The spectra of harmonic layer potential operators on domains with rotationally symmetric conical points
The spectra of harmonic layer potential operators on domains with rotationally symmetric conical points
复制标题
旋转对称锥点域上的谐波层势算符谱
DOI：
10.1016/j.matpur.2017.10.012
复制
发表时间：
2017
期刊：
影响因子：
--
通讯作者：
Karl
中科院分区：
文献类型：
--
作者：
J. Helsing;Karl
本文研究了与Laplacian算子（Neumann-Poincar 'e算子的伴随算子）相联系的双层势的伴随算子，作为一个在具有锥点的区域的边界面上的映射.该算子的谱直接反映了相关传输问题的适定性。特别地，如果域被理解为具有复介电常数的包含体，嵌入在具有单位介电常数的背景介质中，则域的极化率张量在属于能量范数的预解式集合时被定义。我们研究具有有限个旋转对称圆锥点的曲面。在能量空间上，我们证明了本质谱由一个区间组成。上，即平方可积的边界数据，我们表明，本质谱由一个可数的联盟的曲线，其外的Fredholm指数可以计算为一个绕组数相对于本质谱。我们提供了明确的公式，取决于锥点的开口角。我们加强我们的研究与非常精确的数值实验，计算能量空间谱和光谱措施的极化率张量在两个不同的例子。我们的结果表明，在能量空间谱的连续部分，谱测度的密度可以非常迅速地接近零。
We study the adjoint of the double layer potential associated with the Laplacian (the adjoint of the Neumann-Poincar\'e operator), as a map on the boundary surfaceof a domain inwith conical points. The spectrum of this operator directly reflects the well-posedness of related transmission problems across. In particular, if the domain is understood as an inclusion with complex permittivity, embedded in a background medium with unit permittivity, then the polarizability tensor of the domain is well-defined whenbelongs to the resolvent set in energy norm. We study surfacesthat have a finite number of conical points featuring rotational symmetry. On the energy space, we show that the essential spectrum consists of an interval. On, i.e. for square-integrable boundary data, we show that the essential spectrum consists of a countable union of curves, outside of which the Fredholm index can be computed as a winding number with respect to the essential spectrum. We provide explicit formulas, depending on the opening angles of the conical points. We reinforce our study with very precise numerical experiments, computing the energy space spectrum and the spectral measures of the polarizability tensor in two different examples. Our results indicate that the densities of the spectral measures may approach zero extremely rapidly in the continuous part of the energy space spectrum.