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 <artwork name="GAPA31035ei1"> for small positive t, where λ_ν 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.

Keywords:

• Inverse Problems
• Heat Kernel
• Eigenvalues
• Hearing The Shape Of Annular Membranes
• Classical Ideal Gas
• Ams Subject Classifications: 35k99
• 35p05
• 35p99