# Some elements of Sobolev spaces

This page collects some elements of the theory of Lebesgue and Sobolev spaces. For further information, see e.g. L. C. Evans: Partial Differential Equations, R. A. Adams: Sobolev spaces or G. B. Folland: Real analysis.

## Preliminaries

A multi-index is a vector $\alpha\in\mathbb{N}_0^d$ with components $\alpha_1,\dots,\alpha_d\in\mathbb{N}_0 = \{0,1,2,\dots\}$. We write $|\alpha| = \alpha_1+\alpha_2+\dots+\alpha_d$. We use multi-indices as a shorthand notation for mixed partial derivatives: If $u:\Omega\to\mathbb{R}$ is a given, smooth function on some domain $\Omega\subset\mathbb{R}^d$ then $D^\alpha u := \frac{\partial^{|\alpha|}u}{\partial x_1^{\alpha_1}\cdots\partial x_d^{\alpha_d}}.$ For instance, if $d=2$ and $\alpha=(2,1)$ then $D^\alpha u = \dfrac{\partial^3 u}{\partial x_1^2 \partial x_2}$.

We denote by $C^\infty(\Omega)$ the space of all infinitely differentiable functions $\phi:\Omega\to\mathbb{R}$, and by $C_c^\infty(\Omega)$ those $\phi$ in $C^\infty(\Omega)$ with compact support – that is, there exists a compact set $K\subset\Omega$ such that $u\equiv 0$ outside of $K$. Note that if $\Omega$ is open then this implies that $\phi$ and all its derivatives vanish close to the boundary of $\Omega$.

Recall that if $\Omega\subset\mathbb{R}^d$ is open then by the integration-by-parts formula (or Gauss' theorem, or the divergence theorem) we have $\int_\Omega u\nabla v\ dx = \int_{\partial\Omega} uvn\ dx - \int_\Omega v\nabla u\ dx \qquad \forall\ u,v\in C^\infty(\Omega),$ where $n$ is the unit normal on $\partial\Omega$. If either $u$ or $v$ has compact support then the boundary integral drops out, and we can generalize the above to the following:

#### Integration by parts

If $u$ and $v$ are $k$ times differentiable and at least one of them has compact support, then \begin{equation}\label{eq:intbyparts} \int_\Omega u D^\alpha v\ dx = (-1)^{|\alpha|}\int_\Omega vD^\alpha u\ dx \end{equation} for every multi-index $\alpha$ of size $|\alpha|\leq k$.

## The Lebesgue space $L^2(\Omega)$

#### Definition

For a set $\Omega \subset \mathbb{R}^d$ we define $L^2(\Omega) := \left\{ u:\Omega\to\mathbb{R}\ :\ \int_\Omega |u(x)|^2\ dx < \infty\right\}.$ We denote the $L^2$ norm of a function $u\in L^2(\Omega)$ by $\|u\|_{L^2(\Omega)} := \left(\int_\Omega |u(x)|^2\ dx\right)^{1/2}$. Functions $u\in L^2(\Omega)$ are called square integrable functions.

It can be shown that $L^2(\Omega)$ is a Hilbert space when equipped with the inner product $(u,v) := \int_\Omega u(x)v(x)\ dx$. The fact that $(u,v)$ is a finite number follows from Hölder's inequality: $\int_\Omega u(x)v(x)\ dx \leq \|u\|_{L^2(\Omega)}\|v\|_{L^2(\Omega)} \qquad \forall\ u,v\in L^2(\Omega)$ (also called the Cauchy-Schwartz inequality for $L^2(\Omega)$). Note in particular that $(u,u) = \|u\|_{L^2(\Omega)}^2$.

