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

Abstract

The asymptotic expansions of the trace of the heat kernel <artwork name="GAPA31035ei1"> for small positive t, where λν are the eigenvalues of the negative Laplacian <artwork name="GAPA31035ei2"> 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 <artwork name="GAPA31035ei3"> of ∂Ω2 are considered such that <artwork name="GAPA31035ei4"> and <artwork name="GAPA31035ei5"> and where the coefficients <artwork name="GAPA31035ei6"> 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.