arXiv2023
We say that $Γ$, the boundary of a bounded Lipschitz domain, is locally dilation invariant if, at each $x\in Γ$, $Γ$ 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 $α_x\in (0,1)$. 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 is the union of the spectra of related continuous families of operators $K_t$, for $t\in [-π,π]$; 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.