Why define Sobolev spaces $W^{1,p}_0$ as completion of $C^1_c$ and not of $C^1_0$?

Question

Let $C^1_c(\Omega)$ be the once continuously differentiable functions with compact support, and $C^1_0(\Omega)$ once cont' diff' functions which approach zero on the boundary of $\Omega$.

I am wondering why books don't define $W^{1,p}_0$ as the completion of $C^1_0$, and instead go in a roundabout way and prove that $C^1_0\subset W^{1,p}_0$. I mean, when trying to prove that a classical solution which is in $C^2_0$ to some BVP is also a weak solution, it would be trivial if one would define $W^{1,p}$ as I suggested. Instead, the standard definition seems to be as the completion of $C^1_c$, and then it takes some effort to prove that a classical solution is in $W^{1,p}_0$. Is there something I'm missing that makes my suggested definition not good?

Answer 1

Firstly, let us write $H^{1,p}_0$ for said closure, and reserve the notation $W^{1,p}_0$ for the definition by weak derivatives. Some reasons to use $C^1_c$ (in fact $C^\infty_c$) are as follows:

1. Many results for $H^{1,p}_0$ rely on a combination of density arguments and integration-by-parts arguments. Integration by parts in $C^\infty_c$ is inherently easier, since not only the functions, but also all their derivatives vanish on the boundary of the domain.

2. Many other results for $H^{1,p}_0$ rely on convolution. The convolution of a function in $C^\infty_0$ with a Friedrichs mollifier (of sufficiently small support) is supported in the domain if and only if the function is in fact in $C^\infty_c$.

3. It is more useful when giving definitions/assertions via closures, to use a smaller set, not a larger one. Indeed, $$B=\overline{A_0} \implies B=\overline{A_1}$$ whenever $A_0\subset A_1\subset B$. This is particularly important in the statement of Myers–Serrin Theorem that $H^{1,p}=W^{1,p}$, where $H^{1,p}$ is "the smallest possible definition" while $W^{1,p}$ is "the largest possible definition" of the same Sobolev space.

4. Further note that proving that $C^2_0$ solutions are weak solutions to a BVP is merely a consistency check, and it is relevant to theory only to justify its consistency. Weak solutions (and Sobolev spaces) are introduced precisely to study problems that do not have classical solutions.