Spectral determinant on Euclidean isosceles triangle envelopes of fixed area as a function of angles: absolute minimum and small-angle asymptotics

TL;DR

This paper investigates how the determinant of the Friedrichs Laplacian on Euclidean isosceles triangle envelopes varies with angles, revealing asymptotic growth, critical points at equilateral configurations, and area-dependent extremal properties.

Contribution

It provides the first analysis of the determinant's extremal properties on triangle envelopes, including small-angle asymptotics and critical point characterization.

Findings

Determinant grows unbounded as an angle approaches zero.

Equilateral triangle envelope is a critical point of the determinant.

For small areas, the equilateral triangle minimizes the determinant; for large areas, it is a local maximum.

Abstract

We study extremal properties of the determinant of Friederichs selfadjoint Laplacian on the Euclidean isosceles triangle envelopes of fixed area as a function of angles. Small-angle asymptotics show that the determinant grows without any bound as an angle of triangle envelope goes to zero. We prove that the equilateral triangle envelope (the most symmetrical geometry) always gives rise to a critical point of the determinant and find the critical value. Moreover, if the area of envelopes is not too large, then the determinant achieves its absolute minimum only on the equilateral triangle envelope and there are no other critical points, whereas for sufficiently large area the equilateral triangle envelope corresponds to a local maximum of the determinant.