Let $a$, $b \in \mathbb{R}$. Does there exist a $0 < c \in \mathbb{R}$ such that $c(u(a)^2+u(b)^2) \leq \left\lVert u \right\rVert_{H^1}^2$ for all $u \in H^1(a,b)$?
I can't find an explicit bound but I am pretty sure that even $c(u(a)^2+u(b)^2) \leq \left\lVert u \right\rVert_{L^2}^2$ holds for some $c$.
Edit: I formerly wrote $c(u(a)^2+u(b)^2) \leq \left\lVert u \right\rVert_{H^1}$. However I meant to square the right hand side.