Kernels of trace operators via fine continuity

Abstract

Given a closed subset $\hat{I}“$ of ${\mathbb{R}}^n$ that is the support of a measure $\hat{I}¼$, we study the kernels of trace operators from fractional Sobolev spaces $H_p^{\hat{I}}\pm ({\mathbb{R}}^n)$ into the space of $\hat{I}¼$-equivalence classes of functions on $\hat{I}“$. We characterise these kernels as the closure of $C_c^{\infty }({\mathbb{R}}^n\setminus \hat{I}“)$ in $H_p^{\hat{I}}\pm ({\mathbb{R}}^n)$, provided quasi continuous representatives of elements of $H_p^{\hat{I}}\pm ({\mathbb{R}}^n)$ have the following key property: they vanish quasi everywhere on $\hat{I}“$ if and only if they vanish $\hat{I}¼$-almost everywhere on $\hat{I}“$. We establish that this key property holds if the measures satisfy localized upper density conditions. Such measures need not be doubling, in particular the set $\hat{I}“$ may be a finite union of closed sets having different Hausdorff dimensions. We provide corresponding results for spaces $H_p^{\hat{I}}\pm (\hat{I}\text{c◯})$ on domains $\hat{I}\text{c◯}\subset {\mathbb{R}}^n$ satisfying a weakened version of the measure density condition. We observe that the above key property is essential for the convergence of Galerkin integral equation methods, based on integration with respect to the measure $\hat{I}¼$, for certain BVPs in the complement of $\hat{I}“$.