FRACTIONAL ORDER SOBOLEV SPACES FOR THE NEUMANN LAPLACIAN AND THE VECTOR LAPLACIAN

Abstract

. In this paper we study fractional Sobolev spaces characterized by a norm based on eigenfunction expansions. The goal of this paper is twofold. The first one is to define fractional Sobolev spaces of order − 1 ≤ s ≤ 2 equipped with a norm defined in terms of Neumann eigen- function expansions. Due to the zero Neumann trace of Neumann eigenfunctions on a boundary, fractional Sobolev spaces of order 3 / 2 ≤ s ≤ 2 characterized by the norm are the spaces of functions with zero Neumann trace on a boundary. The spaces equipped with the norm are useful for studying cross-sectional traces of solutions to the Helmholtz equation in waveguides with a homogeneous Neumann boundary condition. The sec- ond one is to define fractional Sobolev spaces of order − 1 ≤ s ≤ 1 for vector-valued functions in a simply-connected, bounded and smooth do- main in R 2 . These spaces are defined by a norm based on series expansions in terms of eigenfunctions of the vector Laplacian with boundary condi- tions of zero tangential component or zero normal component. The spaces defined by the norm are important for analyzing cross-sectional traces of time-harmonic electromagnetic fields in perfectly conducting waveguides.