Let $\Pi_0$ be the $L_2$ projector that maps $u \in L_2(I)$ to a constant function, where $I$ is taken as the interval $(0,1)$.
I would like to find an error estimation for $u \in H^{1/2}(I)$, such as
$$ \|u-\Pi_0 u\|_{L_2(I)} \le C |u|_{H^{1/2}(I)} $$
Could anyone give me a hint on concrete value or upper bound for the constant $C$?
I have tried to transfer the constant determination problem into an eigenvalue problem.
Let $V:=\{ v \in H^{1/2}(I), \int_I vdx=0 \}$. Then the eigenvalue problem is:
Find $u\in V$ and $\lambda \in R$ such that, $$ (u,v)_{H^{1/2}(I)} = \lambda (u,v)_{L_2(I)} \mbox{ for } v\in V\:. $$
It seems that the eigen-function can be characterized by the following equation. $$ \int_I \frac{u(y)}{(x-y)^2} dy = \lambda u(x)\:. $$
But, I cannot go on to find the value or the lower bound of the minimal eigenvalue in need.
Any hint will be greatly appreciated.