偏心环空中第一类Steklov-Dirichlet eigenvalue的形状单调性
Shape monotonicity of the first Steklov-Dirichlet eigenvalue on eccentric annuli
摘要：
本文研究了偏心圆环上第一Steklov-Dirichlet特征值关于圆环内、外边界中心间距离$t$的单调性.我们首先证明了$t$中特征值的可微性，并得到了二维及更高维中导数值的整式表达式。然后，我们推导出一个上界的特征值为每个$t$，在二维，由变分制剂。我们还得到了一个下界的特征值，给定的限制，两个边界的环是足够接近的。下界证明的关键是分析第一本征函数在双极坐标下的无穷级数展开式的极限性态。我们还进行了数值实验，表现出单调性的两个维度。
英文摘要：
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.
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