From complete knowledge of the eigenvalues of the negative Laplacian on a bounded domain, one may extract information on the geometry and the boundary conditions by analyzing the asymptotic expansion of a spectral function. Explicit calculations are performed for an equilateral triangular domain with Dirichlet or Neumann boundary conditions, yielding in particular the corner angle terms. In three dimensions, some applications to eigenvalue problems for an equilateral triangular prism are dealt with, including the solid vertex terms.
REFERENCES
1.
H. P. W.
Gottlieb
, “Eigenvalues of the Laplacian for rectilinear regions
,” J. Aust. Math. Soc. Ser. B
29
, 270
–281
(1988
).2.
A.
Pleijel
, “A study of certain Green’s functions with applications in the theory of vibrating membranes
,” Ark. Mat.
2
, 553
–569
(1954
).3.
A.
Pleijel
, “On Green’s functions and the eigenvalue distribution of the three-dimensional membrane equation
,” Skandinav. Mat. Konger
12
, 222
–240
(1954
).4.
M.
Kac
, “Can one hear the shape of a drum?
,” Am. Math. Mon.
73
, 1
–23
(1966
).5.
H. P.
McKean
and I. M.
Singer
, “Curvature and the eigenvalues of the Laplacian
,” J. Diff. Geom.
1
, 43
–69
(1967
).6.
K.
Stewartson
and R. T.
Waechter
, “On hearing the shape of a drum: Further results
,” Proc. Cambridge Philos. Soc.
69
, 353
–363
(1971
).7.
R. T.
Waechter
, “On hearing the shape of a drum: An extension to higher dimensions
,” Proc. Cambridge Philos. Soc.
72
, 439
–447
(1972
).8.
B. D.
Sleeman
and E. M. E.
Zayed
, “An inverse eigenvalue problem for a general convex domain
,” J. Math. Anal. Appl.
94
, 78
–95
(1983
).9.
B. D.
Sleeman
and E. M. E.
Zayed
, “Trace formulas for the eigenvalues of the Laplacian
,” ZAMP J. Appl. Math. Phys.
35
, 106
–115
(1984
).10.
P.
Hsu
, “On the Θ function of a Riemannian manifold with boundary
,” Trans. Am. Math. Soc.
333
, 643
–671
(1992
).11.
E. M. E.
Zayed
and A. I.
Younis
, “An inverse problem for a general convex domain with impedance boundary conditions
,” Q. Appl. Math.
48
, 181
–188
(1990
).12.
E. M. E. Zayed, “An inverse eigenvalue problem for the Laplace operator,” in Ordinary and Partial Differential Equations, edited by W. N. Everitt and B. D. Sleeman, Lecture Notes in Mathematics, Vol. 964 (Springer-Verlag, Berlin, 1982), pp. 718–726.
13.
E. M. E.
Zayed
, “Eigenvalues of the Laplacian: An extension to higher dimensions
,” IMA J. Appl. Math.
33
, 83
–99
(1984
).14.
E. M. E.
Zayed
, “An inverse eigenvalue problem for a general convex domain: An extension to higher dimensions
,” J. Math. Anal. Appl.
112
, 455
–470
(1985
).15.
E. M. E.
Zayed
, “Eigenvalues of the Laplacian for the third boundary value problem
,” J. Aust. Math. Soc. Ser. B
29
, 79
–87
(1987
).16.
E. M. E.
Zayed
, “Eigenvalues of the Laplacian for the third boundary value problem: An extension to higher dimensions
,” J. Math. Anal. Appl.
130
, 78
–96
(1988
).17.
E. M. E.
Zayed
, “Heat equation for an arbitrary doubly connected region in with mixed boundary conditions
,” ZAMP J. Appl. Math. Phys.
40
, 339
–355
(1989
).18.
E. M. E.
Zayed
, “Hearing the shape of a general convex domain
,” J. Math. Anal. Appl.
142
, 170
–187
(1989
).19.
E. M. E.
Zayed
, “On hearing the shape of an arbitrary doubly connected region in
,” J. Aust. Math. Soc. Ser. B
31
, 472
–483
(1990
).20.
E. M. E.
Zayed
, “Heat equation for an arbitrary multiply connected region in with impedance boundary conditions
,” IMA J. Appl. Math.
45
, 233
–241
(1990
).21.
E. M. E.
Zayed
, “Hearing the shape of a general doubly connected domain in with impedance boundary conditions
,” J. Math. Phys.
31
, 2361
–2365
(1990
).22.
E. M. E.
Zayed
, “Hearing the shape of a general doubly connected domain in with mixed boundary conditions
,” ZAMP J. Appl. Math. Phys.
42
, 547
–564
(1991
).23.
E. M. E.
Zayed
, “Heat equation for a general convex domain in with a finite number of piecewise impedance boundary conditions
,” Appl. Anal.
42
, 209
–220
(1991
).24.
M. V.
Den Berg
and S.
Srisatkunarajah
, “Heat equation for a region in with a polygonal boundary
,” J. London Math. Soc.
37
, 119
–127
(1988
).25.
H. P. W.
Gottlieb
, “Eigenvalues of the Laplacian with Neumann boundary conditions
,” J. Aust. Math. Soc Ser. B
26
, 293
–309
(1985
).26.
H. P. W.
Gottlieb
, “Hearing the shape of an annular drum
,” J. Aust. Math. Soc. Ser. B
24
, 435
–438
(1983
).27.
M. A.
Pinsky
, “The eigenvalues of an equilateral triangle
,” SIAM J. Math. Anal.
11
, 819
–827
(1980
);“
Completeness of the eigenfunctions of the equilateral triangle
,” SIAM J. Math. Anal.
16
, 848
–851
(1985
)., SIAM J. Math. Anal.
28.
A. I. Younis, Ph.D. thesis, Zagazig University, Egypt (in preparation).
This content is only available via PDF.
© 1994 American Institute of Physics.
1994
American Institute of Physics
You do not currently have access to this content.
