MathOverflow Asked by Guy Fsone on December 1, 2021
Assume $Omegasubset Bbb R^d$ is Lipschitz open set. Let $pgeq 1$ and $0<sleq 1/p$.
How to prove that $C_c^infty(Omega)$ is dense in $W^{s,p}(Omega)$?
Recall that,
$$|u|^p_{W^{s,p}(Omega)}= iintlimits_{OmegaOmega}frac{|u(x)-u(y)|^p}{|x-y|^{d+sp}}dx dy$$
and
$$W^{s,p}(Omega)= {uin L^p(Omega): |u|^p_{W^{s,p}(Omega)}<infty}$$
equipped with the Banach norm
$$|u|^p_{W^{s,p}(Omega)}= |u|^p_{L^{p}(Omega)}+|u|^p_{W^{s,p}(Omega)}$$
The main difficulty in solving this is to construct a suitable family of cut-off with compact support in $Omega$.
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