
> 
> Define $\mathbb{W} = H\_{0}^{1}(-1,1) \times H\_{0}^{1}(-1,1)$, where $u \in H\_{0}^{1}(-1,1)$ if $u,u^{\prime} \in L^{2}(-1,1)$ and $u(-1) = u(1) = 0$. Consider the sesquilinear form $a: \mathbb{W} \times \mathbb{W} \to \mathbb{C}$ given by
> $$
> a((u,v),(w,z)) = \int\_{-1}^{1}u^{\prime}\overline{w}^{\prime}dx + \int\_{-1}^{1}v^{\prime}\overline{z}^{\prime}dx + \int\_{-1}^{1}u\overline{z}dx + \int\_{-1}^{1}v\overline{w}dx 
> $$
> for $(u,v),(w,z) \in \mathbb{W}$. I want to show that $a$ is coercive, that is, there is $C > 0$ such that
> 
> 
> 


$$
\text{Re}~a((u,v),(u,v)) \geq C \|(u,v)\|\_{W}^{2}, \ \ \forall (u,v) \in \mathbb{W}.
$$


Where $\|(u,v)\|\_{W} = \|u\|\_{H\_{0}^{1}} + \|v\|\_{H\_{0}^{1}}$, such that $\|u\|\_{H\_{0}^{1}} = \|u^{\prime}\|\_{L^{2}}^{2} + \|u\|\_{L^{2}}^{2} $


Let $(u,v) \in \mathbb{W}$ and
$$
a((u,v),(u,v)) = \int\_{-1}^{1}|u^{\prime}|^{2}dx + \int\_{-1}^{1}|v^{\prime}|^{2}dx + \int\_{-1}^{1}u\overline{v}dx + \int\_{-1}^{1}v\overline{u}dx 
$$
But
$$
 \int\_{-1}^{1}u\overline{v}dx + \int\_{-1}^{1}v\overline{u}dx = 2\text{Re}\int\_{-1}^{1}u\overline{v}dx
$$
Then, by Poincaré’s inequality
$$
a((u,v),(u,v)) = \|u^{\prime}\|\_{L^{2}}^2 + \|v^{\prime}\|\_{L^{2}}^2 + 2\text{Re}\int\_{-1}^{1}u\overline{v}dx \geq 2^{-3/2}\|u\|\_{L^{2}(-1,1)}^2 + 2^{-3/2}\|v\|\_{L^{2}(-1,1)}^2 + 2\text{Re}\int\_{-1}^{1}u\overline{v}dx
$$
I know that  

$$
0 \leq (\|u\|\_{L^{2}} - \|v\|\_{L^{2}})^{2} = 
(\|u\|\_{L^{2}}^2 + \|v\|\_{L^{2}}^2) - 2\|u\|\_{L^{2}} \|v\|\_{L^{2}}
$$
Then
$$
2\text{Re}\int\_{-1}^{1}u\overline{v}dx \leq 2\bigg|\int\_{-1}^{1}u\overline{v}dx \bigg| \leq 2\|u\|\_{L^{2}} \|v\|\_{L^{2}} \leq \|u\|\_{L^{2}}^2 + \|v\|\_{L^{2}}^2 
$$


But I need a contrary inequality and not the one above. Help me, please!!



Proof that sesquilinear form in is coercive
The first eigenvalue of the second derivative with Dirichlet b.c. on $(-1,1)$ is $\pi^2/4$ (with eigenfunction $\cos \frac{\pi x}{2}$) and then Poincare' inequality with optimal constant is $\|u\|\_2^2 \leq \frac{4}{\pi^2}\|u'\|\_2^2$. It follows that  

$$2| Re \int\_{-1}^1 u\bar v | \leq 2\|u\|\_2\|v\|\_2 \leq \|u\|\_2^2+\|v\|\_2^2\leq \frac{4}{\pi^2}(\|u'\|\_2^2+\|v'\|\_2^2)$$ which gives $C \geq 1-4/\pi^2$, using on $W$ the norm $\|(u,v)\|^2\_W=\|u'\|\_2^2+\|v'\|\_2^2$.


