Assume $\lambda(P)$ is the first Dirichlet eigenvalue of a regular polygon $P$. Assume $u$ is the corresponding eigenfunction and $\partial_{\nu}u$ its normal derivative on the boundary. Is the following estimate correct?
\begin{eqnarray*}
\lambda(P)\geq \|\nabla u\|^2_{L^{\infty}(\partial P)}\vert P\vert.
\end{eqnarray*}
Here $\vert P\vert$ denotes the area of $P$?