$\int_a^b|u(x)|^2dx \le C\int_a^b|u'(x)|^2dx $ for any $u(x)\in C_c^1([a,b])$
The following questions are what I want to know:
What does the notation $C_c^1([a,b])$ mean? (little c)
condition ' $u(a)=u(b)=0$' is essential to hold poincare inequality? If it is not essential, how can I prove that there is constant c satisfying the inequality?(not necessarily to be best constant)
Usually, little $c$ means "compact support". (See the comment of Eff). However, for the Poincare inequality to hold, the assumption $u \in C^1([a,b])$, $u(a)=u(b)=0$ is enough.
The condition $u(a)=u(b)=0$ is essential. Otherwise, take $u$ to be a constant function. Then $u' \equiv 0$, but $\int_a^b |u|^2 \, dx > 0$.