On the spectrum of the double-layer operator on locally-dilation-invariant Lipschitz domains
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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 Γ_x such that Γ_x=α_xΓ_x, for some α_x∈ (0,1). In this paper we study, for such Γ, the essential spectrum of D_Γ, 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 such Γ is the union of the spectra of related continuous families of operators K_t, for t∈ [-π,π]; moreover, each K_t 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 D_Γ; each approximation is the union of the eigenvalues of finitely many finite matrices arising from Nyström-method approximations to the operators K_t. 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 D_Γ 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.
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