The Asymptotics of the
Two-Dimensional Wave Equation for a General Multi-Connected
Vibrating Membrane with Piecewise Smooth Robin Boundary
Conditions |
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| Authors: | Email author" target="_blank">E?M?E?ZayedEmail author |
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| Institution: | (1) Department of Mathematics, Faculty of Science, Zagazig University, Zagazig, Egypt |
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| Abstract: | Abstract
The asymptotic expansion for small |t| of the trace of the
wave kernel
, where
and
are the eigenvalues of
the negative Laplacian
in the (x
2,x
2)-plane,
is studied for a multi-connected vibrating membrane Ω in
R
2
surrounded by simply connected bounded domains
Ω
j
with smooth boundaries ∂Ω
j
(j = 1, ..., n), where a finite number of piecewise
smooth Robin boundary conditions on the piecewise smooth
components Γ
i
(i = 1+k
j−1, ...,
k
j
) of the boundaries
∂Ω
j
are considered, such that
and
k
0 =
0. The basic problem is to extract information on the geometry
of Ω using the wave equation approach. Some geometric quantities
of Ω (e. g. the area of Ω, the total lengths of its boundary,
the curvature of its boundary, the number of the holes of Ω,
etc.) are determined from the asymptotic expansion of the trace
of the wave kernel
for small |t|. |
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| Keywords: | Inverse problem Wave kernel Eigenvalues Robin boundary conditions Vibrating membrane Hearing the shape of a drum |
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