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Approximation in fractional Sobolev space

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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