#### Examples

$L^2$ contains a very large number of functions, including highly discontinuous and/or singular functions. If $u(x)=|x|^\gamma$ for $x$ in the unit interval $\Omega=(-1,1)$, then $u\in L^2(\Omega)$ provided $\gamma>-1/2$. More generally, in the $d$-dimensional unit ball $\Omega=\{x\in\mathbb{R}^d\ :\ |x|\leq 1\}$, the function $u(x)=|x|^\gamma$ satisfies (using polar coordinates) $\int_\Omega |u(x)|^2\ dx = \int_\Omega |x|^{2\gamma}\ dx = \omega_n\int_0^1 r^{2\gamma} r^{d-1}\ dr < \infty \qquad \text{provided } \gamma>-d/2.$ (Here, $\omega_n$ is the area/volume/hypervolume of the unit sphere in $\mathbb{R}^d$.) Thus, singularities of order less than $d/2$ are square integrable. For example, the function $u(x) = \frac{1}{|x|}$ is square integrable in three dimensions or greater, but not in one or two dimensions.

## Sobolev spaces

### Definitions

We let now $\Omega\subset\mathbb{R}^d$ be an open, bounded, connected set.

#### Definition

The Sobolev space $H^k(\Omega)$ is the space of all functions $u\in L^2(\Omega)$ such that for every multi-index $\alpha\in\mathbb{N}_0^d$ of length $|\alpha|\leq k$ exists a function $v_\alpha\in L^2(\Omega)$ such that $\int_\Omega u(x) D^\alpha \phi(x)\ dx = (-1)^{|\alpha|}\int_\Omega v_\alpha(x)\phi(x)\ dx \qquad \forall\ \phi\in C_c^\infty(\Omega).$ The functions $v_\alpha$ are the weak derivatives of $u$. At the risk of confusion, we denote $D^\alpha u = v_\alpha$. For the sake of convenience we will denote $H^0(\Omega) = L^2(\Omega)$.

#### Remark

Note the similarity between the definition of the weak derivatives and the integration-by-parts formula. In fact, if $u$ also lies in $C^\infty(\bar{\Omega})$ then all of its weak derivatives exist and coincide with its classical derivatives. It is straighforward to show that the weak derivatives of a function are unique (if they exist).

The space $H^k(\Omega)$ is equipped with the norm $\|u\|_{H^k(\Omega)} := \Biggl[\sum_{|\alpha|\leq k} \|D^\alpha u\|_{L^2(\Omega)}^2\Biggr]^{1/2},$ where the sum goes over all multi-indices of length at most $k$. It can be shown that $H^k(\Omega)$ with the norm $\|\cdot\|_{H^k(\Omega)}$ is a Banach space. Equipped with the inner product $(u,v)_{H^k} := \sum_{|\alpha|\leq k} (D^\alpha u, D^\alpha v)$ (where $(u,v)$ is the $L^2$ inner product), the Sobolev space $H^k(\Omega)$ is in fact a Hilbert space.

The $H^k$ semi-norm is the number $|u|_{H^k(\Omega)} := \Biggl[\sum_{|\alpha|=k} \|D^\alpha u\|_{L^2(\Omega)}^2\Biggr]^{1/2}.$ Thus, $\|u\|_{H^k(\Omega)} = \left(\sum_{r=0}^k |u|_{H^r(\Omega)}^2\right)^{1/2}$.

#### Definition

The space $H_0^k(\Omega)$ is defined as the closure of $C_c^\infty(\Omega)$ in $H^k(\Omega)$, or in other words, the space of all $u\in H^k(\Omega)$ for which there exists a sequence $\phi_n\in C_c^\infty(\Omega)$ such that $\|u-\phi_n\|_{H^k(\Omega)} \to 0$ as $n\to\infty$.

It can be shown that an equivalent definition is $H_0^k(\Omega) = \{u\in H^k(\Omega)\ :\ D^\alpha u = 0 \text{ on } \partial\Omega \text{ for every } |\alpha|\leq k-1\}.$ Thus, for instance, $H_0^1(\Omega) = \{u\in H^1(\Omega)\ :\ u = 0 \text{ on } \partial\Omega\}.$

### Inequalities and embeddings

#### Theorem (the Poincaré inequality)

Let $\Omega$ be open and bounded. Then there exists a constant $C>0$, only dependent on $\Omega$, such that $\|u\|_{L^2(\Omega)} \leq C|u|_{H^1(\Omega)} \qquad \forall\ u\in H_0^1(\Omega).$ More generally, for every $k\in\mathbb{N}$ there exists a constant $C>0$ such that $|u|_{H^{r-1}(\Omega)} \leq C|u|_{H^r(\Omega)} \qquad \forall\ r=1,\dots,k \text{ and } u\in H_0^k(\Omega).$ The result also holds if $H_0^k(\Omega)$ is replaced by $H_{\Gamma}^k(\Omega) := \{u\in H^k(\Omega)\ :\ D^\alpha u = 0 \text{ on } \Gamma \text{ for all } |\alpha|\leq k-1\}$, where $\Gamma\subset\partial\Omega$ is only a part of the boundary of $\Omega$.

