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Here $\Omega \subset \mathbb{R}^n$ is a closed disc centred at $0$ with radius $r$.The book I am reading is assuming the Dirichlet boundary condition on $\Omega$ and claiming that the dual of $H^1_0(\Omega)$ is $H^{-1}(\Omega)$. I understand the concept of Sobolev spaces on $\mathbb{R}^n$ or on compact manifolds without boundary. I just need a little help with this notation $H^1_0(\Omega)$. Thanks!

PS: Basically, it is the zero on the subscript that is confusing me.

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    It means the space of $H^1(\Omega)$ functions that are zero on the boundary. The formalization is a bit awkward because elements of $H^1$ are not really functions but "almost everywhere" equivalence classes, but for practical purposes, it just means this. https://en.wikipedia.org/wiki/Sobolev_space#Traces – Peter Franek Aug 22 '15 at 14:01
  • To support @PeterFranek's comment, this paper also defines this notation on page 12: https://www.math.psu.edu/bressan/PSPDF/sobolev-notes.pdf – Ilham Aug 22 '15 at 14:02

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As the commenters said, the subscript $0$ indicates vanishing on the boundary in the Sobolev space sense: that is, it can be approximated in the Sobolev norm by smooth functions with compact support in $\Omega$.

The reason we use $H^1_0$ instead of $H^1$ in the definition of $H^{-1}$ is explained in The dual of the Sobolev space $W^{k,p}$.

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