Spectral isoperimetric inequalities for a class of mixed eigenvalue problems of the Laplacian on triangles and trapezoids

TL;DR

This paper investigates the extremal properties of mixed eigenvalues of the Laplacian on triangles and trapezoids, providing geometric characterizations under specific constraints.

Contribution

It offers new spectral isoperimetric inequalities for mixed Laplacian eigenvalues on triangles and trapezoids, with geometric characterizations of extremal cases.

Findings

Characterization of extremum values of mixed eigenvalues

Spectral inequalities for Laplacian on triangles and trapezoids

Geometric constraints influence eigenvalue extremization

Abstract

In this paper, under suitable geometric constraints, we have successfully obtained characterizations for the extremum values of the functional of mixed eigenvalues of the Laplacian on triangles (or trapezoids) in the Euclidean plane $\mathbb{R}^2$.