On hearing the shape of a bounded domain with Robin boundary conditions
Elsayed M. E. Zayed · IMA Journal of Applied Mathematics · 2000
The asymptotic expansions of the trace of the heat kernel θ(t) = [sum ]∞j=1 exp(-tλj) for small positive t, where {λj}∞j=1 are the eigenvalues of the negative Laplacian -Δn = -[sum ]nk=1 (∂/∂xk)2 in Rn (n = 2 or 3), are studied for a general multiply connected bounded domain Ω which is surrounded by simply connected bounded domains Ωi with smooth boundaries ∂Ωi (i = 1,...,m), where smooth functions Yi (i = 1,...,m) are assuming the Robin boundary conditions (∂―∂ni + Yi)φ = 0 on ∂Ωi. Here ∂/∂ni denote differentiations along the inward-pointing normals to ∂Ωi (i = 1,...,m). Some applications of an ideal gas enclosed in the multiply connected bounded container with Neumann or Robin boundary conditions are given.