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

Abstract

We say that Gamma , the boundary of a bounded Lipschitz domain, is locally dilation invariant if, at each xin Gamma , Gamma is either locally C^1 or locally coincides (in some coordinate system centred at x) with a Lipschitz graph Gamma _x such that Gamma _x=alpha _xGamma _x, for some alpha _xin (0,1). In this paper we study, for such Gamma , the essential spectrum of D_Gamma , the double-layer (or Neumann–Poincaré) operator of potential theory, on L^2(Gamma ). 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, for tin [-pi ,pi ]; moreover, each K_t is compact if Gamma is C^1 except at finitely many points. For the 2D case where, additionally, Gamma is piecewise analytic, we construct convergent sequences of approximations to the essential spectrum of D_Gamma ; 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 Gamma satisfies the well-known spectral radius conjecture, that the essential spectral radius of D_Gamma on L^2(Gamma ) is <1/2 for all Lipschitz Gamma . 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,beta } diffeomorphisms, show that the spectral radius conjecture holds for all Lipschitz curvilinear polyhedra.