In remark 17 of the book [1], it is stated that
$$H_0^1(\mathbb{R}^d \setminus \{0\}) = H^1(\mathbb{R}^d \setminus \{0\})
\tag{1}
$$
for all spatial dimensions $d \geq 2$. Here $H_0^1(\mathbb{R}^d \setminus \{0\})$ is defined to be the closure of
$C_c^{\infty}(\mathbb{R}^d \setminus \{0\})$ in $H^1(\mathbb{R}^d \setminus \{0\})$.
I think I am able to prove this for $d \geq 3$.
In my proof I make use of the fact that, for $d \geq 3$,
$$\frac{n^2}{n^d} \rightarrow 0 \qquad \textrm{as } n \rightarrow \infty$$
Clearly this argument doesn't work with $d=2$.
Does anybody know how
$H_0^1(\mathbb{R}^d \setminus \{0\}) = H^1(\mathbb{R}^d \setminus \{0\})$
can be proven in the $d=2$ case?
Here is my attempted proof.
Introduce a cut-off function $\eta \in C_c^{\infty}(\mathbb{R}^d)$
such that $0 \leq \eta \leq 1$ and
$$
\eta(x) = \begin{cases}
1 & \textrm{ if } |x| \leq 1, \\
0 & \textrm{ if } |x| \geq 2.
\end{cases}
$$
Let $\eta_n(x) = \eta(nx)$. Then, to prove $(1)$, it suffices to show that for any $u \in H^1(\mathbb{R}^d \setminus \{0\})$ we have
$$ \| \eta_n u \|_{H^1(\mathbb{R}^d \setminus \{0\})} \rightarrow 0.$$
However, note that $H^1(\mathbb{R}^d \setminus \{0\})
\cap L^{\infty}(\mathbb{R}^d \setminus \{0\})$
is dense in $H^1(\mathbb{R}^d \setminus \{0\})$
(see e.g. Item 3 of Remark 1.27 in https://math.aalto.fi/~jkkinnun/files/sobolev_spaces.pdf).
Therefore, we can assume without loss of generality that $u \in H^1(\mathbb{R}^d \setminus \{0\})
\cap L^{\infty}(\mathbb{R}^d \setminus \{0\})$.
Now, using dominated convergence, it is easily checked that
$$
\| \eta_n u \|_{L^2(\mathbb{R}^d \setminus \{0\})} \rightarrow 0, \\
\| \eta_n (\nabla u) \|_{L^2(\mathbb{R}^d \setminus \{0\})} \rightarrow 0.
$$
Therefore, since $\nabla (\eta_n u) = (\nabla \eta_n) u + \eta_n (\nabla u)$, it remains to show that
$$
\| (\nabla \eta_n) u \|_{L^2(\mathbb{R}^d \setminus \{0\})} \rightarrow 0.
$$
However, because $u \in L^{\infty}(\mathbb{R}^d \setminus \{0\})$ we have
$$
\| (\nabla \eta_n) u \|_{L^2(\mathbb{R}^d \setminus \{0\})}
\leq
\| \nabla \eta_n \|_{L^2(\mathbb{R}^d \setminus \{0\})}
\| u \|_{L^{\infty}(\mathbb{R}^d \setminus \{0\})}
$$
and therefore it suffices to show that
$$
\| \nabla \eta_n \|_{L^2(\mathbb{R}^d \setminus \{0\})}
\rightarrow 0.
$$
Let $B_n = \{ x \in \mathbb{R}^d : \| x\|\ \leq 2/n \}$. We see that
$$
\| \nabla \eta_n \|_{L^2(\mathbb{R}^d \setminus \{0\})}^2
= \int_{B_n} \| \nabla \eta_n \|^2
\leq n^2 \| \nabla \eta \|_{L^{\infty}(\mathbb{R}^d \setminus \{0\})}
\int_{B_n} 1
\leq C \frac{n^2}{n^d}.
$$
Thus if $d \geq 3$ we get
$\| \nabla \eta_n \|_{L^2(\mathbb{R}^d \setminus \{0\})}
\rightarrow 0$ as desired, completing the proof.
[1]: Brezis, Haim, Functional analysis, Sobolev spaces and partial differential equations, Universitext. New York, NY: Springer (ISBN 978-0-387-70913-0/pbk; 978-0-387-70914-7/ebook). xiii, 599 p. (2011). ZBL1220.46002.
Best Answer
This happens to be an exercise in a different book (Exercise 2.2, S.T. Kuroda’s An Introduction to Scattering Theory) with the hint that in the case $d=2$, we need to find a replacement for $n x$. Let $\eta$ be radial such that $$ \eta(|x|)=\begin{cases} 1 & |x|<1/4 \\ 0 & |x|\ge 1/2\\ \text{smooth }&\text{in between}\end{cases}$$ (So far basically the same as yours) and define for $\sigma>0$, $$ \eta_\sigma(x) :=\eta(|x|^\sigma) \in C^\infty_c(B_{2^{-1/\sigma}}(0))$$ (That’s the weird part.) with the intention to send $\sigma\to 0$. Note that $\eta_\sigma \to 0$ at every $x\neq 0$. Now we continue like your proof with dominated convergence for $d>2$ and end up needing to estimate $\|\nabla\eta_\sigma\|_{L^2}$, and this goes like so: \begin{align} \|\nabla\eta_\sigma\|_{L^2}^2 &= \int_{|x|\le 2^{-1/\sigma}}|(\nabla \eta)(|x|^\sigma)|^2 \sigma^2 |x|^{2\sigma-2}dx \\ &\lesssim_{\|\nabla \eta\|_{L^\infty}}\sigma^2 \int_0^{2^{-1/\sigma}} r^{2\sigma-2}rdr\\&\lesssim \frac{\sigma^2}{2\sigma} r^{2\sigma}\Big|_{r=0}^{2^{-1/\sigma}} \\& \lesssim \sigma \to 0, \end{align}whence the result.
Kuroda, S. T., An introduction to scattering theory, Lecture Notes Series, No. 51. Aarhus: Aarhus Universitet, Matematisk Institut. Not consecutively paged. (1978). ZBL0407.47003.