I know that the below Poincare's inequality is valid in the space $$H_{E}^{2}=\left \{ y\in H^{2}(0,1)\mid y(0)=y_{x}(0)=0 \right \}$$ Let $w(x, t)$, $x ∈ (0, 1)$, $t ∈ (0, ∞)$ satisfies the boundary condition $w(0, t)=0$. Then we have $$\int_{0}^{1}\left | w(x,t) \right |^{2}dx \leq \int_{0}^{1}\left | w_{x}(x,t) \right |^{2}dx$$ $$\left | w(1,t) \right |^{2} \leq \int_{0}^{1}\left | w_{x}(x,t) \right |^{2}dx$$
where the symbole $y_{x}$ denotes the derivative of $y$ with respect to x.
My question is : is this lemma still valid in the space $$H_{r}^{2}=\left \{ y\in H^{2}(0,L)\mid y(L)=y_{x}(L)=0 \right \}$$ by changing the boundary coundation from $0$ to $L\in\mathbb{R}$.
Thanks in advance.