Approximating $u \in H^1$ s.t. $u(T)=0$ with $u_n \in H^1_0$ in the gradient norm?

Question

Let $u \in H^1(0,T)$ with $u(T)=0$. Is it possible to find a sequence $u_n \in H^1_0(0,T)$ such that $\nabla u_n \to \nabla u$ in $L^2$?

I only need the convergence in the gradient.. not the full norm.

I know $H^1_0$ is not dense in $H^1$. But this is not the same question.

Answer 1

This is similar to Thomas' comment: No, it can't work.

Indeed, for each $u_n \in H_0^1$, we have $\int \nabla u_n = 0$, but $\int \nabla u$ might not be zero. It is zero iff $u \in H_0^1$, since $u(T) = 0$.