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

Abstract

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