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

Question

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.

Answer 1

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).