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 $B[u,u]\geq K\|u\|_{H^{1}(\Omega )}^{2}$ 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 $\|u\|_{L^{2}}\leq CB[u,u]$. 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 ${\dfrac {C}{{\frac {G}{C_{p}}}+1}}$