Two-term spectral asymptotics in linear elasticity
Matteo Capoferri, Leonid Friedlander, Michael Levitin, Dmitri Vassiliev
备注：
37 pages, 10 figures. This version corrects typesetting misprints in formula (B.6) in the previous arXiv and published versions. These misprints do not affect the rest of the paper. Authors' computational scripts are available at https://michaellevitin.net/elasticity.html
发表信息：
Journal of Geometric Analysis 33 (2023), article 242
We establish the two-term spectral asymptotics for boundary value problems of linear elasticity on a smooth compact Riemannian manifold of arbitrary dimension. We also present some illustrative examples and give a historical overview of the subject. In particular, we correct erroneous results published in [J. Geom. Anal. 31 (2021), 10164--10193].
Spectral asymptotics for linear elasticity: the case of mixed boundary conditions
Maxwell Institute for Mathematical Sciences（麦克斯韦数学科学研究所）；Department of Mathematics, Heriot-Watt University（赫瑞-瓦特大学数学系）；Department of Mathematics, University of York（约克大学数学系）
We consider the Neumann-Poincar'e (double layer potential) operator in 3D elasicity on a smooth closed surface. Its essential spectrum consists of 3 points. We find the asymptotics of sequences of eigenvalues converging to these three pounts. They are expressed via the eigenvalue distribution for a matrix eigenvalue problem depending on the points of the cotangent bundle of the surface. For the two-sided asymptotics for the distribution of the the union of sequences of eigenvalues converging to the point of the essential spectrum from both sides, an expression is found via the Euler characteri…
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 …
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