Abstract:
We consider the Neumann boundary-value problem of finding the
small-parameter asymptotics of the eigenvalues and eigenfunctions for the
Laplace operator in a singularly perturbed domain consisting of two
bounded domains joined by a thin “handle”. The small parameter is the
diameter of the cross-section of the handle. We show that as the small
parameter tends to zero these eigenvalues converge either to the
eigenvalues corresponding to the domains joined or to the eigenvalues of
the Dirichlet problem for the Sturm–Liouville operator on the segment to
which the thin handle contracts. The main results of this paper
are the complete power small-parameter asymptotics of the eigenvalues and the
corresponding eigenfunctions and explicit formulae for the first terms of
the asymptotics. We consider critical cases generated by the choice of the
place where the thin “handle” is joined to the domains, as well as by
the multiplicity of the eigenvalues corresponding to the domains joined.