An Eigenvalue Optimisation Problem for Triangles and Quadrilaterals

Abstract

We study the following eigenvalue optimisation problem: Among all triangles of a given area, the equilateral triangle has the least principal eigenvalue for the Dirichlet Laplacian. Among all quadrilaterals of a given area, the square has the least principal eigenvalue. This means we want to find the minima of the function mapping a domain to its principal eigenvalue where the domain is any triangle (quadrilateral resp.) of a given area. We study the continuity of this function, i.e, we prove that if a sequence of domains converges in the Hausdorff complement topology, then the eigenvalues of the domains also converge to that of the limit domain. We use a specific algorithm wherein we start from an arbitrary triangle (quadrilateral resp.) of a given area, a sequence of symmetrisation operations yields an equilateral triangle (a square resp.). We further study the convergence of the corresponding sequence of eigenvalues to that of the equilateral triangle (the square resp.) of the same area. The result follows from the fact that the symmetrisation process considered keeps the area unaltered and decreases the eigenvalue.