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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 \|\nabla u\|^2_{L^{\infty}(\partial P)}\vert P\vert\quad? \end{eqnarray*} Here $\vert P\vert$ denotes the area of $P$.

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 $P$.

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 $P$?

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 \|\nabla u\|^2_{L^{\infty}(\partial P)}\vert P\vert\quad? \end{eqnarray*} Here $\vert P\vert$ denotes the area of $P$.

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\|_{L^{\infty}(\partial P)}\vert P\vert. \end{eqnarray*} Here $\vert P\vert$ denotes the area of $P$?

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 $P$?