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Everywhere I look in the literature, Sobolev spaces are defined on an open subset of the real line. What are the technical issues with defining a Sobolev space on a closed subset, i.e. are there problems at the boundary, and does anyone know any good references that cover this?

My main purpose is to prove $H^1([0,T];\mathbb{R}) = \{ x \in L^2([0,T];\mathbb{R}) : ||x'||_{L^2} + \gamma^{2}||x||_{L^2} < \infty \}$ is a reproducing kernel Hilbert space. I can do this for $(0,T)$ and want to know if the proof is transferable to the case of the closed interval $[0,T]$.

Many thanks,

Matthew.

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1 Answer 1

Well, I suspect you defined the Sobolev space on the close interval as the one on the open interval (i.e. using the same functional equation taking the test functions on $\mathscr{C}^{\infty}_0((0,T))$), and then extend every function of it by choosing arbitrary values on the boundary.

In that case $H^1((0,T))=H^1([0,T])$ and your problem is of little concern, because every function of $W^{1,1}((0,T))$ is also continue on the closed interval $[0,T]$ (i.e. they admit finite limit on the boundary). So you can apply, by limit, the same technique you used in the case of the open interval $(0,T)$ (for example using the continuous inclusion of $W^{1,1}(I)$ in $L^{\infty}(I)$).

Anyway, I think the main reason beyond defining Sobolev space on open sets lies in this: $\mathscr{C}^{\infty}_0(\bar{I})=\mathscr{C}^{\infty}(\bar{I})$.

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