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

TL;DR

This paper investigates the essential spectrum of the double-layer operator on locally-dilation-invariant Lipschitz domains, providing new approximation methods and supporting the spectral radius conjecture for such geometries.

Contribution

The paper introduces Floquet-Bloch-type analysis for the essential spectrum and develops convergent approximation sequences, advancing understanding of spectral properties on complex Lipschitz domains.

Findings

Essential spectrum characterized as union of spectra of operator families.

Constructed convergent eigenvalue approximations for piecewise-analytic domains.

Supported the spectral radius conjecture for Lipschitz curvilinear polyhedra.

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.