论文标题
偏心annuli上的第一个steklov-dirichlet特征值的单调性
Shape monotonicity of the first Steklov-Dirichlet eigenvalue on eccentric annuli
论文作者
论文摘要
在本文中，我们调查了偏心环上的第一个dirichlet特征值的单调性相对于距离（即$ t $），即环环的内部和外部边界之间的单调性。我们首先在$ t $中显示了特征值的不同性，并在两个及更高维度中获得衍生值的积分表达式。然后，我们根据各种公式来得出每个$ t $的特征值的上限。鉴于环形的两个边界足够接近，我们还获得了特征值的下限。下边界证明的关键点是分析双极坐标中第一个征函数的无限串联膨胀的极限行为。我们还执行数值实验，这些实验表现出两个维度的单调性。
In this paper, we investigate the monotonicity of the first Steklov--Dirichlet eigenvalue on eccentric annuli with respect to the distance, namely $t$, between the centers of the inner and outer boundaries of an annulus. We first show the differentiability of the eigenvalue in $t$ and obtain an integral expression for the derivative value in two and higher dimensions. We then derive an upper bound of the eigenvalue for each $t$, in two dimensions, by the variational formulation. We also obtain a lower bound of the eigenvalue, given a restriction that the two boundaries of the annulus are sufficiently close. The key point of the proof of the lower bound is in analyzing the limit behavior of an infinite series expansion of the first eigenfunction in bipolar coordinates. We also perform numerical experiments that exhibit the monotonicity for two dimensions.
