On the limiting behaviour of solutions to boundary integral equations associated with time harmonic wave equations for small frequencies

Abstract

AbstractThe treatment of boundary value problems for Helmholtz equation and for the time harmonic Maxwell's equations by boundary integral equations leads to integral equations of the second kind which are uniquely solvable for small positive frequencies λ. However, the integral equations obtained in the limiting case λ = 0 which are related to boundary value problems of potential theory in general are not uniquely solvable since the corresponding boundary value problems are not. By first considering in a general setting of a Banach space X the limiting behaviour of solutions ϕλ to the equation ϕλ – K λ ϕ λ = fλ as λ → 0 where {Kλ: X → X, λ ∈ (0,α)}, α > 0, denotes a family of compact linear operators such that I ‐ Kλ (I identity) is bijective for λ∈(0,α) whilst I ‐ K0 is not and ‖ Kλ – K0‖ →, 0, ‖fλ – f0‖ → 0, λ → 0, and then applying the results to the boundary integral operators, the limiting behaviour of the integral equations is considered. Thus, the results obtained by Mac Camey for the Helmholtz equation are extended to the case of non‐connected boundaries and Werner's results on the integral equations for the Maxwell's equations are extended to the case of multiply connected boundaries.