TITLE: First Dirichlet eigenvalue on regular polygons
QUESTION [5 upvotes]: Assume $\lambda(P)$ is the first Dirichlet eigenvalue of a regular polygon $P$. Let $u$ be the corresponding eigenfunction, normalized by $\|u\|_{L^2(P)}=1$,  and $\partial_{\nu}u$ be its normal derivative on the boundary. Is the following estimate correct:
\begin{eqnarray*}
\lambda(P)\geq \|\partial_{\nu} u\|^2_{L^{\infty}(\partial P)}\vert P\vert\quad?
\end{eqnarray*}
Here $\vert P\vert$ denotes the area of $P$.

REPLY [6 votes]: There is a more general reason why any such statement will fail: If you consider a $P$ consisting of $N$ separate copies of the same basic region $P_0$, then $|P|=N|P_0|$, while everything else in your inequality is independent of $N$, so the inequality will fail for large $N$. (This problem has a degenerate ground state, but of course you could address this by slightly changing the shapes.)
If you connect the components by thin tubes, then you now have a connected $P$ and are still approximately in the situation described above.