Inverse Problems of Boundary Value Problems for Annular Vibrating Membranes with Piecewise Smooth Positive Functions in the Boundary Conditions

The asymptotic expansions of the trace of the heat kernel for small positive t, where λ_ν are the eigenvalues of the negative Laplacian in Rⁿ (n=2 or 3), are studied for a general annular bounded domain Ω with a smooth inner boundary ?Ω₁ and a smooth outer boundary ?Ω₂ where a finite number of piecewise smooth Dirichlet, Neumann and Robin boundary conditions on the components Γ _j (j=1,…,m) of ?Ω₁ and on the components of ?Ω₂ are considered such that and and where the coefficients in the Robin boundary conditions are piecewise smooth positive functions. Some applications of Θ (t) for an ideal gas enclosed in the general annular bounded domain Ω are given.