2019/04/20 by Yoshihisa Miyanishi, Grigori Rozenblum, Miyanishi, Yoshihisa +1
Computer Science · Engineering · Mathematics · #47A75 (primary) #58J50 (secondary) #Advanced Mathematical Modeling in Engineering #Composite Material Mechanics #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.1904.09449
openalex publication_date 2019/04/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the double layer potential (Neumann-Poincaré) 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 et.al., 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 Lamé 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.