functional analysis

Question

Please I have two questions: we take $ V = \{v \in H ^ 1 (0,1) ; v (0) = 0\}$ \

Q1/ the space $V \cap H ^ 2 (0.1 )$ is dense in $ V $ ?

Q2/ the space $V \cap H ^ 2 (0.1)$ is dense in $H ^ 1 (0,1)$ ?

I would need proof. Thank you

Answer 1

Q1/ Yes. If you know the basic fact that $H^2(0,1)$ is dense in $H^1(0,1)$, then just take $v\in V$, then there is $u_n$ a sequence of $H^2$ converging to $v$. Then $\tilde{u}_n:=u_n-u_n(0)$ belongs to $H^2(0,1)\cap V$ and as $u_n(0)\to_{n\to\infty} v(0)=0$ (I guess you know that $H^1$ convergence implies uniform convergence, in dimension 1), you have $\tilde{u}_n$ converges to $v$.

Q2/ No. As $V$ is closed, you cannot reach $H^1$ from functions in $V$.