An inverse problem for a general annular-bounded domain in R2 with mixed boundary conditions and its physical applications

This paper deals with the very interesting problem about the influence of the boundary conditions on the distribution of the eigenvalues of the negative Laplacian in R^2. The asymptotic expansion of the trace of the heat kernel @q(t)=@?"@u"="1^~exp(-t@m"@n), where {@m"@n}"@n"="1^~ are the eigenvalues of the negative Laplacian -@D"2=-@?"k"="1^2(@?/@?x^k)^2 in the (x^1,x^2)-plane, is studied for short-time t of a general annular-bounded domain @W in R^2 together with its smooth inner boundary @?@W"1 and its smooth outer boundary @?@W"2, where a finite number of Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components @C"i(i=1,...,m) of @?@W"1 and on the piecewise smooth components @C"i(i=m+1,...,n) of @?@W"2 such that @?@W"1=@?"i"="1^m@C"i and @?@W"2=@?"i"="m"+"1^n@C"i, are considered. In this paper, one may extract information on the geometry of @W by analyzing the asymptotic expansions of @q(t) for short-time t. Some applications of @q(t) for an ideal gas enclosed in @W are given. Thermodynamic quantities of an ideal gas enclosed in @W are examined. We use an asymptotic expansion for high temperatures to obtain the partition function of an ideal gas showing the leading corrections to the internal energy due to a finite container. We show that the ideal gas cannot feel the shape of its container, although it can feel some geometrical properties of it.

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

[2]  E. Zayed,et al.  An inverse problem of the wave equation for a general annular drum in R3 with Robin boundary conditions , 2001 .

[3]  E. Zayed.  An inverse problem for a general doubly connected bounded domain in R3 with a Finite Number of Piecewise Impedance Boundary Conditions , 1997 .

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

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

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

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

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

[9]  R. Robinett.  Visualizing classical periodic orbits from the quantum energy spectrum via the Fourier transform: Simple infinite well examples , 1997 .

[10]  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 .

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

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

[13]  E. Zayed,et al.  The 3D inverse problem for waves with fractal and general annular bounded domain with piecewise smooth Robin boundary , 2001 .

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

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

[16]  F. Reif,et al.  Fundamentals of Statistical and Thermal Physics , 1965 .

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

[18]  G. Gutiérrez,et al.  Can an ideal gas feel the shape of its container , 1997 .

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

[20]  Eberhard R. Hilf,et al.  Spectra of Finite Systems , 1980 .

[21]  R. Balian,et al.  DISTRIBUTION OF EIGENFREQUENCIES FOR THE WAVE EQUATION IN A FINITE DOMAIN. I. THREE-DIMENSIONAL PROBLEM WITH SMOOTH BOUNDARY SURFACE. , 1970 .

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

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

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

[25]  Elsayed M. E. Zayed.  AN INVERSE EIGENVALUE PROBLEM FOR AN ARBITRARY MULTIPLY CONNECTED BOUNDED REGION IN R2 , 1991 .

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

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