A remark on the order of mixed Dirichlet-Neumann eigenvalues of polygons

TL;DR

This paper investigates the relationship between the lowest eigenvalues of Laplacians with mixed boundary conditions on convex polygons, providing conditions under which one eigenvalue dominates and applying results to triangles, partially confirming a conjecture.

Contribution

It offers a sufficient condition for eigenvalue dominance in mixed boundary problems and applies it to triangles, advancing understanding of spectral properties of polygons.

Findings

Established a sufficient condition for eigenvalue dominance.

Applied the condition to triangles, confirming a conjecture.

Proved a similar result for right triangles.

Abstract

Given the Laplacian on a planar, convex domain with piecewise linear boundary subject to mixed Dirichlet-Neumann boundary conditions, we provide a sufficient condition for its lowest eigenvalue to dominate the lowest eigenvalue of the Laplacian with the complementary boundary conditions (i.e. with Dirichlet replaced by Neumann and vice versa). The application of this result to triangles gives an affirmative partial answer to a recent conjecture. Moreover, we prove a further observation of similar flavor for right triangles.