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I am reading Ciarlets book on elliptic boundary value problems and the finite element method. He defines the Sobolev spaces $H^m_0(\Omega)$ as the completion of the space $\mathcal{D}(\Omega)$ of compactly supported infinitely differentiable real valued functions with respect to the Sobolev norm $$ \| u\|_{m, \Omega} = \left( \int_\Omega \sum_{|\alpha| \leq m} |D^\alpha u|^2 \right)^{1/2} $$

However, under certain requirements on the boundary $\Gamma$ of the domain $\Omega$, the spaces for $ m = 1 $ and $m = 2$ can be characterised as

$$ H^1_0(\Omega) = \left\{ u \in H^1(\Omega) \mid u = 0 \text{ on } \Gamma\right\} \\ H^2_0(\Omega) = \left\{ u \in H^2(\Omega) \mid u = \partial_{n} u = 0 \text{ on } \Gamma\right\} $$ where $n$ is the outward pointing unit normal to $\Gamma$. What is the fundamental difference between these characterisations, and the definition involving the completion of the space? Under what circumstances can I use the characterisation at the bottom?

Thanks for all help in advance! :)

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They are the same, you can find the proof in Adam's book, he considered these two spaces and he denoted them by $H^{m,p}$ and $W^{m,p}$, then he proved the equivalence between these two definitions.

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