We study extremal properties of the determinant of Friedrichs selfadjoint Laplacian on the Euclidean isosceles triangle envelopes of fixed area as a function of angles. We deduce an explicit closed formula for the determinant. Small-angle asymptotics show that the determinant grows without any bound as an angle of a 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 finds the critical value. When the area is below a certain value (approximately 1.92), the equilateral envelope minimizes the determinant. When the area exceeds this value, the equilateral envelope is a local maximum of the determinant.

我们研究了作为角度函数的固定面积的欧几里得等腰三角形包络上 Friedrichs 自伴随拉普拉斯算子的行列式的极值特性。我们为行列式推导出一个明确的封闭公式。小角度渐近线表明，当三角形包络的角度变为零时，行列式无任何限制地增长。我们证明等边三角形包络（最对称的几何图形）总是会产生行列式的临界点并找到临界值。当面积低于某个值（约 1.92）时，等边包络使行列式最小化。当面积超过此值时，等边包络是行列式的局部最大值。