We consider the Laplace operator on a triangle, subject to attractive Robin boundary conditions. We prove that the equilateral triangle is a local maximiser of the lowest eigenvalue among all triangles of a given area provided that the negative boundary parameter is sufficiently small in absolute value, with the smallness depending on the area only. Moreover, using various trial functions, we obtain sufficient conditions for the global optimality of the equilateral triangle under fixed area constraint in the regimes of small and large couplings. We also discuss the constraint of fixed perimeter.

中文翻译：

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三角形最低 Robin 特征值的逆等周不等式

我们考虑三角形上的拉普拉斯算子，并受到有吸引力的 Robin 边界条件的影响。我们证明，只要负边界参数的绝对值足够小，等边三角形是给定区域的所有三角形中最低特征值的局部最大值，且该小值仅取决于面积。此外，使用各种试验函数，我们获得了小耦合和大耦合状态下固定面积约束下等边三角形全局最优性的充分条件。我们还讨论了固定周长的约束。