On hearing the shape of a bounded domain with Robin boundary conditions

Abstract

The asymptotic expansions of the trace of the heat kernel θ(t) = Σ∞ j=1 exp(-tλ j ) for small positive t, where [λ j ]∞ j=1 are the eigenvalues of the negative Laplacian -An = -Σ n k=1 (∂/∂x k ) 2 in R n (n = 2 or 3), are studied for a general multiply connected bounded domain Ω which is surrounded by simply connected bounded domains Ω i with smooth boundaries ∂Ω i (i = 1,…,m), where smooth functions γ i (i = 1,…,m) are assuming the Robin boundary conditions (∂/∂ni + γ i )o = 0 on ∂Ω i . Here ∂/∂ ni denote differentiations along the inward-pointing normals to ∂Ω i (i = 1..., m). Some applications of an ideal gas enclosed in the multiply connected bounded container with Neumann or Robin boundary conditions are given.