Spectral radius of the Neumann–Poincaré operator on non-energy Sobolev spaces

Determine the spectral radius of the projected Neumann–Poincaré operator on the non-Lipschitz self-similar chain when acting on mean-zero Sobolev spaces other than the energy space, and characterize its dependence on the scaling exponent and geometric parameters.

Background

The paper completely characterizes the essential spectrum of the projected Neumann–Poincaré operator on the mean-zero space W^{s,p}_0(Γ) through the operator-valued Floquet symbol κ_α, where α=(d−1)/p−s. In the energy space H^{-1/2}_0(Γ), symmetrization confines the spectrum to the real interval [−1/2,1/2].

For other function spaces, the authors state that neither this spectral bound nor the shape of the essential spectrum is inherited. Determining the spectral radius is therefore unresolved in the general self-similar-chain setting. The explicit annular example may permit computation of the radius as a function of α and the geometric parameters, whereas a general theory would address arbitrary reference shapes and admissible Sobolev exponents.