Talk:Gårding's inequality

Excuse me if I'm wrong (I'm new to the subject) - but the application of Garding's Inequality in the example with Poisson's equation seems to me completely unnecessary, since the statement $$B[u,u] \ge K\|u\|_{H^1(\Omega)}^2$$ in this case is precisely Poincare's inequality (or "Friedrich's inequality" in Wikipedia's terminology). Maybe there exists a better example? Gregor erasmus (talk) 15:03, 14 June 2013 (UTC)

I have the same feeling. Moreover, I think the statement that the two terms can be combined using the Poincare's inequality is wrong, as nothing can be said about C and G (besides that they are >= 0). It could be that the right hand side of $$B[u, u] \geq C \| u \|_{H^{1} (\Omega)}^{2} - G \| u \|_{L^{2} (\Omega)}^{2}$$ is negative for some u.

I agree with the first remark. However, "the statement that the two terms can be combined using the Poincare's inequality" is not wrong in my opinion. Indeed, write $$B[u, u] + G \| u \|_{L^{2} (\Omega)}^{2} \geq C \| u \|_{H^{1} (\Omega)}^{2} $$, and applying Poincaré's inequality, there exists some $$C_{P}$$ such that $$\| u \|_{L^{2}} \leq C B[u,u] $$. This leads to $$\left(\frac{G}{C_p}+1\right)B[u,u] \geq C\|u\|_{H^1(\Omega)}$$. Thus $$K$$ can be taken as $$\dfrac{C}{\frac{G}{C_p}+1}$$