Easy question about $H_0^1$ space

Question

I have some trouble with proper understanding of $H_0^1(0,1)$ space. Consider the following space $$H_D = \{u\in H^1(0,1): u(0) = u(1) = 0\}.$$ What can we say about the connection between $H_D$ and $H^1_0(0,1)$. Is $H_D$ in $H^1_0(0,1)$? In literature stays, that functions in $H_0^1(0,l)$ are interpreted as $$''u = 0\; \text{on}\; \partial\Omega''.$$

Answer 1

Do you know anything about the trace operator? There is a theorem that says that if $U$ is a bounded domain and $\partial U$ is $C^1$ and $u \in W^{1,p}$, then $$ u \in W_0^{1,p}(U) \iff Tu=0 \text{ on } \partial U, $$ where $T$ is the trace operator. You want $p = 2$ here.

Answer 2

In one-dimensions, Sobolev functions have a well-defined continuous representative. Therefore, they are well-defined at every point of their domain.