Faber-Krahn inequality for polygons

Problem's Description

Let $\lambda_1$ denotes the first eigenvalue of the Laplace operator with Dirichlet boundary conditions. The famous Faber-Krahn inequality asserts that the ball minimizes $\lambda_1$ among open sets with a given volume. A natural question is to ask whether the regular $N$-gon also minimizes $\lambda_1$ among all polygons with $N$ sides and given area. Using Steiner symmetrization, G. Polya proves that it is true for triangles and quadrilaterals, but the problem is still completely open for $N\geq 5$.

1. Book Extremum Problems for eigenvalues of Elliptic Operators

   • Antoine Henrot