On the spectrum of the double-layer operator on locally-dilation-invariant Lipschitz domains

Abstract

We say that Γ, the boundary of a bounded Lipschitz domain, is locally dilation invariant if, at each x ∈ Γ, Γ is either locally C 1 or locally coincides (in some coordinate system centred at x) with a Lipschitz graph Γx such that Γx = αxΓx, for some αx ∈ (0, 1). In this paper we study, for such Γ, the essential spectrum of DΓ, the double-layer (or Neumann-Poincaré) operator of potential theory, on L 2 (Γ). We show, via localisation and Floquet-Bloch-type arguments, that this essential spectrum is the union of the spectra of related continuous families of operators Kt, for t ∈ [−π, π]; moreover, each Kt is compact if Γ is C 1 except at finitely many points. For the 2D case where, additionally, Γ is piecewise analytic, we construct convergent sequences of approximations to the essential spectrum of DΓ; each approximation is the union of the eigenvalues of finitely many finite matrices arising from Nyström-method approximations to the operators Kt. Through error estimates with explicit constants, we also construct functionals that determine whether any particular locally-dilation-invariant piecewise-analytic Γ satisfies the well-known spectral radius conjecture, that the essential spectral radius of DΓ on L 2 (Γ) is < 1/2 for all Lipschitz Γ. We illustrate this theory with examples; for each we show that the essential spectral radius is < 1/2, providing additional support for the conjecture. We also, via new results on the invariance of the essential spectral radius under locally-conformal C 1,β diffeomorphisms, show that the spectral radius conjecture holds for all Lipschitz curvilinear polyhedra.