TY - JOUR

T1 - Spectral Properties of the Neumann-Poincaré Operator in 3D Elasticity

AU - Miyanishi, Yoshihisa

AU - Rozenblum, Grigori

PY - 2021/6/1

Y1 - 2021/6/1

N2 - We consider the adjoint double layer potential (Neumann-Poincaré (NP)) 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 nonhomogeneity. 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.

AB - We consider the adjoint double layer potential (Neumann-Poincaré (NP)) 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 nonhomogeneity. 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.

KW - теория упругости

KW - потенциал

KW - Спектр

UR - http://www.scopus.com/inward/record.url?scp=85116819443&partnerID=8YFLogxK

U2 - 10.1093/imrn/rnz341

DO - 10.1093/imrn/rnz341

M3 - Article

AN - SCOPUS:85116819443

VL - 2021

SP - 8715

EP - 8740

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

IS - 11

ER -