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Generalized Poincare Inequality

Consider \Omega \subset \Bbb{R}^N a bounded domain with Lipschitz boundary. If H is a non-zero, closed subspace in H^1(\Omega), which does not contain the non-zero constant functions, then there is a constant C>0, depending on \Omega, such that \|u \|_{L^2}\leq C \| |\nabla u | \|_{L^2},\ \forall u \in H.

Note that this generalizes the usual Poincare inequality, which says that the above inequality holds for some C>0 on the space H_0^1(\Omega), a space which does not contain the non-zero constant functions.

Proof: Suppose that the given inequality is not true. Then for every n \in \Bbb{N}^* there exists v_n \in H such that \displaystyle \frac{1}{n} > \frac{ \| | \nabla v_n | \|_{L^2}}{\|v_n \|_{L^2}}. Consider the sequence (u_n) \subset H defined by u_n=v_n / \|v_n \|_{L^2}. From the previous inequality it follows that \| | \nabla u_n | \|_{L^2} < 1/n, therefore \| |\nabla u_n | \|_{L^2} \to 0 as n \to \infty.

On the other hand we have \|u_n\|_{H^1}^2=1+\| | \nabla u_n | \|_{L^2}^2, which implies that (u_n) is bounded in H^1(\Omega). Then there is a subsequence denoted without loss of generality (u_n) which converges weakly to u \in H^1(\Omega). Since H is closed, it follows that u \in H. Since the inclusion H^1(\Omega) \subset L^2(\Omega) is compact, it follows that (u_n) has a subsequence which converges strongly in L^2(\Omega) to the same u. Without loss of generality, denote this sequence by (u_n). Then, using the weak-sequential lower semicontinuity of \|\cdot \|_{H^1} we have

\displaystyle \|u\|_{L^2}+\| |\nabla u| \|_{L^2}^2=\|u\|_{H^1}\leq \liminf_{n \to \infty} \|u_n\|_{H_1}^2=

\displaystyle =\liminf_{n \to \infty}(\|u_n\|_{L^2}^2+\| |\nabla u_n|\|_{L^2}^2)=\|u\|_{L^2}^2.

From the above, we get that \| |\nabla u| \|_{L^2}=0 and therefore u is constant. Moreover, \|u\|=\lim \|u_n\|=1, and we have found a non-zero constant u\in H. This contradicts the definition of H.

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