Short-time Asymptotics of the two-Dimensional wave Equation for an Annular Vibrating Membrane with Piecewise smooth Boundary Conditions

Από το τεύχος 49 του περιοδικού Δελτίο της Ελληνικής Μαθηματικής Εταιρίας

This paper deals with the very interesting problem about the influence of piecewise smooth boundary conditions on the distribution of the eigenvalues of the negative Laplacian in . The asymptotic expansion of the trace of the wave operator 2121?()exp()tiννμμ?==?Σ t for small | and |t1i=?, where 1{}ννμ?= are the eigenvalues of the negative Laplacian 221()xκκ=??Δ=??Σ in the (12,)xx? plane, is studied for an annular vibrating membrane Ω in together with its smooth inner boundary and its smooth outer boundary . In the present paper a finite number of Dirichlet, Neumann and Robin boundary conditions on the piecewise smooth components of 2R1?Ω2?Ω(1,,jjmΓ=K ) 1?Ω and on the piecewise smooth components (1,jjmn,) Γ=+K of such that 2?Ω11mjj=?Ω=ΓU and 21njjm=+?Ω=ΓUis considered. The basic problem is to extract information on the geometry of the annular vibrating Ω from complete knowledge of its eigenvalues using the wave equation approach by analyzing the asymptotic expansions of the spectral function ?()tμ for small .

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