The wave equation approach to an inverse problem for a general multi-connected domain in R2 with mixed boundary conditions

The spectral function @[email protected]^(t)[email protected]?"J"="1^~exp([email protected]"J^1^/^2) where @m"J"J"="1^~ are the eigenvalues of the negative Laplacian [email protected]"[email protected]?"@u"="1^[email protected]?/@?x^@u^2 in R^2 is studied for small |t| for a variety of domains, where -~

[1]  H. Gottlieb,et al.  Eigenvalues of the Laplacian with Neumann boundary conditions , 1985, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[2]  Michel L. Lapidus,et al.  Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture , 1991 .

[3]  H. McKean,et al.  Curvature and the Eigenvalues of the Laplacian , 1967 .

[4]  E. Zayed,et al.  An inverse problem for a general convex domain with impedance boundary conditions , 1990 .

[5]  R. Courant,et al.  Methods of Mathematical Physics , 1962 .

[6]  Eigenvalues of the Laplacian for the third boundary value problem , 1987, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[7]  H. Gottlieb.  Hearing the shape of an annular drum , 1983, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[8]  E. Zayed.  Heat equation for an arbitrary doubly-connected region inR2 with mixed boundary conditions , 1989 .

[9]  René Carmona,et al.  Can one hear the dimension of a fractal? , 1986 .

[10]  Kudrolli,et al.  Experiments on not "hearing the shape" of drums. , 1994, Physical review letters.

[11]  E. Zayed,et al.  The wave equation approach to the two-dimensional inverse problem for a general bounded domain with piecewise smooth mixed boundary conditions , 2001 .

[12]  B. Sleeman,et al.  An inverse eigenvalue problem for a general convex domain , 1983 .

[13]  E. Zayed.  The Asymptotics of the Heat Semigroup for a General Bounded Domain with Mixed Boundary Conditions , 2000 .

[14]  L. Hörmander,et al.  The spectral function of an elliptic operator , 1968 .

[15]  E. Zayed.  An inverse problem for a general multiply connected bounded domain , 1995 .

[16]  E. Zayed.  Hearing the shape of a general convex domain , 1989 .

[17]  K. Stewartson,et al.  On hearing the shape of a drum: further results , 1971, Mathematical Proceedings of the Cambridge Philosophical Society.

[18]  Eigenvalues of the Laplacian for rectilinear regions , 1988, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[19]  David L. Webb,et al.  One cannot hear the shape of a drum , 1992, math/9207215.

[20]  E. Zayed.  Short—time Asymptotics of the Heat Kernel of the Laplacian for a Multiply-connected Domain in R 2 with Robin Boundary Conditions , 2001 .

[21]  E. Zayed,et al.  On hearing the shape of rectilinear regions , 1994 .

[22]  Lance Smith,et al.  The asymptotics of the heat equation for a boundary value problem , 1981 .

[23]  M. Kac.  Can One Hear the Shape of a Drum , 1966 .

[24]  Å. Pleijel.  A study of certain Green's functions with applications in the theory of vibrating membranes , 1954 .

[25]  B. Sleeman,et al.  Trace formulae for the eigenvalues of the Laplacian , 1984 .