Small question about strong convergence

Question

I have a small question

I have that $ \lambda_1 ||v_0||^2_{L^2(0,1)}=||v_0||^2_{H^1_0(0,1)}$ and that $v_n\rightarrow v_0$ on $L^2$ (strongly) From the Poincaré inequality i have that $\lambda_1||v_n||_{L^2(0,1)}\leq ||v_n||_{H^1_0(0,1)} $

$v_n$ is weakly convergent to $v_0$ on $H^1_0$

can i say that $v_n\rightarrow v_0$ on $H^1_0(0,1)$ (strongly)?

Thank you .

Answer 1

No. Take any function $v_0$ satisfying Poincaré inequality with equality. Then take your favorite sequence $x_n\in H^1_0(0,1)$ with $x_n \to 0$ in $L^2(0,1)$, $x_n \rightharpoonup 0$ in $H^1_0(0,1)$ but not strongly converging in $H^1_0(\Omega)$.

If you know $\lambda_1 \|v_n\|_{L^2(0,1)} =\|v_n\|_{H^1_0(0,1)}$ then you can prove strong convergence of $v_n$ by weak convergence plus convergence of norms.