## Generalized Poincare Inequality

Consider a bounded domain with Lipschitz boundary. If is a non-zero, closed subspace in , which does not contain the non-zero constant functions, then there is a constant , depending on , such that .

Note that this generalizes the usual Poincare inequality, which says that the above inequality holds for some on the space , a space which does not contain the non-zero constant functions.

**Proof: **Suppose that the given inequality is not true. Then for every there exists such that . Consider the sequence defined by . From the previous inequality it follows that , therefore as .

On the other hand we have , which implies that is bounded in . Then there is a subsequence denoted without loss of generality which converges weakly to . Since is closed, it follows that . Since the inclusion is compact, it follows that has a subsequence which converges strongly in to the same . Without loss of generality, denote this sequence by . Then, using the weak-sequential lower semicontinuity of we have

.

From the above, we get that and therefore is constant. Moreover, , and we have found a non-zero constant . This contradicts the definition of .