We discuss some of the geometric properties, such as the foliated Schwarz symmetry, the monotonicity along the axial and the affine-radial directions, of the first eigenfunctions of a Zaremba problem for the Laplace operator on annular domains. Together with the shape calculus, these fine geometric properties help us to prove that the first eigenvalue is strictly decreasing as the inner ball moves towards the boundary of the outer ball.

中文翻译：

---

通过特征函数的几何形状的形状变化结果

我们讨论了环形域上拉普拉斯算子的 Zaremba 问题的第一特征函数的一些几何性质，例如叶状 Schwarz 对称性、沿轴向和仿射径向方向的单调性。连同形状演算，这些精细的几何性质帮助我们证明第一特征值随着内球移向外球的边界而严格减小。