Show that the bilinear form is coercive?

Question

Consider $I=(0,1)$ and $k>0$. Let $B(u,v):H^1_0(I)×H^1_0(I)\to \mathbb{R}$ be a bilinear form defined by $$B(u,v)=\int_{0}^{1}u'v'+kuv,~v\in H^1_0(I)$$ I have tried to show that $B$ is coercive, so I want $B(u,u)\ge \alpha \|u\|^2_{H^1}$ for some $\alpha$ but I could not get rid of $k$ to get the result.

Thank you all.

Answer 1

Use Poincaré inequality (https://en.wikipedia.org/wiki/Poincar%C3%A9_inequality) and you have that for every $u \in H_0^1(I)$,

$$\left\|u\right\|_{H^1(I)} \le \alpha \left\|u'\right\|_2$$ Now as $k> 0$, $$B(u,u) \ge \left\|u'\right\|_2^2 \ge \frac{1}{\alpha^2 } \left\|u\right\|_{H^1(I)}^2 $$