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We study extremal properties of the determinant of Friederichs selfadjointLaplacian on the Euclidean isosceles triangle envelopes of fixed area as afunction of angles. Small-angle asymptotics show that the determinant growswithout any bound as an angle of triangle envelope goes to zero. We prove thatthe equilateral triangle envelope (the most symmetrical geometry) always givesrise to a critical point of the determinant and find the critical value.Moreover, if the area of envelopes is not too large, then the determinantachieves its absolute minimum only on the equilateral triangle envelope andthere are no other critical points, whereas for sufficiently large area theequilateral triangle envelope corresponds to a local maximum of thedeterminant.

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