An inverse problem for a general convex domain with impedance boundary conditions
Authors:
E. M. E. Zayed and A. I. Younis
Journal:
Quart. Appl. Math. 48 (1990), 181-188
MSC:
Primary 35R30; Secondary 35C99, 35P05, 58G25
DOI:
https://doi.org/10.1090/qam/1040241
MathSciNet review:
MR1040241
Full-text PDF Free Access
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Abstract: The spectral function $\theta \left ( t \right ) = \sum \nolimits _{n = 1}^\infty {\exp \left ( { - t{\lambda _n}} \right )}$, where $\left \{ {{\lambda _n}} \right \}_{n = 1}^\infty$ are the eigenvalues of the Laplace operator $\Delta = \sum \nolimits _{i = 1}^2 {{{\left ( {\partial /\partial {x^i}} \right )}^2}}$ in the ${x^1}{x^2}$-plane, is studied for a general convex domain $\Omega \subseteq {R^2}$ with a smooth boundary $\partial \Omega$ together with a finite number of piecewise smooth impedance boundary conditions on the parts ${\Gamma _{1,...,}}{\Gamma _m}$ of $\partial \Omega$ such that $\partial \Omega = U_{j = 1}^m{\Gamma _j}$.
- Mark Kac, Can one hear the shape of a drum?, Amer. Math. Monthly 73 (1966), no. 4, 1–23. MR 201237, DOI https://doi.org/10.2307/2313748
- H. P. McKean Jr. and I. M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geometry 1 (1967), no. 1, 43–69. MR 217739
- Åke Pleijel, A study of certain Green’s functions with applications in the theory of vibrating membranes, Ark. Mat. 2 (1954), 553–569. MR 61257, DOI https://doi.org/10.1007/BF02591229
- B. D. Sleeman and E. M. E. Zayed, An inverse eigenvalue problem for a general convex domain, J. Math. Anal. Appl. 94 (1983), no. 1, 78–95. MR 701450, DOI https://doi.org/10.1016/0022-247X%2883%2990006-9
- Lance Smith, The asymptotics of the heat equation for a boundary value problem, Invent. Math. 63 (1981), no. 3, 467–493. MR 620680, DOI https://doi.org/10.1007/BF01389065
K. Stewartson and R. T. Waechter, On hearing the shape of a drum: further results, Proc. Camb. Phil. Soc. 69, 353–363 (1971)
- Elsayed M. E. Zayed, Hearing the shape of a general convex domain, J. Math. Anal. Appl. 142 (1989), no. 1, 170–187. MR 1011418, DOI https://doi.org/10.1016/0022-247X%2889%2990173-X
M. Kac, Can one hear the shape of a drum?, Amer. Math. Monthly 73, No. 4, part II, 1–23 (1966)
H. P. McKean, Jr., and I. M. Singer, Curvature and the eigenvalues of the Laplacian, J. Differential Geom. 1, 43–69 (1967)
Ȧ. Pleijel, A study of certain Green’s functions with applications in the theory of vibrating membranes, Ark. Mat. 2, 553–569 (1954)
B. D. Sleeman and E. M. E. Zayed, An inverse eigenvalue problem for a general convex domain, J. Math. Anal. Appl. 94 (1), 78–95 (1983)
L. Smith, The asymptotics of the heat equation for a boundary value problem, Invent. Math. 63, 467–493 (1981)
K. Stewartson and R. T. Waechter, On hearing the shape of a drum: further results, Proc. Camb. Phil. Soc. 69, 353–363 (1971)
E. M. E. Zayed, Hearing the shape of a general convex domain, J. Math. Anal. Appl. 142 (1), 170–187 (1989)
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Article copyright:
© Copyright 1990
American Mathematical Society