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An Isaac Newton Institute Workshop

The Theory of Highly Oscillatory Problems

Spectral results for membrane with perturbed stiffness and mass density

Authors: Natalia Babych (University of Bath, United Kingdom), Yuri Golovaty (Lviv National University, Ukraine)


We study the spectrum of a nonhomogeneous membrane that consists of two parts with strongly different stiffness and mass density. The small parameter describes the quotient of stiffness coefficients. The M-th power of the parameter is comparable to the ratio of densities. We show that the asymptotic behaviour of eigenvalues and eigenfunctions depends on rate M. We distinguish three cases M<1, M=1 and M>1.

The strong resolvent convergence of perturbed operators leads to loss of the completeness of limit eigenfunction system in both cases when M is different from 1. Therefore the limit operators describe only a part of the prelimit membrane vibrations. With an eye to close this gap we use the WKB-asymptotic expansions with a quantized small parameter to prove the existence of other kind of eigenvibrations, namely high frequency vibrations.

In the critical case M=1 the limit operator is a nonself-adjoint one, nevertheless the perturbed operators are self-adjoint in a certain topology. The multiplicity of spectrum and structure of root spaces are investigated.

Complete asymptotic expansions for eigenelements are constructed and justified in each case of the perturbations.