#### Remark

As a consequence of the Poincaré inequality, there is a constant $C>0$ such that $\|u\|_{H^1(\Omega)} \leq C|u|_{H^1(\Omega)} \qquad \forall\ u\in H_0^1(\Omega),$ and similarly for the $H^k$ norm. This inequality is used extensively for proving coercivity of bilinear forms on $H_0^1(\Omega)$.

#### Definition

Let $\Omega\subset\mathbb{R}^d$ be an open set. We say that $\Omega$ has a $C^1$ boundary if $\partial\Omega$ it can be parametrized locally by a differentiable function. More precisely, for every $x_0\in\partial\Omega$ there exists an open neighborhood $U\subset\mathbb{R}^d$ of $x_0$, an open set $V\subset\mathbb{R}^{d-1}$ and a differentiable, bijective function $\phi:V\to U\cap\partial\Omega$.

Similarly, $\Omega$ has Lipschitz boundary if $\partial\Omega$ can be parametrized locally by a Lipschitz function.

Intuitively, a domain has $C^1$ (or Lipschitz) boundary if for every $x_0\in\partial\Omega$, after rotating the domain, the boundary $\partial\Omega$ in the vicinity of $x_0$ is the graph of a $C^1$ (or Lipschitz continuous) function. Note that domains with Lipschitz boundaries can have "kinks", so any polygon (or polyhedron in 3D) has Lipschitz boundary.

#### Theorem (Sobolev embedding theorem)

Let $\Omega\subset\mathbb{R}^d$ be an open, bounded subset with a $C^1$ boundary, and let $m$ be a nonnegative integer. Then $H^k(\Omega) \subset C^m(\Omega)$ provided $k > m + d/2$.

For instance, $H^1$ functions on the real line are continuous ($k=1$, $d=1$ and $m=0$ in the above theorem), and $H^3$ functions in the plane are differentiable ($k=3$, $d=2$ and $m=1$).

#### Remark

To gain some intuition into the Sobolev embedding theorem, consider the function $u(x)=|x|^\gamma$ for $x$ in a neighborhood of the origin. This function is continuous if $\gamma\geq0$ and discontinuous for $\gamma<0$. Moreover, its $r$-th partial derivatives all behave like $|x|^{\gamma-r}$. Hence, $\gamma \geq m \quad \Rightarrow \quad u\in C^m.$ On the other hand, assume that $u\in H^k$ for some integer $k$. Then all $k$-th partial derivatives of $u$ lie in $L^2$, which (by the example in the section on Lebesgue spaces) happens if and only if $\gamma-k > -d/2$. Thus, if we ensure that $k > m+d/2$ then $\gamma > k-d/2 > m+d/2-d/2 = m \quad \Rightarrow \quad u\in C^m.$

### Approximation by smooth functions

Note that, by definition, $C_c^\infty(\Omega)$ is dense in $H^k_0(\Omega)$ for any $k\in\mathbb{N}$. A similar result holds for $H^k(\Omega)$:

#### Theorem

Let $\Omega\subset\mathbb{R}^d$ be an open, bounded subset with Lipschitz boundary (see the definition in the previous section). Then $C^\infty(\bar{\Omega})$ is dense in $H^k(\Omega)$ for any $k\in\mathbb{N}$. In other words, for any $u\in H^k(\Omega)$ there are functions $\phi_1,\phi_2,\ldots \in C^\infty(\bar{\Omega})$ such that $\|u-\phi_n\|_{H^k(\Omega)} \to 0$ as $n\to\infty$.