Spectral asymptotics of the Dirichlet-to-Neumann map on multiply connecteddomains in ℝd

Abstract

We study the spectral asymptotics of the Dirichlet-to-Neumann operator Λγ on a multiply connected, bounded, domain Ω ⊂ ℝd, d ≥ 3, associated with the uniformlyelliptic operator Lγ = − ∑i,j = 1d ∂i γij∂j, where γ is a smooth,positive-definite, symmetric matrix-valued function on Ω. We prove thatthe operator is approximately diagonal in the sense that Λγ = Dγ + Rγ, where Dγ is a direct sum ofoperators, each of which acts on one boundary component only, and Rγis a smoothing operator. This representation follows from the fact that theγ-harmonic extensions of eigenfunctions of Λγ vanish rapidly away from the boundary. Using this representation,we study the inverse problem of determining the number of holes in thebody, that is, the number of the connected components of the boundary,by using the high-energy spectral asymptotics of Λγ.