Sobolev spaces on non open sets

Question

in the definition of Wikipedia and several books the Sobolevspace $W^{k,p}(\Omega)$ is defined on a open subset $\Omega\subset \mathbb{R}^d$. Why does $\Omega$ have to be open? Why is [0,1] not possible?

Best wishes and thanks for answers :)

Answer 1

You can, if you wish, define a Sobolev space on a non-open set $[0,1]$. For example:

A function $f:[0,1]\to \mathbb R$ belongs to $W^{1,p}([0,1])$ if there exists $g\in L^p([0,1])$ and $c\in \mathbb R$ such that $f(x) = c+\int_0^x g$ for a.e. $x$.

The definition is the easy part. The hard part is figuring out what to do with this definition. Has it really created a new function space? What is its purpose?

Sobolev spaces are useful for studying PDE. And PDE naturally hold on an open domain, with some conditions on the boundary of that domain. So it makes sense to look at Sobolev spaces there. Also, from the PDE point of view a Sobolev function is just a distribution whose $k$th derivative happens to be represented by an $L^p$ function. Distributions are defined as linear functionals on test functions; test functions are defined as $C^\infty$ functions with additional properties; the concept of $C^\infty$ calls for an open set.

All that said, Sobolev spaces are being defined and studied on all sorts of sets and spaces, e.g., on the Sierpiński gasket:

E.g., this paper concerns such spaces. There are also several definitions for Sobolev spaces (especially of 1st order) on general metric measure spaces; here is a recent one.