Poincaré inequality in application of Lax-Milgram

Question

I was just reading the Wikipedia article about Garding's inequality, where an application is described, in which the conditions of the Lax-Milgram theorem are required. For some bilinear form one has Garding's inequality,

\begin{equation} B[u,u] \geq C\|u\|_{H^1(\Omega)}^2 - G\| u \|_{L^2(\Omega)} \ \ \ \forall u \in H_0^1(\Omega), \end{equation}

with positive constants $C$ and $G$. Now it is said, that by Poincaré's inequality there is another positive $K>0$, such that

\begin{equation} B[u,u] \geq K\|u\|_{H^1(\Omega)}^2 \ \ \ \forall u \in H_0^1(\Omega). \end{equation}

Now I'm wondering, when we apply Poincaré, i.e. $\| u \|_{L^2(\Omega}\leq \tilde{C}\|\nabla u\|_{L^2(\Omega)} \leq \tilde{C}\| u\|_{H^1(\Omega)}$, couldn't it happen that $C < G\tilde{C}$, so that

\begin{equation} B[u,u] < 0, \end{equation}

at least for some $u \in H_0^1(\Omega)$? Probably I'm missing am really simple argument right now, but could someone explain to me why the Wikipedia article is correct?

Answer 1

The reasoning on the wikipedia page is only valid for $$B[u,v] = \int \nabla u \cdot \nabla v \,\mathrm{d}x.$$ See also the last lines on the wikipedia page mentioned.