Density of $C_c^\infty([0,1])$ in $H_0^1(0,1)$

Question

I'm trying to prove that $C_c^\infty([0,1])$ is dense in $H_0^1(0,1) = \{ f \in H^1(0,1) : f(0) = f(1) =0\}$ for the usual Sobolev norm $\Vert f \Vert = \Vert f \Vert_{L^2(0,1))} + \Vert f' \Vert_{L^2(0,1)}$. So far, I have no clue on how to proceed, so any help is greatly appreciated !

Answer 1

Let $\phi \in C_c^{\infty}([0,1])$ be fixed with $0\leq \phi \leq 1$. Choose $\{g_n\} \subset C_c^{\infty}([0,1])$ such that $\|g_n-f'\|_2 \to 0$. Define $f_n(x)=\int_0^{x}g_n(t)\, dt -c_n \int_0^{x}\phi (t)\, dt$ where $c_n=\frac {\int_0^{1} g_n(t)\, dt} {\int_0^{1}\phi(t)\, dt}$. I leave it to you to verify that $\{f_n\} \subset C_c^{\infty}([0,1])$ and $f_n\ \to f$ in $H_0^{1}(0,1)$.