Sobolev Space

See more in

• Sobolev space - Wikipedia
• PDE notes page 38-39
• Sobolev embedding theorem - Wikipedia

1. Weak Partial Derivative

We first note that $\alpha = \left(\alpha_1, \ldots, \alpha_n\right)$ is called multi-index of order $|\alpha| = \sum_{i=1}^{n} \alpha_i$ and we use the notition that

$$D^\alpha f = \frac{\partial^{|\alpha|} f}{\partial x_1^{\alpha_1} \ldots \partial x_n^{\alpha_n}}$$

for $f =$ in $\mathbb{R}^n$.

Now we come to the definition of weak partial derivative. Given function $u, v \in \mathrm{L}^1_{\alpha}(\Omega)$ on compact set $\Omega$, for all infinitely differentiable functions with compact support $\Omega$: $\varphi \in \mathcal{C}_0^\infty(\Omega)$, if

$$\int_{\Omega} v \varphi \mathrm{d} x=(-1)^{|\alpha|} \int_{\Omega} u D^{\alpha} \varphi \mathrm{d} x$$

we call $v$ the weak $|\alpha|$-th partial derivative of $u$, denote as $v = D^\alpha u$.

2. Sobolev Space

The Sobolev spaces $W^{k, p}(\Omega)$ combine the concepts of weak differentiability and Lebesgue norms. In the one-dimensional case the Sobolev space $W^{k, p}(\mathbb{R}) \text { for } 1 \leqslant p \leqslant \infty$ is defined as the subset of functions $f \text { in } L^{p}(\mathbb{R})$ such that $f$ and its weak derivatives up to order $k$ have a finite $L^p$ norm.

With this definition, the Sobolev spaces admit a natural norm,

$$\|f\|_{k, p}=\left(\sum_{i=0}^{k}\left\|f^{(i)}\right\|_{p}^{p}\right)^{\frac{1}{p}}=\left(\sum_{i=0}^{k} \int\left|f^{(i)}(t)\right|^{p} d t\right)^{\frac{1}{p}}$$

Equipped with the norm $\|\cdot\|_{k,p},W^{k,p}$ becomes a Banach space. It turns out that it is enough to take only the first and last in the sequence, i.e., the norm defined by

$$\left\|f^{(k)}\right\|_{p}+\|f\|_{p}$$

is equivalent to the norm above (i.e. the induced topologies of the norms are the same).

3. Sobolev Embeddings

It is a natural question to ask if a Sobolev function is continuous or even continuously differentiable. Roughly speaking, sufficiently many weak derivatives or large p result in a classical derivative. This idea is generalized and made precise in the Sobolev embedding theorem.

Let $W^{k,p}\left(\mathbf{R}^{n}\right)$ denote the Sobolev space consisting of all real-valued functions on $\mathbf{R}^{n}$ whose first $k$ weak derivatives are functions in $L^{p} .$ Here $k$ is a non-negative integer and $1 \leq p<\infty .$ The first part of the Sobolem states that if $k>\ell$ and $1 \leq p<q<\infty$ are two real numbers such that $(k-\ell) p<n$ and:

$$\frac{1}{p}-\frac{k}{n}=\frac{1}{q}-\frac{\ell}{n}$$

then

$$W^{k, p}\left(\mathbf{R}^{n}\right) \subseteq W^{\ell, q}\left(\mathbf{R}^{n}\right)$$

4. Sobolev Embeddings Applications

In the special case of $k=1$ and $\ell=0,$ Sobolev embedding gives

$$W^{1, p}\left(\mathbf{R}^{n}\right) \subseteq L^{p^{*}}\left(\mathbf{R}^{n}\right)$$

where $p^{*}$ is the Sobolev conjugate of $p,$ given by

$$\frac{1}{p^{*}}=\frac{1}{p}-\frac{1}{n}$$

This special case of the Sobolev embedding is a direct consequence of the Gagliardo–Nirenberg–Sobolev inequality.

The second part of the Sobolev embedding theorem applies to embeddings in Hölder spaces $C^{r, a}\left(\mathbf{R}^{n}\right)$. If $n<p$ and

$$\frac{1}{p}-\frac{k}{n}=-\frac{r+\alpha}{n}$$

with $\alpha \in(0,1]$ then one has the embedding

$$W^{k, p}\left(\mathbf{R}^{n}\right) \subset C^{r, \alpha}\left(\mathbf{R}^{n}\right)$$

This part of the Sobolev embedding is a direct consequence of Morrey's inequality. Intuitively, this inclusion expresses the fact that the existence of sufficiently many weak derivatives implies some continuity of the classical derivatives.