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Is the space of compactly supported functions of class 2: ${\cal C}_c^2(\Omega)$ dense in the Sobolev space $H^2(\Omega)$? under some smoothness assumptions on the a bounded open set $\Omega$ of $R^n.$

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No. Functions in $C_c^2(\Omega)$ vanish in the neighborhood of the boundary. Hence, you cannot approximate functions with non-zero traces.

However, (by definition) $C_c^2(\Omega)$ is dense in $H_0^2(\Omega)$.

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