On an angle-averaged Neumann-to-Dirichlet map for thin filaments

Abstract

We consider the Laplace equation in the exterior of a thin filament in $\mathbb{R}^3$ and perform a detailed decomposition of a notion of slender body Neumann-to-Dirichlet (NtD) and Dirichlet-to-Neumann (DtN) maps along the filament surface. The decomposition is motivated by a filament evolution equation in Stokes flow for which the Laplace setting serves as an important toy problem. Given a general curved, closed filament with constant radius $\epsilon>0$, we show that both the slender body DtN and NtD maps may be decomposed into the corresponding operator about a straight, periodic filament plus lower order remainders. For the straight filament, both the slender body NtD and DtN maps are given by explicit Fourier multipliers and it is straightforward to compute their mapping properties. The remainder terms are lower order in the sense that they are small with respect to $\epsilon$ or smoother. While the strategy here is meant to serve as a blueprint for the Stokes setting, the Laplace problem may be of independent interest.