Consider the Laplacian eigenvalue problem $-\Delta u = \lambda u$ on $\Omega$ with Dirichlet boundary conditions. Let $\lambda_1$ denote the first eigenvalue. The following theorem is well known:

(Faber-Krahn) Let $c$ be a positive number and $B$ the ball of volume $c$. Then $$\lambda_1(B) = \min\{\lambda_1(\Omega), \Omega\ \text{open subset of}\ \mathbb{R}^n, |\Omega| = c\}.$$

I am considering the question of minimizing $\lambda_1$ in the class of polygons with a given number $N$ as sides. If we denote by $\mathcal{P}_N$ the class of plane polygons with at most $N$ edges, then it is known that the problem $$\min\{\lambda_1(\Omega), \Omega \in \mathcal{P}_N, |\Omega| = a\}$$ has a solution. This one has exactly $N$ edges. For the case $N=3$, it has been proven the equilateral triangle minimizes $\lambda_1$. For $N=4$, the square minimizes $\lambda_1$.

Question: Is there a general result that the regular $N$-gon have the least first eigenvalue among all the $N$-gons of given area for $N \geq 5$?

Consider the Laplacian eigenvalue problem $-\Delta u = \lambda u$ on $\Omega$ with Dirichlet boundary conditions. Let $\lambda_1$ denote the first eigenvalue. The following theorem is well known:

(Faber-Krahn) Let $c$ be a positive number and $B$ the ball of volume $c$. Then $$\lambda_1(B) = \min\{\lambda_1(\Omega), \Omega\ \text{open subset of}\ \mathbb{R}^n, |\Omega| = c\}.$$

I am considering the question of minimizing $\lambda_1$ in the class of polygons with a given number $N$ as sides. If we denote by $\mathcal{P}_N$ the class of plane polygons with at most $N$ edges, then it is known that the problem $$\min\{\lambda_1(\Omega), \Omega \in \mathcal{P}_N, |\Omega| = a\}$$ has a solution. This one has exactly $N$ edges. For the case $N=3$, it has been proven the equilateral triangle minimizes $\lambda_1$. For $N=4$, the square minimizes $\lambda_1$. (Both proof uses the properties of the Steiner symmetrization of $\Omega$. The original proof was due to Pólya. Unfortuantely I could not find the original paper, but the proof can also be found in Extremum problems for Eigenvalues of Elliptic Operators by Henrot.)

Question: Is there a general result that the regular $N$-gon have the least first eigenvalue among all the $N$-gons of given area for $N \geq 5$?

Consider the Laplacian eigenvalue problem $-\Delta u = \lambda u$ on $\Omega$ with Dirichlet boundary conditions. Let $\lambda_1$ denote the first eigenvalue. The following theorem is well known:

(Faber-Krahn) Let $c$ be a positive number and $B$ the ball of volume $c$. Then $$\lambda_1(B) = \min\{\lambda_1(\Omega), \Omega\ \text{open subset of}\ \mathbb{R}^n, |\Omega| = c\}.$$

I am considering the question of minimizing $\lambda_1$ in the class of polygons with a given number $N$ as sides. If we denote by $\mathcal{P}_N$ the class of plane polygons with at most $N$ edges, then it is known that the problem $$\min\{\lambda_1(\Omega), \Omega \in \mathcal{P}_N, |\Omega| = a\}$$ has a solution. This one has exactly $N$ edges. For the case $N=3$, it has been proven the equilateral triangle minimizes $\lambda_1$. For $N=4$, the square minimizes $\lambda_1$.

Question: Is there a general result that the regular N-gone have the least first eigenvalue among all the N-gones of given area for $N \geq 5$?

Consider the Laplacian eigenvalue problem $-\Delta u = \lambda u$ on $\Omega$ with Dirichlet boundary conditions. Let $\lambda_1$ denote the first eigenvalue. The following theorem is well known:

(Faber-Krahn) Let $c$ be a positive number and $B$ the ball of volume $c$. Then $$\lambda_1(B) = \min\{\lambda_1(\Omega), \Omega\ \text{open subset of}\ \mathbb{R}^n, |\Omega| = c\}.$$

I am considering the question of minimizing $\lambda_1$ in the class of polygons with a given number $N$ as sides. If we denote by $\mathcal{P}_N$ the class of plane polygons with at most $N$ edges, then it is known that the problem $$\min\{\lambda_1(\Omega), \Omega \in \mathcal{P}_N, |\Omega| = a\}$$ has a solution. This one has exactly $N$ edges. For the case $N=3$, it has been proven the equilateral triangle minimizes $\lambda_1$. For $N=4$, the square minimizes $\lambda_1$.

Question: Is there a general result that the regular $N$-gon have the least first eigenvalue among all the $N$-gons of given area for $N \geq 5$?