Let $\Omega \subset \mathbb{R}^d$ be a bounded domain with Lipschitz boundary, and consider $H^m(\Omega) = W^{m,2}(\Omega)$, the $m$-th order Sobolev space on $\Omega$, for $m \geq 1$. We define the closed subspace $H^m_0(\Omega)$ as the closure of compactly supported smooth functions in the $H^m(\Omega)$ norm.

I've heard that in general, $H_0^m(\Omega) \neq H^m(\Omega)$, but I have no intuition for why. Can someone provide an example of an element of $H^m(\Omega)$ which is not in $H_0^m(\Omega)$ for, say, $m=1$ and some choice of $\Omega$?

  • 3
    $\begingroup$ The 0 is suggestive notation for zero boundary values (of the function itself and all derivatives up to order $m-1$). For example the constant function with value 1 everywhere is not in H_0^m. $\endgroup$ – Bananach Apr 9 '17 at 6:53

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