Density of compactly supported functions on a clopen set

Question

Let $C^\infty_c((0,1])$ be the set of compactly supported functions on $(0,1]$. and denote by $H^1(0,1)$ the standard Sobolev space.

Is $C^\infty_c((0,1])$ dense in $\{u \in H^1(0,1), \quad u(0)=0\}$?

Answer 1

The space $C_c^\infty((0,1])$ is dense in your space. Here is the idea of the proof:

Take $u$ in your space. Use a classical extension operator to extend the function $u$ to $H^1_0((0,2))$. Approximate this function by smooth functions from $C_c^\infty((0,2))$. Their restrictions to $(0,1]$ are in $C_c^\infty((0,1])$ and approximate $u$.