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 $x\in \Gamma$, $\Gamma$ is either locally $C^1$ or locally coincides (in some coordinate system centred at $x$) with a Lipschitz graph $\Gamma_x$ such that $\Gamma_x=\alpha_x\Gamma_x$, for some $\alpha_x\in (0,1)$. In this paper we study, for such $\Gamma$, the essential spectrum of $D_\Gamma$, the double-layer (or Neumann-Poincar\'e) 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 $t\in [-\pi,\pi]$; 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 Nystr\"om-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.