Spectral Characterization of the Trace Spaces ${H^s({\partial \Omega})}$

Abstract

This paper defines certain scales of trace spaces ${H^s({\partial \Omega})}$ using harmonic Steklov eigenfunction expansions. The approach is intrinsic and applies to bounded regions in ${\mathbb R}^n$ for which standard imbedding results hold. In particular it suffices that the boundary of the region be a finite disjoint union of Lipschitz surfaces. The definition generalizes the classical definitions that require the boundary to consist of smooth manifolds. The description depends on a special inner product on ${H^1(\Omega)}$, certain completeness theorems for Steklov eigenfunctions, and special properties of the harmonic Steklov eigenfunctions. The characterization provides explicit formulae for the inner products and norms of a function in ${H^s({\partial \Omega})}$ and allows the description of specific orthonormal bases for these spaces. For $s < 0$, the spaces are obtained by duality from the case for $s > 0$.