Ytukyg

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
${Tin mathcal{D}'(D): D^alpha Tin L^2(D) forall |alpha|=1}$?

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,
   $$ lVertvarphirVert_{L^2(D)} leq C lVertnabla varphirVert_{L^2(D)}. $$
   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 $lVertnabla^kcdotrVert_{L^p(D)}$ 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).