On mixed Dirichlet-Neumann eigenvalues of triangles

TL;DR

This paper investigates the ordering of mixed Dirichlet-Neumann eigenvalues on triangles and rhombi, providing classifications based on boundary conditions and symmetry, with conjectures extending to more general cases.

Contribution

It establishes a systematic order for mixed eigenvalues on right triangles and rhombi, and proposes conjectures for broader triangle configurations.

Findings

Ordered eigenvalues for right triangles with various boundary conditions.

Classified eigenvalues of rhombi by symmetry and antisymmetry.

Proposed conjecture for eigenvalue ordering in general triangles.

Abstract

We order lowest mixed Dirichlet-Neumann eigenvalues of right triangles according to which sides we apply the Dirichlet conditions. It is generally true that Dirichlet condition on a superset leads to larger eigenvalues, but it is nontrivial to compare e.g. the mixed cases on triangles with just one Dirichlet side. As a consequence of that order we also classify the lowest Neumann and Dirichlet eigenvalues of rhombi according to their symmetry/antisymmetry with respect to the diagonal. We also give an order for the mixed Dirichlet-Neumann eigenvalues on arbitrary triangle, assuming two Dirichlet sides. The single Dirichlet side case is conjectured to also have appropriate order, following right triangular case.