right>=\sum_{|\alpha|\leq k}\int_{\Omega}\overline{\D^{\alpha}u}\cdot \D^{\alpha}v.

Lemma 1.3. $(W^{k,p}(\Omega),||\cdot||_{k,p})$ is a Banach space and $(W^{k,2}(\Omega),\left<\cdot,\cdot\right>_{k})$ a Hilbert space.

Proof. We proceed in steps:

$||\cdot||_{k,p}$ is a norm: Homogeneity and $||u||_{k,p}=0\Rightarrow u=0$ a.e. is clear. The triangle inequality in the case $p=\infty$ is immediate; for $p<\infty$ and $u,v\in W^{k,p}(\Omega)$ , we use Minkowski's inequality and the triangle inequality for $L^{p}$ functions:

$\begin{aligned} ||u+v||_{k,p}=\left( \sum_{|\alpha|\leq k} ||\D^{\alpha}u + \D^{\alpha}v||_{p}^{p} \right)^{1/p}&=\left(\int_{\Omega}\sum_{|\alpha|\leq k}|\D^{\alpha}u + \D^{\alpha}v|^{p} \right)^{1/p}\\ &\leq \left(\int_{\Omega}\left( \left(\sum_{|\alpha|\leq k}|\D^{\alpha}u|^{p}\right)^{1/p} + \left(\sum_{|\alpha|\leq k}|\D^{\alpha}v|^{p}\right)^{1/p} \right)^{p} \right)^{1/p}\\ &=\left\Vert\left(\sum_{|\alpha|\leq k}|\D^{\alpha}u|^{p}\right)^{1/p} + \left(\sum_{|\alpha|\leq k}|\D^{\alpha}v|^{p}\right)^{1/p}\right\Vert_{p}\\ &\leq ||u||_{k,p} + ||v||_{k,p}\end{aligned}$

Completeness: Suppose $\{u_{n}\}_{n=1}^{\infty}\subset W^{k,p}(\Omega)$ is a Cauchy sequence:

$||u_{n}-u_{m}||_{k,p}\xrightarrow{n,m\rightarrow\infty}0\Leftrightarrow(\forall |\alpha|\leq k)\ \ ||\D^{\alpha}u_{n}-\D^{\alpha}u_{m}||_{p}\xrightarrow{n,m\rightarrow\infty}0.$
By the completeness of $L^{p}(\Omega)$ , we may, for each $\alpha$ with $|\alpha|\leq k$ , find a $v^{\alpha}\in L^{p}(\Omega)$ such that $||\D^{\alpha}u_{n}-v^{\alpha}||_{p}\xrightarrow{n\rightarrow\infty}0$ . Now, by definition of the weak derivative, we have that for each $\varphi\in \cs(\Omega)$ ,

$\int_{\Omega}u_{n}\cdot\D^{\alpha}\varphi = (-1)^{|\alpha|}\int_{\Omega}\D^{\alpha}u_{n}\cdot \varphi.$
Taking limits on both sides of this equation (why?), we obtain

$\int_{\Omega}v^{0}\cdot \D^{\alpha}\varphi = (-1)^{|\alpha|}\int_{\Omega}v^{\alpha}\cdot\varphi,$
i.e. $v^{\alpha}=\D^{\alpha}v^{0}$ for all such $\alpha$ so that $||\D^{\alpha}u_{n}-\D^{\alpha}v^{0}||_{p}\xrightarrow{n\rightarrow\infty}0\Rightarrow ||u_{n}-v^{0}||_{k,p}\xrightarrow{n\rightarrow\infty}0$ .

Remark. If we wish to differentiate between Sobolev norms corresponding to Sobolev spaces over different domains, we shall write $||\cdot||_{k,p,\Omega}$ for the norm on $W^{k,p}(\Omega)$ ; likewise, we shall sometimes write $||\cdot||_{p,\Omega}$ for the norm on $L^{p}(\Omega)$ .