Questions tagged [sobolev-spaces]

Sobolev spaces are function spaces generalizing the Lebesgue spaces. Whereas elements of Lebesgue spaces have certain integrability condition imposed on them, derivatives of functions in a Sobolev space are also required to be sufficiently integrable: that is, we require all (weak) partial derivatives of the function up to a certain order belong to a certain fixed Lebesgue space.

In more detail, let $U \subseteq \mathbb{R}^n$ be an open set. A weak $\alpha$th partial derivative $D^\alpha f$ of $f$ is a function $g\in L^1_{\mathrm{loc}}(U)$ such that $$\int_U f D^\alpha \phi \, dx = (-1)^{|\alpha|} \int g\phi \, dx$$ for each compactly supported smooth function $\phi \in C^\infty_c(U)$. the Sobolev space $W^{k, p}(U)$ consists of those functions $f\in L^p(U)$ such that for every multi-index $\alpha$ of length at most $k$, every weak partial derivative $D^{\alpha}f$ exists and is an element of $L^p$. The Sobolev spaces are equipped with norms defined by

$$\|u\|_{W^{k,p}(U)} = \begin{cases} \left( \sum_{|\alpha| \le k} \|D^{\alpha} u\|_{L^p(U)}^p \right)^{1/p} & p < \infty, \\ \max_{|\alpha| \le k} \|D^{\alpha}\|_{L^{\infty}(U)} &p=\infty .\end{cases}$$

$(W^{k,p}(U),\|\cdot\|_{W^{k,p}(U)})$ are Banach spaces for each $k\in\mathbb N$, and each $p\in[1,\infty]$. The norms measure both the size and the regularity of a function.

The basic fundamental result of Sobolev spaces is the Sobolev embedding theorem. In words, it says that (1) if $kp<n$, having $k$ weak derivatives in $L^p$ places your function in a better Lebesgue space $L^{p^*}$, where $\frac1{p^*} = \frac1p - \frac kn$, and (2) if $kp>n$, then your function is not only in $L^\infty$ but also has a continuous representative in some space of Hölder continuous functions. In particular, sufficiently many weak derivatives means that your function is in fact classically differentiable.

Reference: Sobolev space.