Equivalence between Sobolev norm and Sobolev-Slobodeckij norm for integer-order spaces

1 minute read

This short post explores the equivalence between Sobolev norm and Sobolev-Slobodeckij norm for $W^{s,p}(\Omega)$ when $s$ is an integer.

Question

Take $W^{1,2} = H^1$ for example. If we still use Slobodeckij norm (which is normally defined for a fractional Sobolev space) as follows for a $u\in H^1(\Omega)$ with the exponent being in integer: $$ [u]_{1,\Omega}^2:= \int_{\Omega} \int_{\Omega} \frac{|u(x) - u(y)|^2}{|x-y|^{n+2}} \,dxdy. $$ Is this equivalent to the Sobolev semi-norm: $$ |u|_{1,\Omega}^2:= \int_{\Omega} |Du|^2 dx\,? $$

A simple argument

My educated guess to the question above is “No”, because when we divide $\overline{\Omega} = \overline{\Omega}_1 \cup \overline{\Omega}_2$ into two non-overlapping subdomain, for the Sobolev semi-norm it is just adding broken norms on the subdomains:

\[|u|_{1,\Omega}^2 = |u|_{1,\Omega_1}^2 + |u|_{1,\Omega_2}^2.\]

However for the Slobodeckij norm, it has some cross term like:

\[\int_{\Omega_1} \int_{\Omega_2} \frac{|u(x) - u(y)|^2}{|x-y|^{n+2}} \,dxdy,\]

and apparently,

\[[u]_{1,\Omega}^2 \neq [u]_{1,\Omega_1}^2 + [u]_{1,\Omega_2}^2.\]

I am guessing the equivalence relies on the regularities of the $\Omega$, but I could not find any reference on this. Someone on Math.SE gave the reference1.

The reference

The correct behavior of fractional norms in the limit $s \to 1^-$ is

\[\lim_{s \to 1^-} (1 - s)[u]_{s, p}^p = K(p, N)\|\nabla u\|_p^p,\]

where $K(p, N)$ is a constant depending on $p$ and $N$. If $\Omega_1$ and $\Omega_2$ are disjoint, it means that $|x - y| \geq d > 0$ when $x \in \Omega_1$ and $y \in \Omega_2$, therefore the cross term will tend to zero. As shown in a paper by Bourgain, Brezis, and Mironescu2.

J. Bourgain, H. Brezis, P. Mironescu. Another look at Sobolev spaces, in: J.L. Menaldi, E. Rofman, A. Sulem (Eds.), Optimal Control and Partial Differential Equations, A Volume in Honour of A. Bensoussan’s 60th Birthday, IOS Press, 2001, pp. 439-455.

  1. user345872 (https://math.stackexchange.com/users/345872/user345872), Equivalence between Sobolev norm and Sobolev-Slobodeckij norm for $W^{s,p}(\Omega)$ when $s$ is an integer, URL (version: 2016-07-29): https://math.stackexchange.com/q/1875552 ↩

  2. http://hal.upmc.fr/file/index/docid/747692/filename/another_look_sobolev_spaces_2000.pdf ↩

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