Talk:Gårding's inequality
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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 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)
![{\displaystyle B[u,u]\geq K\|u\|_{H^{1}(\Omega )}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cf1634073052d549058699792137e5d819bb1aad)
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 is negative for some u.
![{\displaystyle B[u,u]\geq C\|u\|_{H^{1}(\Omega )}^{2}-G\|u\|_{L^{2}(\Omega )}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a1edd8b09e410d694761f47f1be3ef9e4b8903b)
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 , and applying Poincaré's inequality, there exists some such that . This leads to . Thus can be taken as
![{\displaystyle B[u,u]+G\|u\|_{L^{2}(\Omega )}^{2}\geq C\|u\|_{H^{1}(\Omega )}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f39a6ab4fdef4f04867db804f19be5c6c78c5119)
![{\displaystyle \|u\|_{L^{2}}\leq CB[u,u]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96fab8ca17e868e5562cfec7ce62cc8f8b929598)
![{\displaystyle \left({\frac {G}{C_{p}}}+1\right)B[u,u]\geq C\|u\|_{H^{1}(\Omega )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bac8e3253c4247adf5451bfa961ae65d6b8144e2)
