We consider the double layer potential (Neumann-Poincare) operator appearing in 3-dimensional elasticity. We show that the recent result about the polynomial compactness of this operator for the case of a homogeneous media follows without additional calculations from previous considerations by Agranovich this http URL., based upon pseudodifferential operators. Further on, we define the NP operator for the case of a nonhomogeneous isotropic media and show that its properties depend crucially on the character of non-homogeneity. If the Lame parameters are constant along the boundary, the NP operator is still polynomially compact. On the other hand, if these parameters are not constant, two or more intervals of continuous spectrum may appear, so the NP operator ceases to be polynomially compact. However, after a certain modification, it becomes polynomially compact again. Finally, we evaluate the rate of convergence of discrete eigenvalues of the NP operator to the tips of the essential spectrum.

中文翻译：

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3D 弹性中 Neumann-Poincaré 算子的光谱特性

我们考虑出现在 3 维弹性中的双层势（Neumann-Poincare）算子。我们表明，关于该算子在均匀介质情况下的多项式紧凑性的最新结果遵循 Agranovich 这个 http URL 之前的考虑，没有额外的计算，基于伪微分算子。此外，我们为非均匀各向同性介质的情况定义了 NP 算子，并表明其性质主要取决于非均匀性的特征。如果 Lame 参数沿边界不变，则 NP 算子仍然是多项式紧致的。另一方面，如果这些参数不是常数，则可能会出现两个或更多的连续谱区间，因此 NP 算子不再是多项式紧的。但是经过一定的修改后，它再次变得多项式紧凑。最后，我们评估了 NP 算子的离散特征值对基本谱尖端的收敛速度。