# completion of $C^\infty_0(D)$ w.r.t $\|\cdot\|_\nabla$

Let $$D$$ be an unbounded domain in $$\mathbb{R}^n$$. Consider the set $$C^\infty_c(D)$$ with two different norms: $$\|\cdot\|_\nabla$$ and $$\|\cdot\|_\nabla + \|\cdot\|_{L^2}$$.

It is known that when $$D$$ is bounded, the two norms are comparable. Hence their completions are the same.

Question 1. Are they the same when $$D$$ is unbounded? Is there any book talking about that?

Question 2. Is it true that the completion of $$(C^\infty_0(D),\|\cdot\|)$$ can be seen as the set $$\{T\in \mathcal{D}'(D): D^\alpha T\in L^2(D) \forall |\alpha|=1\}$$?

I've searched in Haim Brezis and Evans' books but couldn't find any.

Not sure if it is related. What I care is when $$D$$ is a proper subset of the plane and when its boundary is non-smooth.

Thanks.

In general it depends; there are a few things you can say, but I'm not aware of any general characterisation. These spaces are usually denoted by $$\dot W^{k,p}(D)$$ in the literature, but I don't know of any references that treats them in detail.
1. The equivalence of norms boils down to establishing a Poincaré inequality for the given domain $$D,$$ namely that there exists $$C> 0$$ such that for all $$\varphi \in C^{\infty}_c(D)$$ we have, $$\lVert\varphi\rVert_{L^2(D)} \leq C \lVert\nabla \varphi\rVert_{L^2(D)}.$$ As noted in the comments, there are many variants and not all require boundedness. The classical cases are when $$D$$ has finite width, and when it has bounded Lebesgue measure (so $$\mathcal{L}^n(D) < \infty$$). These certainly aren't the only cases however; for example you probably perturb a finite-width domain by an ambient diffeomorphism to get a domain which no longer has finite width, but where the Poincaré inequality still holds.
2. In the reverse direction, if there are balls $$B_{r_i}(x_i) \subset D$$ with $$r_i \rightarrow \infty,$$ then the norms are inequivalent (this is an exercise from Leoni's text "A first course in Sobolev spaces").
3. The homogenous Sobolev spaces are defined as $$\mathring{W}^{k,p}(D) = \{ u \in \mathcal{D}'(D) : \nabla^{\alpha}u \in L^p(D) \ \forall \,|\alpha|=k\}.$$ Note that this space can be identified as a subspace of $$L^1_{\mathrm{loc}}(D)$$ by Sobolev embedding. However the completion of $$C^{\infty}_c(D)$$ will never coincide with space, because $$\lVert\nabla^k\cdot\rVert_{L^p(D)}$$ is not a norm on the homogenous spaces (they contain constant functions).
4. If $$D = \mathbb R^n$$ with $$n \geq 3,$$ then there exists $$C>0$$ such that, $$\int_{\mathbb R^n} \frac{|u(x)|^2}{|x|^2}\,\mathrm{d}x \leq C \int_{\mathbb R^n} |\nabla u(x)|^2\,\mathrm{d}x.$$ This follows by writing $$|x|^{-2} = \frac1{n-2}\mathrm{div}(\frac{x}{|x|^2}),$$ invoking the divergence theorem to move the derivative onto $$u$$ and applying Cauchy-Schwarz. Inequalities of this form are usually called Hardy-type inequalities. I don't claim any kind of characterisation, but this may be a good starting point to investigate (also note that the Sobolev inequality always holds).