# The Sobolev Spaces $H_0^s(\Omega)$

Let $\bar{\Omega}$ be a smooth, compact manifold with boundary; we denote the interior by $\Omega$. We can suppose $\bar{\Omega}$ is contained in a compact, smooth manifold $M$, with $\partial\Omega$ a smooth hypersurface. For $s\geq0$, we define $H_0^s(\Omega)$ to consist of the closure of $C_0^\infty(\Omega)$ in $H^s(\Omega)$. For $s=k$ a nonnegative integer, how can I show that $$H_0^k(\Omega)=\{u\in H^k(M):\text{supp }u\subset\bar{\Omega}\}?$$ I am trying to write a rigorous proof of that, and I just need a subtle pointer. Do you guys have any ideas? Thanks in advance!

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## 1 Answer

Notation: $D=\{u\in H^k(M):\text{supp }u\subset\bar{\Omega}\}$. Why $D$? Cause I like the letter D... =)

To prove that two sets are the same, it's usually a good idea to prove that one is a subset of the other and viceversa.

First you can easily show that if $u\in H^k_0$, than for sure supp $u\subset \bar{\Omega}$ (argue by contradiction); this proves $H^k_0\subset D$.

To prove that $D\subset H^k_0$ use the definition of $H^k$ (that is the closure of $C^\infty$ with respect to the $H^k$ norm) and prove that, for a given $u\in D$, any approximating sequence $\varphi_n\in D$ must have a trace on $\partial \Omega$ that vanishes as $n$ goes to infinity. Then try to approximate each of the $\varphi_n$ with a function $\psi_n\in D$. Then conclude.

I hope I didn't give too many details and let you some steps to have fun with... ;-)

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