Inequalities between the lowest eigenvalues of Laplacians with mixed boundary conditions

TL;DR

This paper investigates inequalities between the lowest eigenvalues of Laplacians with mixed boundary conditions on convex domains, providing new results and partial proofs related to a conjecture on triangles.

Contribution

It establishes inequalities between eigenvalues for different boundary conditions and proves parts of a conjecture on mixed eigenvalues of triangles.

Findings

Proved inequalities between lowest eigenvalues for different boundary conditions.

Established partial results supporting a conjecture on triangles.

Enhanced understanding of spectral properties of Laplacians with mixed conditions.

Abstract

The eigenvalue problem for the Laplacian on bounded, planar, convex domains with mixed boundary conditions is considered, where a Dirichlet boundary condition is imposed on a part of the boundary and a Neumann boundary condition on its complement. Given two different such choices of boundary conditions for the same domain, we prove inequalities between their lowest eigenvalues. As a special case, we prove parts of a conjecture on the order of mixed eigenvalues of triangles.