We consider the Laplace operator in a thin tube of
with a Dirichlet condition on its boundary. We study asymptotically the spectrum of
such an operator as the thickness of the tube's cross section goes to zero. In particular we
analyse how the energy levels depend simultaneously on the curvature of the tube's central axis
and on the rotation of the cross section with respect to the Frenet frame. The main argument is a
-convergence theorem for a suitable sequence of quadratic energies.
The paper deals with a Dirichlet spectral problem for an elliptic operator with -periodic coefficients in a 3D bounded domain of small thickness . We study the asymptotic behavior of the spectrum as and tend to zero. This asymptotic behavior depends crucially on whether and are of the same order ( ≈ ), or is much less than ( =
< 1), or is much greater than ( =
> 1). We consider all three cases.
The paper deals with a Dirichlet spectral problem for an elliptic operator with
-periodic coefficients in a 3D bounded domain of small thickness
. We study the asymptotic behavior of the spectrum as
and tend to zero. This asymptotic behavior depends
crucially on whether and are of the same order
( ≈ ), or is much less than
( =
< 1),
or is much greater than
( =
...
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